PRICING MODELS FOR INFLATION LINKED
 DERIVATIVES IN AN ILLIQUID MARKET
 by
 Thibaut Zafack Takadong
 Programme in Advanced Mathematics of Finance
 School of Computational and Applied Mathematics
 University of the Witwatersrand
 A dissertation submitted for the degree of
 Master of Science
 September 2008
ABSTRACT
 Recent  nancial crises have highlighted the sensitivity and vulnerability of  nancial markets
 to in ation, which reduces the value of money and a ects the net returns of  nancial instru-
 ments. In response to this, investors who are concerned with maintaining their investment?s
 purchasing power rather than its market value are resorting to in ation linked (IL) products
 to hedge their in ation risk. Consequently, the in ation market has been rapidly growing for
 the last decade and has further great potential growth worldwide. It is highly probable that
 in ation linked derivatives will eventually be as common as conventional products. Another
 cause of the in ation market boost is the growing extension of the time frame of  nancial
 transactions, which has generated an increase in in ation expectation; since 1980 the av-
 erage time to maturity of long-dated transactions went from one decade to three decades.
 This is, in part, due to the ageing population in the developed world. This research inves-
 tigates some alternative models in order to improve the match between model prices and
 observed prices in the American and South African in ation markets. It takes into account
 the relative illiquidity of IL products. The main tools used are L evy distributions, macroe-
 conomic factors, no-arbitrage and pricing kernel models. L evy processes can replicate the
 behaviour of the return innovations of a wide range of  nancial securities. Adding a stochas-
 tic time change to the L evy process randomises the market clock, thus generating stochastic
 volatilities, higher stochastic return moments and eventually stochastic skewness. These are
 observed stylised facts most conventional models do not achieve. Moreover, in contrast to
 the hidden factor approach, each L evy process component and its stochastic time change
 can readily be assigned an economic meaning. This explicit economic mapping facilitates
 the interpretation of current models and provides a more intuitive approach to building
 new models that capture other observed behaviours. Finally, L evy processes also provide
 tractable formulas for derivative pricing and market estimations. In general, in ation is a
 consequence of macroeconomic factors. Exogenous dynamics of the most signi cant of these
 factors are used to deduce the endogenous in ation dynamics in some of the considered
 models. In these cases, the calibration of the pricing kernel models requires little historical
data on IL derivatives. In fact, the required macroeconomic historical data is easily available
 because of the current national and international legislation.
 Key words: market illiquidity, in ation linked products, L evy processes, pricing kernel,
 macroeconomic factors.
ACKNOWLEDGMENTS
 TO MY DAD AND MY MOM.
 There will not have been a calibration to the South African market without the assistance
 of Shameer Sukha, Simona Levet and Nicolette Roussos. Each of them was very helpful and
 provided some sample data for the calibration.
 Many thanks to David Taylor, coordinator of the Program of Advanced Mathematics of
 Finance at Wits, for admitting me in the program and for the great courses (both theoretical
 and practical) he makes sure are always available. I was impressed by the quality of the
 curriculum and learned a lot from them.
 I will also like to thank my \colleagues" in the laboratory at Wits. They created an ideal en-
 vironment for research and took the time to proof read each draft of the current manuscript.
 Numerous mistakes were thus corrected and the multiple discussion we had were always
 fruitful. Any mistake left is entirely due to me.
 Last but not least, I can not thank enough my supervisor Raouf Ghomrasni for all the hard
 work he put in and for all the proof reading, critics and pushing he kept doing.
DECLARATION
 This is to certify that
 (i) the thesis comprises only my original work toward the MSc, except where indicated
 in the Preface,
 (ii) due acknowledgement has been made in the text to all other material used.
 I hereby certify that this thesis was independently written by me. No material was used
 other than that referred to. Sources directly quoted and ideas used, including  gures, tables,
 sketches, drawings and photos, have been correctly denoted. Those not otherwise indicated
 belong to the author.
Contents
 Table of Contents vi
 List of Figures ix
 List of Tables xii
 1 In ation Modelling 4
 1.1 In ation Market . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
 1.1.1 IL Products Issuers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
 1.1.2 IL Products Investors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
 1.2 In ation Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
 1.2.1 In ation Linked Bonds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
 1.2.2 Zero-Coupon In ation Swaps . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
 1.2.3 Year-on-Year Swap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
 1.2.4 In ation Caps, Floors and Swaptions . . . . . . . . . . . . . . . . . . . . . . . 10
 1.3 In ation Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
 1.3.1 Interest Rate Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
 1.3.2 Foreign Exchange Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
 1.3.3 Macro- nance models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
 1.3.4 Stochastic monetary economy models . . . . . . . . . . . . . . . . . . . . . . 17
 1.3.5 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
 2 The L evy Process Framework 20
 2.1 L evy Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
 2.2 Ito^ Formula for L evy Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
 2.3 Some Useful Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
 2.4 Examples of L evy Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
 2.4.1 The Generalized Hyperbolic Distribution . . . . . . . . . . . . . . . . . . . . 37
 2.4.2 The Hyperbolic Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
 2.4.3 The Normal Inverse Gaussian Distribution . . . . . . . . . . . . . . . . . . . . 40
 2.4.4 The Variance Gamma Distribution . . . . . . . . . . . . . . . . . . . . . . . . 41
 2.4.5 The GH Skew Student?s t Distribution . . . . . . . . . . . . . . . . . . . . . . 42
 2.5 Option Pricing Using the Fast Fourier Transform . . . . . . . . . . . . . . . . . . . . 43
 3 Heath-Jarrow-Morton Model 47
 3.1 The Extended HJM Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
 3.2 In ation Linked Swaps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
 3.2.1 Zero Coupon In ation Indexed Swap . . . . . . . . . . . . . . . . . . . . . . . 62
 3.2.2 Year-on-Year In ation Indexed Swaps . . . . . . . . . . . . . . . . . . . . . . 64
 3.3 In ation Linked Caplets/Floorlets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
 3.3.1 Lognormally Distributed CPI . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
 vi
CONTENTS vii
 3.3.2 Pricing with the Bilateral Laplace Transform . . . . . . . . . . . . . . . . . . 71
 3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
 4 Stochastic Monetary Economy Models 75
 4.1 L evy process distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
 4.1.1 Log-separable utility function . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
 4.1.2 Separable power utility function . . . . . . . . . . . . . . . . . . . . . . . . . 84
 4.1.3 Arithmetic Brownian distribution . . . . . . . . . . . . . . . . . . . . . . . . . 89
 4.2 Exponential L evy distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
 4.2.1 Log-separable utility function . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
 4.2.2 Power utility function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
 4.2.3 Geometric Brownian distribution . . . . . . . . . . . . . . . . . . . . . . . . . 101
 4.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
 5 Reverse Engineering 104
 5.1 In ation Breakeven Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
 5.2 Theoretical framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
 5.3 Exponential L evy Gross return . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
 5.4 Exponential L evy In ation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
 5.5 Parameter estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
 6 Empirical Study and Calibration 119
 6.1 Empirical study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
 6.1.1 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
 6.1.2 Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
 6.1.3 Hypothesis Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
 6.1.4 Goodness of Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
 6.1.5 Data Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
 6.1.6 Increasing the data size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
 6.1.7 L evy Distributions? Parameter Estimation . . . . . . . . . . . . . . . . . . . . 137
 6.1.8 Forward rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
 6.1.9 South African data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
 6.1.10 United State of America data . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
 6.2 Option pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
 6.2.1 Maximum-Likelihood Estimator . . . . . . . . . . . . . . . . . . . . . . . . . 160
 6.2.2 Discretisation of the FFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
 6.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
 A Empirical Study SA 163
 A.1 Normal innovations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
 A.1.1 Monthly SA CPI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
 A.1.2 Consumer Price Index (CPIX) . . . . . . . . . . . . . . . . . . . . . . . . . . 163
 A.1.3 Money Supply aggregate M1A . . . . . . . . . . . . . . . . . . . . . . . . . . 168
 A.1.4 Money Supply aggregate M1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
 A.1.5 Money Supply aggregate M2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
 A.2 Student t innovations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
 A.2.1 SA CPIX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
 A.3 Forward estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
CONTENTS viii
 B Empirical Study US 177
 B.1 Normal innovations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
 B.1.1 Consumer Price Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
 B.1.2 Consumer Price Index (End half) . . . . . . . . . . . . . . . . . . . . . . . . . 184
 B.1.3 Money Supply aggregate M1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
 B.2 Student t innovations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
 B.3 Yield Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
 Bibliography 219
List of Figures
 1.1 Evolution of the annual South African and Ghanian CPI [104]. . . . . . . . . . . . . 5
 1.2 Evolution of the annual UK and US CPI [104]. . . . . . . . . . . . . . . . . . . . . . 5
 2.1 Modi ed Bessel function of the third kind. . . . . . . . . . . . . . . . . . . . . 38
 2.2 Probability density function and sample path for NIG. . . . . . . . . . . . . . . . . . 41
 2.3 Probability density function of some VG processes. . . . . . . . . . . . . . . 42
 2.4 Sample paths of a VG process with  = 0:2,  = 0:5 and  = 0:25. . . . . . . . . . . 43
 6.1 Monthly SA CPI correlograms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
 6.2 Monthly SA CPI partial correlograms. . . . . . . . . . . . . . . . . . . . . . . . . . . 123
 6.3 SA CPI raw returns Kolmogorov-Smirnov test. . . . . . . . . . . . . . . . . . 126
 6.4 Filtered vs raw SA CPI data series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
 6.5 Filtered monthly SA CPI correlograms. . . . . . . . . . . . . . . . . . . . . . . . . . 132
 6.6 Filtered monthly SA CPI partial correlograms. . . . . . . . . . . . . . . . . . . . . . 133
 6.7 Daily SA CPI (partial) correlograms. . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
 6.8 Daily  ltered SA CPI (partial) correlograms. . . . . . . . . . . . . . . . . . . . . . . 136
 6.9 Daily raw vs  ltered SA CPI (normal innovations). . . . . . . . . . . . . . . . . . . . 136
 6.10 Daily raw vs  ltered SA CPI (student-t innovqtions). . . . . . . . . . . . . . . . . . . 137
 6.11 Monthly SA CPI probability and QQ plots: Empirical vs normal. . . . . . . . . . . . 138
 6.12 Monthly SA CPI probability plots: Empirical vs L evy. . . . . . . . . . . . . . . . . . 138
 6.13 Filtered daily SA CPI 2005 2008 probability plots. . . . . . . . . . . . . . . . . . . 140
 6.14 Filtered daily SA CPI probability and QQ plots: Empirical vs normal. . . . . . . . . 141
 6.15 Filtered daily SA CPI probability plots: Empirical vs L evy. . . . . . . . . . . . . . . 141
 6.16 Nominal yield and interpolated \-ln" on the 30th May 2008. . . . . . . . . . . . . . . 145
 6.17 SA centred empirical daily return and regression line 29th May 2008. . . . . . . . . . 146
 6.18 SA estimated L1 between the 31st July 2000 and the 29th May 2008. . . . . . . . . . 147
 6.19 Estimated L1: Empirical vs normal. . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
 6.20 Estimated L1 probability plots: Empirical vs L evy. . . . . . . . . . . . . . . . . . . . 148
 6.21 Monthly SA CPIX (1997-2008) correlograms. . . . . . . . . . . . . . . . . . . . . . . 150
 6.22 Monthly SA CPIX (1997-2008) partial correlograms. . . . . . . . . . . . . . . . . . . 150
 6.23 Filtered vs raw monthly SA CPIX (1997-2008) data series. . . . . . . . . . . . . . . . 151
 6.24 Filtered monthly SA CPIX (1997-2008) correlograms. . . . . . . . . . . . . . . . . . 152
 6.25 Filtered monthly SA CPIX (1997-2008) partial correlograms. . . . . . . . . . . . . . 152
 6.26 Filtered monthly SA CPIX probability and QQ plots: Empirical vs normal. . . . . . 153
 6.27 Filtered monthly SA CPIX probability plots: Empirical vs L evy. . . . . . . . . . . . 153
 6.28 Nominal yield curve: Empirical vs model. . . . . . . . . . . . . . . . . . . . . . . . . 158
 6.29 Real yield curve: Empirical vs model. . . . . . . . . . . . . . . . . . . . . . . . . . . 158
 A.1 Monthly SA CPI probability and QQ plots: Empirical vs normal. . . . . . . . . . . . 164
 A.2 Monthly SA CPI probability plots: Empirical vs L evy. . . . . . . . . . . . . . . . . . 164
 ix
LIST OF FIGURES x
 A.3 Monthly SA CPI QQ plots (normality assumption). . . . . . . . . . . . . . . . . . . 165
 A.4 Filtered daily SA CPI QQ plots. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
 A.5 Filtered vs raw monthly SA CPIX (1997-2008) data series. . . . . . . . . . . . . . . . 167
 A.6 Filtered monthly SA CPIX (1997-2008) correlograms. . . . . . . . . . . . . . . . . . 167
 A.7 Filtered monthly SA CPIX (1997-2008) partial correlograms. . . . . . . . . . . . . . 167
 A.8 Filtered vs raw monthly SA M1A (1979-2007) data series. . . . . . . . . . . . . . . . 168
 A.9 Filtered monthly SA M1A (1979-2007) correlograms. . . . . . . . . . . . . . . . . . . 169
 A.10 Filtered vs raw monthly SA M1 (1965-2007) data series. . . . . . . . . . . . . . . . . 169
 A.11 Filtered monthly SA M1 (1965-2007) correlograms. . . . . . . . . . . . . . . . . . . . 169
 A.12 Monthly SA Money Supply M2 (1965-2007) correlograms. . . . . . . . . . . . . . . . 170
 A.13 Monthly SA Money Supply M2 (1965-2007) partial correlograms. . . . . . . . . . . . 170
 A.14 Filtered vs raw monthly SA M2 (1965-2007) data series. . . . . . . . . . . . . . . . . 171
 A.15 Filtered monthly SA M2 (1965-2007) correlograms. . . . . . . . . . . . . . . . . . . . 172
 A.16 Filtered monthly SA M2 (1965-2007) partial correlograms. . . . . . . . . . . . . . . . 172
 A.17 Monthly SA Money Supply M3 (1965-2007) correlograms. . . . . . . . . . . . . . . . 172
 A.18 Monthly SA Money Supply M3 (1965-2007) partial correlograms. . . . . . . . . . . . 173
 A.19 Filtered vs raw monthly SA M3 (1965-2007) data series. . . . . . . . . . . . . . . . . 174
 A.20 Filtered monthly SA M3 (1965-2007) correlograms. . . . . . . . . . . . . . . . . . . . 174
 A.21 Filtered monthly SA M3 (1965-2007) partial correlograms. . . . . . . . . . . . . . . . 175
 A.22 Probability plots: Empirical vs model. . . . . . . . . . . . . . . . . . . . . . . . . . . 175
 A.23 Estimated L1 QQ plots. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
 B.1 Monthly USA CPI (1821-2007) correlograms. . . . . . . . . . . . . . . . . . . . . . . 178
 B.2 Monthly USA CPI (1821-2007) partial correlograms. . . . . . . . . . . . . . . . . . . 178
 B.3 Filtered vs raw monthly USA CPI (1821-2007) data series . . . . . . . . . . . . . . . 178
 B.4 Filtered monthly USA CPI (1821-2007) correlograms. . . . . . . . . . . . . . . . . . 179
 B.5 Filtered monthly USA CPI (1821-2007) partial correlograms. . . . . . . . . . . . . . 180
 B.6 Monthly USA CPI (1821-2007) probability plots: Empirical vs normal. . . . . . . . . 180
 B.7 Monthly USA CPI (1821-2007) probability plots: Empirical vs L evy. . . . . . . . . . 181
 B.8 Monthly US CPI QQ plots (individually). . . . . . . . . . . . . . . . . . . . . . . . . 182
 B.9 Monthly US CPI QQ plots. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
 B.10 Monthly USA CPI (1937-2007) correlograms. . . . . . . . . . . . . . . . . . . . . . . 184
 B.11 Monthly USA CPI (1937-2007) partial correlograms. . . . . . . . . . . . . . . . . . . 184
 B.12 Filtered vs raw monthly USA CPI (1937-2007) data series . . . . . . . . . . . . . . . 185
 B.13 Filtered monthly USA CPI (1937-2007) correlograms. . . . . . . . . . . . . . . . . . 185
 B.14 Filtered monthly USA CPI (1937-2007) partial correlograms. . . . . . . . . . . . . . 186
 B.15 Monthly USA CPI (1937-2007) probability plots: Empirical vs normal. . . . . . . . . 186
 B.16 Monthly USA CPI (1937-2007) probability plots: Empirical vs L evy. . . . . . . . . . 187
 B.17 Monthly US CPI QQ plots. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
 B.18 Monthly US CPI QQ plots (individually). . . . . . . . . . . . . . . . . . . . . . . . . 188
 B.19 Weekly USA Money Supply M1 (1981-2008) correlograms. . . . . . . . . . . . . . . . 189
 B.20 Weekly USA Money Supply M1 (1981-2008) partial correlograms. . . . . . . . . . . . 189
 B.21 Filtered vs raw weekly USA M1 (1981-2008) data series. . . . . . . . . . . . . . . . . 190
 B.22 Filtered weekly USA M1 (1981-2008) correlograms. . . . . . . . . . . . . . . . . . . . 191
 B.23 Filtered weekly USA M1 (1981-2008) partial correlograms. . . . . . . . . . . . . . . . 191
 B.24 Weekly USA M1 (1981-2008) probability plots: Empirical vs normal. . . . . . . . . . 192
 B.25 Weekly USA M1 (1981-2008) probability plots: Empirical vs L evy. . . . . . . . . . . 192
 B.26 Weekly US M1 QQ plots: Empirical vs normal. . . . . . . . . . . . . . . . . . . . . . 193
 B.27 Weekly US M1 QQ plots. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
 B.28 Seasonally adjusted weekly USA Money Supply M1 (1981-2008) correlograms. . . . . 195
 B.29 Seasonally adjusted weekly USA Money Supply M1 (1981-2008) partial correlograms. 195
 B.30 Filtered vs raw weekly USA M1 Adj. (1981-2008) data series. . . . . . . . . . . . . . 196
LIST OF FIGURES xi
 B.31 Filtered weekly USA M1 Adj. (1981-2008) correlograms. . . . . . . . . . . . . . . . . 197
 B.32 Filtered weekly USA M1 Adj. (1981-2008) partial correlograms. . . . . . . . . . . . . 197
 B.33 Weekly USA Adj. M1 (1981-2008) probability plots: Empirical vs normal. . . . . . . 198
 B.34 Weekly USA Adj. M1 (1981-2008) probability plots: Empirical vs L evy. . . . . . . . 198
 B.35 Weekly US M1 Adj. QQ plots: Empirical vs normal. . . . . . . . . . . . . . . . . . . 199
 B.36 Weekly US M1 Adj. QQ plots. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
 B.37 Filtered vs raw weekly USA M1 (1981-2008) data series. . . . . . . . . . . . . . . . . 202
 B.38 Weekly USA M1 (1981-2008) probability plots: Empirical vs L evy. . . . . . . . . . . 202
 B.39 Filtered vs raw weekly USA M1 Adj. (1981-2008) data series. . . . . . . . . . . . . . 203
 B.40 Weekly USA Adj. M1 (1981-2008) probability plots: Empirical vs normal. . . . . . . 203
 B.41 Weekly USA Adj. M1 (1981-2008) probability plots: Empirical vs L evy. . . . . . . . 204
 B.42 Filtered vs raw weekly USA M2 (1981-2008) data series. . . . . . . . . . . . . . . . . 204
 B.43 Estimated L1: Empirical vs normal. . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
 B.44 Estimated L1 probability plots: Empirical vs L evy. . . . . . . . . . . . . . . . . . . . 205
 B.45 Estimated L1 QQ plots. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
 B.46 Estimated L1: Empirical vs normal. . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
 B.47 Estimated L1 probability plots: Empirical vs L evy. . . . . . . . . . . . . . . . . . . . 208
 B.48 US real curve: Estimated L1 QQ plots. . . . . . . . . . . . . . . . . . . . . . . . . . . 209
List of Tables
 6.1 Ljung-Box-Pierce Q-test for SA CPI raw and squared returns. . . . . . . . . . . . . . 127
 6.2 Engle?s ARCH test results for SA CPI raw and squared returns. . . . . . . . . . . . 128
 6.3 Monthly SA CPI L evy distributions? estimated parameters. . . . . . . . . . . . . . . 139
 6.4 Estimated parameters for empirical L1 for SA nominal forward rate. . . . . . . . . . 146
 6.5 Ljung-Box-Pierce Q-test for SA Monthly CPIX (1997-2008) raw and squared returns. 151
 6.6 Engle?s ARCH test results for SA Monthly CPIX (1997-2008) raw and squared returns.151
 6.7 Estimated parameters for monthly SA CPIX log returns. . . . . . . . . . . . . . . . . 154
 6.8 Descriptive statistics of S.A. Data series log returns. . . . . . . . . . . . . . . . . . . 154
 6.9 Chi squared Pearson?s test. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
 6.10 Descriptive statistics of USA Data series log returns. . . . . . . . . . . . . . . . . . . 155
 6.11 Hypothesis Tests of US Data series log returns. . . . . . . . . . . . . . . . . . . . . . 156
 6.12 Estimated parameters for US nominal forward rate. . . . . . . . . . . . . . . . . . . 159
 6.13 Estimated parameters for US real forward rate. . . . . . . . . . . . . . . . . . . . . . 159
 6.14 Kolmogorov-Smirnov test. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
 A.1 Monthly SA CPI L evy distributions? estimated parameters. . . . . . . . . . . . . . . 168
 A.2 Ljung-Box-Pierce Q-test for SA Monthly M2 (1965-2007) raw and squared returns. . 170
 A.3 Engle?s ARCH test results for SA M2 (1965-2007) raw and squared returns. . . . . . 171
 A.4 Ljung-Box-Pierce Q-test for SA Monthly M3 (1965-2007) raw and squared returns. . 173
 A.5 Engle?s ARCH test results for SA M3 (1965-2007) raw and squared returns. . . . . . 173
 B.1 Ljung-Box-Pierce Q-test for USA CPI (1821-2007) raw and squared returns. . . . . . 179
 B.2 Engle?s ARCH test results for USA CPI (1821-2007) raw and squared returns. . . . . 179
 B.3 Estimated parameters for USA CPI (1821-2007). . . . . . . . . . . . . . . . . . . . . 181
 B.4 Ljung-Box-Pierce Q-test for USA CPI (1937-2007) raw and squared returns. . . . . . 184
 B.5 Engle?s ARCH test results for USA CPI (1937-2007) raw and squared returns. . . . . 185
 B.6 Estimated parameters for USA CPI (1937-2007). . . . . . . . . . . . . . . . . . . . . 186
 B.7 Ljung-Box-Pierce Q-test for USA Weekly M1 (1981-2008) raw and squared returns. . 190
 B.8 Engle?s ARCH test results for USA M1 (1981-2008) raw and squared returns. . . . . 190
 B.9 Estimated parameters for weekly USA M1 (1981-2008). . . . . . . . . . . . . . . . . 193
 B.10 Ljung-Box-Pierce Q-test for seasonally adjusted USA Weekly M2 (1981-2008) raw
 and squared returns. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
 B.11 Engle?s ARCH test results for seasonally adjusted USA M1 (1981-2008) raw and
 squared returns. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
 B.12 Estimated parameters for USA Adj. M1 (1981-2008). . . . . . . . . . . . . . . . . . . 199
 B.13 Estimated parameters for empirical L1 for SA nominal forward rate. . . . . . . . . . 207
 B.14 Estimated parameters for empirical L1 for US real forward rate. . . . . . . . . . . . 210
 xii
Preface
 For the last decades major economies worldwide have been experiencing a constantly increasing
 in ation rate. Combined with historically low interest rates and high money growth (more than
 18% per year for China [74]), these place the Central Banks far behind the current in ation/interest
 rate curve. Moreover, if Central Banks  ght in ation by raising interest rates, their currency will
 strengthen, but they will lose market shares. The fact that in ation is rising and should keep
 increasing steadily for the coming years, if not decades; has led to the introduction of a new type of
  nancial instrument.
 Instead of preserving the investment?s (nominal) value, these instruments guarantee its purchasing
 power throughout the years at a certain threshold. These securities are termed in ation linked (IL)
 or in ation indexed (II) derivatives and have their payo linked to a price index, i.e. the prices of
 goods and services.
 Furthermore, IL securities are often more pro table than their corresponding nominal (i.e. conven-
 tional) counterparts. This is because in ation expectation is mostly non-negative (especially for long
 maturities e.g. 10, 20 years and more) due to  uctuations in supply and demand. For example1 not
 long ago a \normal" car had no air-conditioner, nor CD player. Nowadays, because \most" recent
 cars have both air conditioning and CD (and even MP3) players they are more expensive; that is
 in ation. In the meantime, wages did not necessary follow; thus to buy a car people take credit.
 Since future money gets used today, the amount of money available today is \virtually" increased.
 But, from the time value of money, R100 moved back to today is not \really" worth R100 today.
 Hence disrupting the equilibrium between overall value of money in circulation and overall value
 of \goods" being produced. To restore this equilibrium, the intrinsic value (purchasing power) of
 money needs to be devaluated. This imbalance gets propagated in other sectors like oil, food, etc.
 1These are simpli ed examples, an extensive coverage can be found in Economics books and on the web [105].
 1
LIST OF TABLES 2
 Assuming the features of a car didn?t change over time, again because of technological evolution in
 the car production system, over time, more cars get produced during a given period of time. Hence
 supply \exceeds" demand, to restore the equilibrium between supply and demand, the price of cars
 should go down. If this happened, the car industry will not get rewarded for their achievements.
 Thus they won?t be aiming for constant improvement. In the previous example setting to maintain
 the equilibrium between supply and demand, the price of new cars should be identical to old ones?
 (without air conditioning and player) and the same problem is faced. Governments generally imple-
 ment schemes to maintain the prices of services and goods \almost" constant, thus contributing to
 in ation.
 The major drawback of the in ation market is its relative illiquidity just as the interest rate market
 at its beginning. And similarly, to the latter market, the in ation market should experience a fast
 growth in the coming years; eventually rejoining the interest rate market among the key components
 of the  nancial world.
 Mathematical models in Finance literature and those used by investors are mainly based on Brownian
 motion although it is known that real-life  nancial data provides a di erent statistical behaviour
 than that implied by these models [8, 27, 109, 55, 9]. Recent empirical studies [48, 35] have proven
 that L evy processes are better distributions than the normal distribution for models in Finance in
 general and for returns distribution in particular, because of their accuracy and  exibility. Moreover,
 L evy processes do not only improve the  t of the distribution, but they give a more realistic and
 intuitive picture of the price movements at the microstructure level [46]. This study investigates
 how general L evy processes can be applied to the in ation market?s models.
 The emphasis in this study lies on the application of L evy processes in in ation models, particularly
 the pricing of IL bonds, swaps, caps and  oors. In this process, both no-arbitrage and pricing kernel
 models are considered. In pricing kernel models, the main question is the modelling of the stochastic
 process governing the prices of contingent claims. An alternative approach to compensating for
 market illiquidity is through macroeconomic factors speci cation, which is not exclusive of previous
 approaches.
 Here is a brief overview of each chapter. Chapter 1 brie y surveys some aspects of in ation and
 existing in ation derivative models. Chapter 2 reviews generalised hyperbolic distributions and
 L evy processes? characteristics, with focus on how L evy processes can be used to generalise the
 classical structural approach due to Merton [83]. Chapter 3 generalises the Heath Jarrow Morton
LIST OF TABLES 3
 (HJM) approach proposed by Jarrow and Yildirim [71] for IL products? pricing and later extended
 by Hinnerich [63] for marked point processes. Moreover, the new framework is used to price IL
 swaps, caps and  oors.
 Chapters 4 and 5 introduce two pricing kernel frameworks. The former framework built by Hughston
 and Macrina [68] is a stochastic monetary economy structure to price IL securities. While the latter
 chapter does a reverse engineering [8] of the nominal and real pricing kernels from bond prices (IL
 and nominal) and in ation. The latter model is an improvement on the previous model because
 it does not use the agent?s utility functions, thus avoiding the widely documented discrepancies
 between representative agent theories and observed asset prices [61].
 Chapter 6 starts via an empirical study of the in ation market?s data both for the South African and
 the American markets. Note that the former market is in a developing country, i.e. more illiquid
 than the latter market which is in a developed country. The study looks at the L evy distributions
  t against the normal distribution  t. It comes out that there is always at least a L evy distribution
 that performs better than the normal distribution. Moreover, L evy distributions have more degrees
 of freedom than their Gaussian counterpart thanks to their increased number of parameters, which
 make them more  exible and robust for calibration purposes. In the second part, Chapter 6 provides
 calibration tools used in the other chapters for option prices? calibration.
Chapter 1
 In ation Modelling
 As of December 2003 there had been eleven issuances of Treasury In ation Protected Securities
 (TIPS) by the US treasury. TIPS are meant to preserve the purchasing power of investors instead
 of the nominal value of their investment. Since their  rst appearance in the 18th century, it is only
 during the last decade that they have become more and more popular. Almost non-existent in 2001,
 the in ation market grew to about e50bn notional through the broker market in 2004, doubling its
 value in 2003 [72].
 In ation is a persistent increase in the price of products and services; it is synonym to a persistent
 decline in the purchase power of money. The opposite price?s movement, a decrease in the price of
 products is called de ation. Although de ation might seem desired, in ation and de ation con ict
 with the Central Bank objective to stabilize prices through their monetary policy.
 De nition 1.1. In ation (resp. de ation) is de ned as the relative increase (resp. decrease) of the
 level of the consumer price index over a period of time.
 The generalized and constant rise in the prices of goods has generated a growing interest for products
 whose value is linked to a price index and thus \indirectly" to in ation. These instruments are
 referred to as in ation linked (IL) or in ation indexed (II) derivatives. They are meant to preserve
 an investor?s purchasing power at a certain level throughout the years. This is achieved by linking
 their pay-o to the growth rate of prices.
 The present chapter gives an overview of the in ation market and existing frameworks for IL deriva-
 tives pricing. Section 1.1 describes the in ation market and its main players. Section 1.2 reviews
 the most commonly traded IL instruments and their main features. And  nally, Section 1.3 presents
 4
1.1 In ation Market 5
 some existing frameworks for the pricing of IL products.
 1.1 In ation Market
 In ation is a measure of the variation of the price of a prede ned basket of goods and services. Obvi-
 ously, the composition of this reference basket greatly impacts the value of the in ation. According
 to the basket?s components and their respective weight, a variety of in ation indexes has been de-
  ned. Examples of in ation indexes are the Consumer Price Index (CPI), the Retail Price Index
 (RPI), the Euro-zone Harmonised Index of Consumer Prices (Euro-HICP) and the Gross Domestic
 Product (GDP).
 Usually in ation is not high enough to be noticed over a short period of time. Nevertheless a
 straightforward computation shows that if over 30 years we have an averaged in ation of 1% per
 annum, then R100 at initial time will have the purchasing power of R74 and only R48 if in ation
 averages 2:5%. Taking a look at the South African CPI (+11:7% y=y in May 2008 [98]) and the
 US CPI (+0:8% in May 2008 [106]), there is reason to be concerned by in ation especially in South
 Africa (SA).
 (a) SA annual CPI (2000 = 100) 1960 2005 (b) Ghana annual CPI (2000 = 100) 1964 2005
 Figure 1.1 Evolution of the annual South African and Ghanian CPI [104].
 (a) US annual CPI (2000 = 100) 1960 2005 (b) UK annual CPI (2000 = 100) 1960 2005
 Figure 1.2 Evolution of the annual UK and US CPI [104].
 Figures 1.1 and 1.2 present the CPI evolution over about forty years for some developed (UK, USA)
1.1 In ation Market 6
 and emerging (Ghana, SA) countries. Note that these CPI are all normalised at 100 in 2000. As
 was expected, the lowest growth rate of 557% in 45 years is in a developed country, US; and the
 highest growth rate of 4048287% in 41 years is observed in an emerging country, Ghana. The
 Ghanian in ation is more than 7000 times the American in ation over a longer period of time.
 Moreover, the in ation rate of 1444% over 45 years for UK is still almost 3000 times smaller than
 that of Ghana. This is without including the high in ation rate observed during the last years which
 should impact more emerging countries because they do not have in place the structure to e ciently
 perform in ation rate targeting. South Africa whose economy can be said to be in-between that of
 a developed country and an emerging country has an in ation rate of 4152% in 45 years which is
 only about three times that of UK. However, looking at the estimations in the previous paragraph,
 there is still reason to be worried. These statistics suggest that developed countries should hedge
 their in ation risk and emerging countries must hedge their in ation risk.
 A  nancial product exists and persists because of the supply and demand. This implies two cor-
 responding groups of players, respectively payers=issuers and receivers=investors. Governments and
 private corporations constitute the main IL products issuers [38]; while the main investors in IL
 derivatives are pension funds and retail investors.
 1.1.1 IL Products Issuers
 In ation indexed products issuers should have some IL liabilities whose risk they want to share
 through IL products. Some of the countries issuing IL securities had high in ation prior to this
 initiative (Mexico and Brazil with respectively 114:8% and 69:2% during the 1950s and 1960s hy-
 perin ation period), but that was not the case for most.
 Government
 A government might issue IL products for several reasons. Firstly, a government can in uence
 in ation (by reducing public cost through public debt?s interest rate or premium) and thus bene t
 from issuing IL bonds. Secondly, IL bonds are adjusted to in ation whereas the conventional nominal
 bonds bear the risk of loosing value in real terms over time due to in ation. Another way to see this
 is through the fact that the conventional nominal interest rate on government?s bonds is similar to
 the real interest rate plus in ation. This relationship is referred to as the Fisher hypothesis1.
 1See Equation 1.1
1.1 In ation Market 7
 For instance, suppose that a government can issue either conventional nominal bonds with yield 8%
 or IL bonds with a real yield of 3% with the same maturity. The features of these two bonds imply
 that the market expectation of in ation over the lifetime of the bonds is 5%. However, if the realised
 in ation turns out to be 3%, then the government will just have a debt of 6% to repay with IL bonds
 as opposed to a \ xed" 8% with conventional bonds. In the event that in ation turns out higher
 than what had been expected, conventional bond issuance would of course have been the cheaper
 alternative.
 Secondly, given that a government can in uence in ation, issuance of in ation-indexed securities
 is a proof of its determination to  ght in ation. In case of in ation, investors can transfer their
 losses through the purchased IL bonds. The involved risks taken by the government shows its  rm
 intention to dampen in ation. Besides, the government?s performance in controlling in ation can be
 gauged through IL products which provide a direct measure of real interest rate necessary to some
 decision makers. Prior to the issuance of IL securities in the US there was no direct means to study
 real interest rates [32].
 Private Corporations
 Private sector entities elect to issue indexed rather than nominal debt mainly for identical reasons
 as governments. Corporate treasurers judging that the expected in ation (as priced by the market)
 is too high will consider the issuance of in ation-indexed bonds more attractive. Moreover, the
 diversi cation of the company?s debt portfolio implied by IL derivatives issuance and the improved
 risk-return characteristics is also very appealing [38].
 The main non-governmental issuers of IL products are insurance companies. Due to the increasing
 risk of in ation and diminishing pension2 payments, insurance companies have started selling IL
 products to take over some if not all of the in ation risk from their customers [111].
 1.1.2 IL Products Investors
 With the global aging of the world?s population, pension funds and the saving system they represent
 is becoming capital for the economy. The key variable for any pension plan bene ciary is not the
 nominal amount of the pension payment, but the purchasing power guaranteed thereof.
 In a standard contribution pension plan, the plan member bears a considerable risk due to in ation.
 2See next section 1.1.2.
1.2 In ation Products 8
 A distinct feature of pension funds, when compared to other  nancial institutions such as banks
 for example, is the very long maturity of their liabilities [49]. The typical duration of pension fund
 liabilities currently lies over a period of 30 years or more during which the pension bene t acquiring
 power might diminish. In fact, many plan members may not be aware that the bene ts they will
 obtain from a classical, non-IL pension plan may not be su cient to carry their expenses in the
 future, as price levels may have increased due to in ation. A simple calculation shows that an
 annual in ation rate of 1:5% over 30 years will reduce the real value of R100; 000 then to R63; 546.
 It therefore makes sense to link pension products to in ation.
 At an individual level, IL products or structured products can also be used by agents in the market
 to hedge the risk due to in ation.
 1.2 In ation Products
 The most commonly traded in ation linked securities are bonds, swaps, caps and  oors. This section
 reviews the main attributes of these instruments. A more detailed covering of IL securities can be
 found in [72, 38].
 To lighten the text, the nominal currency in this section will be the South African ZAR or R.
 1.2.1 In ation Linked Bonds
 A conventional (nominal) bond Pn(t; T ) represents the value at time t of an instrument that pays
 R1 at maturity T . The corresponding IL bond PIL(t; T ) represents the value (in ZAR) at time t of
 an instrument that pays RI(T ) at maturity T . If the IL bond?s pay-o is measured in unit of I(t)
 at time t, then it also pays 1 at T . When ignoring the units, its pay-o is similar to that of the
 nominal bond.
 When an IL bond?s value is divided by the price index, the corresponding real bond?s value is
 obtained:
 Pr(t; T ) =
 PIL(t; T )
 I(t)
 ;
 where the unit of a real bond, Pr(t; T ), is goods and services.
 Though real bond?s value can be deduced from IL bond?s (nominal) value, only nominal and IL
 bonds are e ectively traded on the market. Moreover, although IL bonds are quoted on the market
 in term of real yield, real bonds only exist by construction and are abstract.
1.2 In ation Products 9
 If rn(t) is the nominal interest rate, then under the risk neutral measure Q
 Pn(t; T ) = E
 Q
 t
 "
 exp
  
 
Z T
 t
 rn(s)ds
 !#
 :
 Similarly, if rr(t) is the real interest rate, then under Q
 Pr(t; T ) = E
 Q
 t
 "
 exp
  
 
Z T
 t
 rr(s)ds
 !#
 ;
 and the in ation linked bond is de ned by
 PIL(t; T ) = Pr(t; T )I(t):
 A  rst order approximation of the relationship between the interest rates, under the nominal risk
 neutral measure, is given by the following equation known as the Fisher equation3 [86]:
 rn(t; T ) = rr(t; T ) + i
 e
 t (t; T ); (1.1)
 where rn(t; T ) (resp. rr(t; T )) is the nominal (resp. real) interest rate for the time interval [t; T ] and
 iet (t; T ) is the expected in ation rate over [t; T ] at time t.
 The di erence between the nominal and in ation yields is referred to as the in ation breakeven rate or
 in ation compensation. From the Fisher equation, the breakeven rate is a \good" approximation of
 the expected in ation. However, the relationship between nominal and real yield is more complicated
 and better estimated using the expectation hypothesis according to which
 rn(t) = rr(t) + i
 e
 t (t; T ) + PrIL(t);
 where PrIL(t) is the in ation risk premium. It is the additional return IL issuers need to pay on
 nominal bonds compared with IL bonds and depends on the volatility of in ation (higher volatil-
 ity leads to higher premium) and risk-averseness of investors (the more risk-averse the higher the
 premium) [38].
 Similarly to the interest rate market whose participants? pool is expanded beyond traders in interest
 rate through other interest rate derivatives (swaps, caps,  oors, etc); IL derivatives have added
  exibility to the in ation market and given new opportunities to investors.
 3The full relationship between the nominal and real interest rates is restated in Equation 5.6 with a detailed
 derivation.
1.2 In ation Products 10
 1.2.2 Zero-Coupon In ation Swaps
 Zero-coupon in ation swaps are considered the building block of the in ation market because of
 their simplicity, their transparency and the new investment opportunities they generate [38]. A
 zero-coupon in ation swap can be used to convert a nominal bond into a corresponding IL bond or
 to preserve the purchasing power of its notional with respect to a given in ation index.
 By locking in a zero-coupon in ation swap, the participants agree to exchange the change in the
 in ation index over a period [t; T ] against a speci ed compounded interest rate. If t is the contract
 signature date (i.e. It is known at the signature of the contract), then the swap is spot starting. If
 t is instead a future date (i.e. It not yet known), then the swap is forward starting.
 Let N denote the notional of the swap. There is no cash  ow initially. At maturity T , the payer pays
 the net increase in in ation over the swap?s life N
  
IT
 It
  1
  
. The receiver pays the  xed amount
 corresponding to a prede ned annual compound rate b, N
  
(1 + b)T t  1
  
. The rate b is referred
 to as the breakeven swap rate and quoted in the market.
 1.2.3 Year-on-Year Swap
 The year-on-year (y-o-y) in ation swap is a variant of the zero-coupon swap with multiple payments
 (typically annually) over the term of the contract. Let [Ti 1; Ti] denote a sub-period of an y-o-y
 in ation swap with notional N . At Ti, the swap payer pays N
  
IT
 It
  1
  
and receives N
 b
 p
 where b
 is a pre-agreed zero coupon rate and p its annual periodicity.
 1.2.4 In ation Caps, Floors and Swaptions
 In ation caps,  oors and swaptions are in ation volatility products. In ation caps and  oors are
 mainly use to set boundaries of an investment?s pay-o (i.e. limit losses or bene t). For example
 along with an IL bond, an IL  oor on the notional is usually bought as protection against eventual
 de ation.
 An in ation cap (resp.  oor) is a collection of caplets (resp.  oorlets) each of which is a call
 (resp. put) on a zero-coupon swap. A caplet (resp.  oorlet) pays the di erence with respect to a
 (compounded) strike in case in ation turns out to be higher (resp. lower) than this pre-speci ed
 strike. A caplet written over the period [t; T ] with notional N and strike K pays a maturity
 Pay-o = N max
  
 
 
IT
 It
  1 K
  
; 0
  
;
1.3 In ation Models 11
 where  = 1 for a caplet and  =  1 for a  oorlet.
 Just as in ation swaps, IL caps and  oors can be spot starting or forward starting.
 A swaption is an option to enter into a forward starting in ation swap (zero-coupon, year-on-year,
 etc) at a pre-speci ed coupon.
 1.3 In ation Models
 In ation has an analogy both with interest rates and Foreign Exchange (FOREX) [23, 67]. Given
 the diversity of methods existing to model these two, some \good" approaches were just tailored
 to IL securities. Most of the IL pricing frameworks use either the foreign currency analogy or the
 pricing kernel. These methodologies are respectively similar to the short-rate and the pricing kernel
 for interest rate derivatives. This section starts by a brief review of common interest rate models
 which are later implicitly used in the foreign currency analogy and pricing kernel frameworks.
 1.3.1 Interest Rate Approach
 Because of its similarities to the interest rate (de ned as a percentage increment to an index) in ation
 models were  rst tailored to interest rate models. This subsection is entirely based on the paper by
 Fischer Black [18].
 At the beginning of in ation theory, it was either modeled as a normal, a lognormal or a square root
 process. Using a normal process, the volatility of the change in the interest rate does not depend
 on the rate, though it may depend on time. When a lognormal distribution is used, the volatility
 of the fractional change in the interest rate does not depend on the rate. And with the square root
 process, the ratio of the variance of the change in the interest rate to the rate does not depend on
 the rate, so the volatility of the change in the rate is proportional to the square root of the rate at
 a given time. Of course, mean reversion can be inserted in any of the previous models.
 The normal process implies that as the rate goes toward zero, the interest rate volatility does not
 decline. This contradicts the observed fact that volatility seems to decline with the rate, but it could
 be considered as an acceptable  aw. Apart from this de ciency, in a normal distribution the nominal
 interest rate has a non-zero probability of being negative. Though in ation can be negative, the
 nominal rate is always non-negative. After all, people can hold currency: they would rather keep
 currency under their mattresses than hold instruments bearing negative interest rates.
1.3 In ation Models 12
 The lognormal process assumes that the nominal rate is non-negative, especially that it is always
 non-zero. However, in the 1930s the US nominal interest rate fell to zero and there are other such
 historical cases. Furthermore, a lognormal distribution implies that as the rate approaches zero the
 volatility falls very rapidly. Whereas, from market observations when the volatility falls, it does not
 seem to fall this rapidly.
 The square root process is the most complex of all three and is halfway between the normal and the
 lognormal. The short rate will be non-zero if the mean reversion is quite strong or the short rate
 drift is large enough. However, when none of these conditions is satis ed, it is possible for the rate
 to become zero, we then have to decide whether zero is a re ecting barrier or an absorbing barrier.
 Assuming that zero is a re ecting barrier implies that the rate will \bounce" at zero while if zero is
 an absorbing barrier, we must assign a probability for the rate becoming positive again; thus having
 more complexity. Of the three alternatives, the absorbing barrier seems the most realistic [18].
 1.3.2 Foreign Exchange Approach
 The Foreign Exchange (FOREX) approach to modelling in ation is the most used nowadays. It is
 based on the foreign currency analogy in which real and nominal rates are assimilated to currencies
 in respectively the foreign and domestic economies, and the CPI is similar to the exchange rate [23].
 The reference framework to price IL securities is due to Jarrow and Yildirim [71]. The following
 subsections present this model and extension by Mercurio [81] and Belgrade-Benhamou-Koehler [14].
 Jarrow and Yildirim Model
 The most quoted foreign currency analogy implementation is due to Jarrow and Yildirim [71] and
 is based on a Heath-Jarrow-Morton (HJM) model. In analogy with the HJM model of foreign
 currency they build a three-factor, arbitrage-free term structure model by modeling the dynamics of
 the real and nominal instantaneous forward interest rates and the in ation. The underlying sources
 of randomness are allowed to be correlated and the instantaneous forward rates are  tted to the
 market data.
 Under the real world  ltered probability space ( ;F ; (Ft)t 0;P), the Jarrow and Yildirim model is
1.3 In ation Models 13
 described by:
 dfn(t; T ) =  n(t; T )dt+ &n(t; T )dWn
 dfr(t; T ) =  r(t; T )dt+ &r(t; T )dWr
 dI(t) = I(t) (t)dt+  II(t)dWI
 with I(0) = I0 > 0, and
 fx(0; T ) = f
 M
 x (0; T ); x 2 fn; rg
 where
 i. f(t; T ) represents an instantaneous forward rate with maturity T at t and I(t) represents the
 in ation rate at time t;
 ii. The Brownian motions (Wk), with k = n; r; I standing respectively for nominal, real and in a-
 tion, have correlation  n;r,  n;I and  r;I ;
 iii.  n,  r,  are adapted processes;
 iv. &n, &r are deterministic functions;
 v.  I is a positive constant;
 vi. fMn (0; T ), f
 M
 r (0; T ) are the nominal and real instantaneous forward rates observed in the market
 at time 0 for maturity T respectively.
 Hence Jarrow and Yildirim assumed both nominal and real (instantaneous) rates to be normally
 distributed under their respective risk neutral measures. Then using the no-arbitrage principle and
 taking forward rate volatilities of the form4  k(t; T ) =  ke ak(T t) with k = n; r, they proved that
 the real and nominal rates are Ornstein-Uhlenbeck5 processes under the nominal measure Qn, and
 that the in ation index I(t), at each time t, is lognormally distributed under Qn.
 Since the real and nominal rates evolve following a Gaussian distribution, closed form solutions
 can be computed. However, this model has several drawbacks such as the di culty to estimate
 parameters from market data and the possibility of interest rate becoming negative.
 4For statistical reasons.
 5i.e. of the form drt =   (rt   )dt+  dWt; where  ;  ;  are parameters and Wt denotes a Brownian motion.
1.3 In ation Models 14
 Mercurio Market Model
 Mercurio [81] proposed two variants of market models as an alternative to JY [71] and equivalent
 to Belgrade et al. [14] for pricing year on year in ation indexed swaps (YYIIS). Resorting to the
 lognormal LIBOR model, the  rst market model considers the nominal and the real rates to follow
 a lognormal distribution and the forward in ation index to follow geometric Brownian motion. The
 YYIIS price is a function of the nominal and real forward rates? (instantaneous) volatilities and
 their correlations, for each cash  ows payment time; the correlations between real forward rates and
 forward in ation indices, again for each cash  ows payment time. This YYIIS pricing formula is
 more complicated both in terms of input parameters and in terms of the calculations involved than
 the JY one. Nevertheless, it can be solved using numerical integration and, as is typical in a market
 model, the input parameters can be determined more easily than in the JY approach. But this
 new formula still depends on the volatility of real rates which may be hard to estimate. Given this
 de ciency, Mercurio developed a second market model to overcome this estimation issue [81].
 He obtained a pricing formula for YYIIS combining the advantage of a fully-analytical formula
 with that of a market-model approach which does not depend on the real rates volatility anymore.
 The price of YYIIS depends on the (instantaneous) volatilities of the forward in ation indexes and
 their correlations, the (instantaneous) volatilities of nominal forward rates and the instantaneous
 correlations between forward in ation indices and nominal forward rates. The drawback of this model
 is that it is based in a rough approximation for long maturities, especially when the correlations
 between forward rates and in ation are non zero; the formula is exact when these correlations are
 zero [81].
 Mercurio has shown that these three models produced similar results when calibrating with market
 data although they di ered when away-from-the-money6 instruments are considered [81].
 A consolidated practice in all developed options markets is to include some kind of smile e ect
 when simultaneously pricing caps with di erent strikes; to achieve this Mercurio and Moreni [82] as
 in Heston [62] introduce stochastic volatility in the forward CPI dynamics under the spot LIBOR
 measure. The cap prices obtained are a good approximation of the model?s price.
 6Not at-the-money, i.e. either in-the-money or out-of-the-money.
1.3 In ation Models 15
 Belgrade-Benhamou-Koehler Model
 The Belgrade-Benhamou-Koehler (BBK) [14] model was designed precisely to solve, using the no-
 arbitrage principle, the two major disadvantages of the JY model: the lack of link between zero
 coupon bonds and year-on-year swaps and the non-observable parameters. This model has two
 main objectives; to be simple (i.e. to have only few parameters) and to be robust (i.e. to replicate
 market prices). The main assumption made by BBK is that the market model for in ation considers
 forward in ation index returns as a di usion with deterministic volatility structure. Under the risk
 neutral probability measure Q, this index follows geometric Brownian motion with deterministic
 drift and volatility. In their paper, they consider three di erent functional forms of volatility (con-
 stant, exponentially decaying and adjusted exponentially). They present a method to parametrize
 the volatility structure to include the market data of caps/ oors. They also perform a convexity
 adjustment of the in ation swaps derived from the di erence of martingale measures between the
 numerator and the denominator. Given that it is not possible to estimate implicit correlations from
 the market data, they suggest some boundary conditions which for certain model hypotheses (for
 example constant volatility structure) give unrealistic results. This model is more suited in markets
 where there is enough information from zero-coupon and year-on-year swaps. It is important to be
 aware that to derive the model some approximations were done in the process so the solution is not
 exact. Another drawback of this model is that it is computationally intensive.
 1.3.3 Macro- nance models
 Because the JY framework is based on the HJM model, it can only perform cross-sectional  tting
 and thus can not estimate the in ation risk premium. However, this limitation can be overcome
 by using a macro- nance model of the term structure. Such a model is characterised by the fact
 that it uses macroeconomic factors to improve the coherence between the model output and the
 observed term structure on the market. Such models form a subclass of the a ne term structure
 models (i.e. tractability and closed form solutions for asset pricing under certain restrictions). A
 macro- nance model can, in general, estimate both the correlation between the real and nominal
 interest rates and the risk premium \endogenously". These models di er by the complexity used to
 include the macroeconomic factors in the conventional short rate models. This subsection provides
 a brief overview of their general properties based on Piazzesi [6].
 In a no-arbitrage framework holding a zero coupon bond over a certain period of time [t; T ] is
1.3 In ation Models 16
 equivalent to the return of an average risk free short term rate during the same period under the
 risk neutral measure Q:
 P (t; T ) = EQ
 "
 exp
  
 
Z T
 t
 r(s)ds
 !#
 ; (1.2)
 where P (t; T ) is the price of a zero coupon bond of maturity T at any time t 2 [0; T ]. The two main
 components of this model are:
 i. the change of measure from the real world P, where the input data is measured and the risk
 neutral measure Q, where the pricing is actually done because of the properties it has and;
 ii. the short rate dynamics.
 For an a ne model the short rate is of the form
 r(t) = R(xt) , with xt 2 D  Rn;
 where R(x) is a ne and xt is an a ne di usion process under Q and the solution of a stochastic
 di erential equation of the form
 dxt =  (xt)dt+  (xt)dWt where Wt is a standard Brownian motion.
 Under some regularity conditions, the corresponding guessed closed form solution for pricing zero
 coupon bonds is a ne in the state variables and of the form:
 P (t; T ) = exp [A(t; T ) B(t; T )xt] ;
 with some restrictions on A(t; T ) and B(t; T ). Solving this system of equations gives the short rates
 dynamics. The change of measure is obtained through the pricing kernel and hence the model is
 complete.
 The pricing kernel or stochastic discount factor  = ( t)t 0 is de ned by
 P (t; T ) =
 E[ T ]
  t
 8 t 2 [0; T ]:
 With the short rate model attributes presented, let us discuss the macroeconomic side of the model.
 The short rate?s dynamics is modelled because it drives the entire yield curve through Equation (1.2).
 From a macroeconomic point of view the short rate can be governed by the Taylor rule [101, 5]. The
 Taylor rule gives the interest rate change a central bank should make in response to a divergence in
 in ation or economic growth. The less complex Fisher equation can also be used to introduce the
1.3 In ation Models 17
 macroeconomic factors [4, 30]. Making the assumption that agents maximise their utility is also part
 of the macroeconomic model and translates the reactions to the central bank behavior, the in ation
 gap (di erence between the realized in ation and the expected in ation), etc. The speci city of each
 micro- nance model is observable through the way the macroeconomic variables are inserted in the
 term structure model.
 1.3.4 Stochastic monetary economy models
 Similarly to the micro- nance models described in the previous subsection, the \stochastic monetary
 economy models" proposed by Hughston and Macrina [68] use macroeconomic factors and the pricing
 kernel to price IL securities. However, the latter framework does not assume linearity7 of the macro-
  nance models; instead, it assumes a positive \nominal" interest rate and the underlying pricing
 kernel that was advocated for by Flesaker and Hughston (FH) [54].
 Assuming Nt is the conventional num eraire, the corresponding pricing kernel is given by  t =
  t
 Nt
 in
 the real world probability measure8, where ( t)t 0 denotes the Radon-Nikodym density martingale
 transforming the real world measure into the risk neutral measure. The latter equation implies that
 under the real probability measure the asset price process multiplied with the pricing kernel process
 is a martingale. The process ( t)t 0 is a decreasing and positive supermartingale (i.e.  t   t+h
 with h > 0) thus ensures interest rate positivity.
 The IL framework built by Hughston and Macrina is based on the assumption that in ation is a
 purely monetary phenomenon. Thus the in uence of  uctuations in wages, supply and demand,
 foreign exchange and employment, etc. on in ation is not treated directly, but is rather re ected in
 the change of the rates of consumption and money supply, and the liquidity bene t of money supply.
 In a discrete time9 model, let the nominal money supply, the aggregate consumption and the nominal
 liquidity bene t be denoted respectively by (fMigi 0), (fkigi 1) and (f igi 0). At time ti, the real
 bene t (in units of goods and services) provided by the money supply is de ned by [68]
 li =
  iMi
 Ci
 for i  0:
 Considering a wealth function of the form
 W = E
 "
 NX
 n=0
  n(Cnkn +  nMn)
 #
 ;
 7i.e. that the models form a subclass of a ne term structure models.
 8This formula is fulling derived in Chapter 4
 9The formulas in continuous time have also been derived and are similar to those obtained in discrete time.
1.3 In ation Models 18
 where U( ;  ) is a bivariate utility function, the CPI (fCigi 0), pricing kernel (f igi 0) and fair price
 of IL instruments (H = fHigi 0) are determined by maximising a consumer investor?s function of
 the form
 J = E
 "
 NX
 n=0
 e  tnU(kn; ln)
 #
 :
 Example 1. (i) Considering a log-separable utility function of the form
 U(x; y) = A ln (x) +B ln (y);
 where A and B are two non-negative constants; the pricing kernel, the CPI and the pricing
 formula for an IL security are respectively
 Cn =
 A
 B
  nMn
 kn
 ;
  n =
 Be  tn
   nMn
 ;
 H0 =  0M0e
   tjE
  
Hj
  jMj
  
:
 (ii) Considering p; q 2]  1; 1]nf0g, two non-negative constants A and B, and a separable power
 utility function of the form
 U(x; y) =
 A
 p
 xp +
 B
 q
 yq;
 gives
 Cn =
  
A
 B
  1 q  nMn
 k(1 q)=(1 p)n
 ;
  n =
 B
 1
 1 q
 A
 q
 1 q
 k
 q
 1 q (1 p)
 n
   nMn
 ;
 H0 =
  0M0
 kq(1 p)=(1 q)0
 e  tjE
 "
 Hjk
 q(1 p)=(1 q)
 j
  jMj
 #
 :
 Note that the formulas obtained are not directly functions of any IL derivative?s price on the market.
 Therefore, this pricing methodology could be a solution to pricing IL products with in ation market
 illiquidity. The performances of this framework are further investigated in Chapter 4.
 1.3.5 Calibration
 The most commonly used data for calibration is that of US market because of its quality (the Federal
 Reserve publishes constant maturity US treasury bond yield data) and its time span, which is longer
1.3 In ation Models 19
 than for most of other countries. Even though there are not zero coupon bond rates available on
 the market, they can be deduced from market data. The common technique is through a bootstrap
 and an interpolation of coupon bearing bonds (and eventually swaps) often ignoring the seasonality;
 however this manipulation can introduce some measurement errors.
 There is no standard estimation method since estimating both yield and macro data is dependent on
 the number of parameters assumed in the model. Nevertheless in order to simplify the calculations,
 researchers usually impose all restrictions on the parameters before the estimation process; this can
 reduce the number of parameters to compute (e.g. symmetry in a matrix).
 Apart from US data, this study also uses South African data to evaluate the performances of the
 models both in a developed economy and a developing economy. All the models are based on L evy
 processes (see Chapter 2) to improve the  t. The parameter estimation is mainly done with the
 maximum likelihood method. Chapter 6 presents in detail the calibration process.
Chapter 2
 The L evy Process Framework
 L evy processes are basically processes with stationary and independent increments. They are an
 excellent tool to model distributions in mathematical  nance for four main reasons. First, they are
 the simplest class of processes with jumps. The latter become more obvious the smaller the time step
 considered between market observations. Second, they are part of both semimartingales and Markov
 processes with an additional robust mathematical structure. Third, some important processes like
 Brownian motion, Poisson process, stable and self-decomposable processes are special cases of L evy
 processes. Finally, they have been successfully applied to mathematical  nance, Physics and other
  elds both for research and practical usage [7, 70].
 This chapter starts by reviewing elements of L evy processes in Section 2.1. The remaining sections
 are devoted to more advanced topics for option pricing. Sections 2.2 and 2.3 present the Ito^ formula
 for L evy processes, the Girsanov change of measure and other tools which will be used later on.
 Section 2.4 describes the General Hyperbolic (GH) distribution and other subclasses considered for
 the calibration. Finally, Section 2.5 examines option valuation using the Fast Fourier transform [26].
 A more detailed presentation, both mathematical and practical, can be found in [70, 88, 35].
 2.1 L evy Processes
 The following basic assumption is made throughout this thesis.
 Assumption 1. Let ( ;F ;F;P) with F = (Ft)t 0 be a  ltered probability space satisfying the usual
 conditions, that is:
 20
2.1 L evy Processes 21
 (i) ( ;F ;P) is complete.
 (ii) All the null sets of F are contained in F0, i.e. all impossible events are known beforehand.
 (iii) F is a right continuous  ltration:
 Fs  Ft  F are  -algebra for s; t 2 R+; s  t; and Ft =
 \
 s>t
 Fs for all t  0:
 Furthermore, assumption is made that
 F =  
0
 @
 [
 t 0
 Ft
 1
 A
 This allows to specify a change of probability measure from P to Q by giving the density process
 (Zt)t 0, where Zt =
 dQ
 dP
  
 
 
 
Ft
 :
 The following de nition of L evy processes is from Applebaum (2004).
 De nition 2.1. An adapted stochastic process X = (Xt)t 0 on a  ltered probability space ( ;F ;F;P)
 taking values on Rd such that X0 = 0 is called a L evy process if:
 (i) X has increments independent of the past, i.e. Xt Xs is independent of Fs for 0  s < t <1.
 (ii) X has stationary increments, i.e. the distribution of Xs+t  Xs does not depend on s or
 equivalently Xs+t  Xs
 d
 = Xt where
 d
 = stands for the equality in distribution.
 (iii) Xt is continuous in probability or stochastically continuous, i.e.
 8 t  0; 8 " > 0 lim
 s!t
 P [jXt  Xsj > "] = 0:
 If the process X satis es all the previous conditions, then it can be shown (See Theorem 30 in [93])
 that there exists a transformation Y = (Yt)t 0 of X = (Xt)t 0 (i.e. P(Yt 6= Xt) = 0 for all t  0)
 with the following property
 (iv) For almost every ! 2  , the function t 7 ! X(t; w) is c adl ag (from the French \continue  a
 droite, limite  a gauche") that is everywhere right-continuous with left limit.
 This transformation is again a L evy process. Because this transformation is always possible, the
 latter condition is generally included among the characteristics of a L evy process.
 De nition 2.2. 1: Processes meeting only conditions (i) and (ii) are called processes with station-
 ary independent increments (PIIS) [70].
2.1 L evy Processes 22
 2: Processes meeting conditions (i), (iii) and (iv) are called time-inhomogeneous L evy processes.
 Example 2. (i) A standard Brownian motion in Rd is a L evy process.
 (ii) The Poisson process (Nt)t 0 with intensity  > 0 is a L evy process with values in N[f0g such
 that
 P(Nt = n) =
 ( t)n
 n!
 e  t:
 De nition 2.3. A probability distribution X is said to be in nitely divisible if for any positive integer
 n, there exists n independent and identically distributed (i.i.d.) random variables Yi; i = 1; 2;    ; n,
 such that Y1 + Y2 +    + Yn has distribution X.
 If (Xt)t 0 is a L evy process, the distribution of Xt, for any t > 0 is in nitely divisible. Hence Xt
 can be decomposed into n i.i.d. parts each having the same distribution with appropriately scaled
 parameters.
 The characteristic function of a L evy process X is of the form
 E[eiz Xt ] = et X(z); z 2 Rd
 where  X( ) : Rd ! R is the corresponding characteristic exponent. Since the log-characteristic
 function is linear in t and Xt is in nitely divisible, the distribution of Xt is fully determined by the
 distribution of X1.
 De nition 2.4. Let (Xt)t 0 be a L evy process on Rd. The jump size at time t  0 is de ned by
  Xt = Xt  Xt :
 Considering the family of Borel sets B(Rd), the Poisson random measure  : R+  Rd   ! R is
 de ned for every U 2 B(Rd) whose closure does not contain 0 by
  (t; U) =  (t; U; !) =
 X
 s:0<s t
  U ( Xs);
 where the indicator or characteristic function with respect to U ,  U is de ned by
  U (x) =
 8
 <
 :
 1; x 2 U
 0; x =2 U
 with U  Rd. In other words,  (t; U) is the number of jumps of size  Xs 2 U which occur before
 or at time t. Henceforth, the randomness parameter ! will be omitted among functions? parameters
 as in the previous equation.
2.1 L evy Processes 23
 De nition 2.5. Let (Xt)t 0 be a L evy process on Rd, then the measure  : Rd ! R de ned by
  (A) = E [#ft 2 [0; 1]g :  Xt 6= 0; Xt 2 A] ; A 2 B(Rd)
 is called L evy measure. It might be interpreted as the average number of jumps per unit time of the
 underlying processes (See [35] Equation (3:10)).
 An in nite-activity process is one with an in nite number of jumps in any  nite time interval. For
 an in nite-activity process,  (A) remains  nite for A such that 0 =2 A. An in nite-activity process,
 with measure  , has a  nite number of jumps on Rd n f0g, but may have an in nite number of
 jumps with measure zero. The sum of jumps becomes an in nite series. The integrability conditions
 imposed in the next theorem, guarantee the convergence of this in nite series.
 Theorem 2.6 (The Ito^-L evy decomposition). If X is a L evy process on Rd, then for every t  0,
 Xt has the decomposition:
 dXt =  dt+ dWt +
 Z
 jzj R
 z (dt; dz) +
 Z
 jzj<R
 z(   )(dt; dz) (2.1)
 with
  the constant R 2 [0;1]. In  nancial literature, R = 1 is commonly used for simpli cation.
  the vector  2 Rd.
  the random factor (Wt)t 0 is a d-dimensional Brownian motion with covariance matrix c.
  the Poisson random measure  on R+  R n f0g has compensator  (dt; dz) =  (dz)dt such as
 Z
 Rd
 (jzj2 ^ 1) (dz) <1.
 The previous theorem splits a L evy process into respectively a deterministic component (predictable
 part), a pure Brownian motion (random part), a Poisson integral (the large jumps) and a com-
 pensated Poisson integral (the small jumps). The  rst two parts constitute a Brownian motion
 with drift which is the continuous part of the L evy process. Similarly, the last two parts form the
 discontinuous part of the process.
 Note that the Brownian motion and Poisson measure are independent.
 The triplet (c;  ;  ) is called the L evy characteristics triplet (or for short L evy triplet) of X. By
 Corollary II:4:19 in [70], L evy process are semimartingales and their characteristics are deterministic.
2.2 Ito^ Formula for L evy Processes 24
 The previous decomposition of L evy processes can be extended to the case when its parameters are
 time and space dependent in the form
 dXt =  (t)dt+  (t)dWt +
 Z
 R
  (t; z) (dt; dz) (2.2)
 where
  (dt; dz) =
 8
 <
 :
 (   )(dt; dz); jzj < R
  (dt; dz); jzj  R
 for some constant R 2 [0;1]. Such processes are referred to as Ito^-L evy processes.
 2.2 Ito^ Formula for L evy Processes
 The dynamics of a su ciently smooth function of L evy processes can be deduced from those of
 the underlying processes through Ito^?s formula. This section reviews the Ito^ formula for Ito^-L evy
 processes and other closely related results that are of great use afterwards.
 Theorem 2.7 (The one-dimensional Ito^ formula [88]). Let (Xt)t 0 be a real valued Ito^-L evy process
 of the form
 dXt =  (t)dt+  (t)dWt +
 Z
 R
  (t; z) (dt; dz) (2.3)
 where
  (dt; dz) =
 8
 <
 :
  (dt; dz)  (dz)dt; jzj < R
  (dt; dz); jzj  R
 for some constant R 2 [0;1].
 Let f 2 C1;2(R2) and Y = (Yt)t 0 such that Yt = f(t;Xt), then the dynamics of the Ito^-L evy process
 Yt are given by
 dYt =
 @f
 @t
 (t;Xt)dt+
 @f
 @x
 (t;Xt) [ (t)dt+  (t)dWt] +
 1
 2
  2(t)
 @2f
 @x2
 (t;Xt)dt
 +
 Z
 jzj<R
  
f (t;Xt +  (t; z)) f (t;Xt ) 
@f
 @x
 (t;Xt )  (t; z)
  
 (dt; dz)
 +
 Z
 R
 ff (t;Xt +  (t; z)) f (t;Xt )g  (dt; dz):
 The following corollaries give some applications of Ito^?s formula for some simple functions. These
 functions are largely encountered in  nancial Mathematics as will be seen later on.
2.2 Ito^ Formula for L evy Processes 25
 Corollary 2.8. Let X = (Xt)t 0 denote a strictly positive real valued Ito^-L evy process with dynamics
 given by
 dXt =  (t)dt+  (t)dWt +
 Z
 R
  (t; z) (dt; dz):
 The L evy process Y = (Yt)t 0 de ned by Yt =
 1
 Xt
 for t  0 has dynamics
 dYt =  Y
 2
 t
  
 (t)  2(t)Yt
  
dt Y 2t  (t)dWt +
 Z
 jzj<R
  
Yt 
1 +  (t; z)Yt 
 Yt + Y
 2
 t  (t; z)
  
 (dt; dz) +
 Z
 R
  
Yt 
1 + Yt  (t; z)
  Yt 
 
 (dt; dz):
 Proof. Consider f(t;Xt) =
 1
 Xt
 , then
 @f
 @t
 = 0,
 @f
 @x
 =  
1
 X2t
 and
 @2f
 @2x
 =
 2
 X3t
 . By the one-
 dimensional Ito^ formula (Theorem 2.7)
 dYt =  
1
 X2t
 [ (t)dt+  (t)dWt] +
 1
 2
  2(t)
 2
 X3t
 dt+
 Z
 R
 ff (t;Xt +  (t; z)) f (t;Xt )g  (dt; dz)
 +
 Z
 jzj<R
  
f (t;Xt +  (t; z)) f (t;Xt ) +
 1
 X2t 
 (t; z)
  
 (dt; dz)
 =  
1
 X2t
 [ (t)dt+  (t)dWt] +  
2(t)
 1
 X3t
 dt+
 Z
 jzj<R
  
1
 Xt +  (t; z)
  
1
 Xt 
+
 1
 X2t 
 (t; z)
  
 (dt; dz)
 +
 Z
 R
  
1
 Xt +  (t; z)
  
1
 Xt 
 
 (dt; dz)
 =  
1
 X2t
  
 (t)  2(t)
 1
 Xt
  
dt 
1
 X2t
  (t)dWt +
 Z
 jzj<R
  
1
 Xt +  (t; z)
  
1
 Xt 
+
 1
 X2t 
 (t; z)
  
 (dt; dz)
 +
 Z
 R
  
1
 Xt +  (t; z)
  
1
 Xt 
 
 (dt; dz)
 =  Y 2t
  
 (t)  2(t)Yt
  
dt Y 2t  (t)dWt +
 Z
 jzj<R
  
Yt 
1 +  (t; z)Yt 
 Yt + Y
 2
 t  (t; z)
  
 (dt; dz)
 +
 Z
 R
  
Yt 
1 + Yt  (t; z)
  Yt 
 
 (dt; dz):
 Corollary 2.9. Let X = (Xt)t 0 denote an Ito^-L evy process with dynamics given by
 dXt
 Xt 
=  (t)dt+  (t)dWt +
 Z
 R
  (t; z) (dt; dz):
 Now consider the process Y = (Yt)t 0 de ned by Yt =
 1
 Xt
 for t  0, then the dynamics of Yt are
 dYt
 Yt 
=
  
  (t) +  2(t)
  
dt+
 Z
 jzj<R
  2(t; z)
 1 +  (t; z)
  (dt; dz)  (t)dWt  
Z
 R
  (t; z)
 1 +  (t; z)
  (dt; dz):
2.2 Ito^ Formula for L evy Processes 26
 Proof. Consider f(t;Xt) =
 1
 Xt
 , then
 @f
 @t
 = 0,
 @f
 @x
 =  
1
 X2t
 and
 @2f
 @2x
 =
 2
 X3t
 . By the one-
 dimensional Ito^ formula
 dYt =  
Xt
 X2t
 [ (t)dt+  (t)dWt] +
 1
 2
 X2t  
2(t)
 2
 X3t
 dt
 +
 Z
 jzj<R
  
f (t; (1 +  (t; z))Xt ) f (t;Xt ) +
 1
 X2t 
Xt (t; z)
  
 (dt; dz)
 +
 Z
 R
 ff (t; (1 +  (t; z))Xt ) f (t;Xt )g  (dt; dz)
 dYt =  
Xt
 X2t
 [ (t)dt+  (t)dWt] +
 1
 2
 X2t  
2(t)
 2
 X3t
 dt+
 Z
 jzj<R
  
1
 Xt 
1
 1 +  (t; z)
  
1
 Xt 
+
 1
 X2t 
Xt (t; z)
  
 (dt; dz) +
 Z
 R
  
1
 Xt 
1
 1 +  (t; z)
  
1
 Xt 
 
 (dt; dz)
 =  
1
 Xt
 [ (t)dt+  (t)dWt] +  
2(t)
 1
 Xt
 dt+
 Z
 jzj<R
  
1
 Xt 
1
 1 +  (t; z)
  
1
 Xt 
+
 1
 Xt
  (t; z)
  
 (dt; dz) +
 Z
 R
  
1
 Xt 
1
 1 +  (t; z)
  
1
 Xt 
 
 (dt; dz):
 Thus
 dYt =  
1
 Xt
 [ (t)dt+  (t)dWt] +  
2(t)
 1
 Xt
 dt+
 1
 Xt
 Z
 jzj<R
  
1
 1 +  (t; z)
  1 +  (t; z)
  
 (dt; dz)
 +
 1
 Xt
 Z
 R
  
1
 1 +  (t; z)
  1
  
 (dt; dz):
 From Yt =
 1
 Xt
 ,
 dYt
 Yt 
=  [ (t)dt+  (t)dWt] +  
2(t)dt+
 Z
 jzj<R
  
1
 1 +  (t; z)
  1 +  (t; z)
  
 (dt; dz)
 +
 Z
 R
  
1
 1 +  (t; z)
  1
  
 (dt; dz)
 dYt
 Yt 
=  [ (t)dt+  (t)dWt] +  
2(t)dt+
 Z
 jzj<R
  2(t; z)
 1 +  (t; z)
  (dt; dz) 
Z
 R
  (t; z)
 1 +  (t; z)
  (dt; dz)
 =
  
  (t) +  2(t)
  
dt  (t)dWt +
 Z
 jzj<R
  2(t; z)
 1 +  (t; z)
  (dt; dz) 
Z
 R
  (t; z)
 1 +  (t; z)
  (dt; dz):
 Corollary 2.10. Let X = (Xt)t 0 denote an Ito^-L evy process with dynamics given by
 dXt
 Xt 
=  (t)dt+  (t)dWt +
 Z
 R
  (t; z) (dt; dz):
2.2 Ito^ Formula for L evy Processes 27
 Now consider the process Y = (Yt)t 0 de ned by Yt = lnXt for t  0, then the dynamics of Yt are
 dYt =
  
 (t) 
1
 2
  2(t)
  
dt+
 Z
 jzj<R
 fln[1 +  (t; z)]  (t; z)g (dt; dz)
 + (t)dWt +
 Z
 R
 ln[1 +  (t; z)] (dt; dz):
 Proof. Consider f(t;Xt) = lnXt, then
 @f
 @t
 = 0,
 @f
 @x
 =
 1
 Xt
 and
 @2f
 @2x
 =  
1
 X2t
 . By the one-
 dimensional Ito^ formula
 dYt =
 Xt 
Xt
 [ (t)dt+  (t)dWt] 
1
 2
  2(t)
 X2t 
X2t
 dt
 +
 Z
 jzj<R
  
ln [Xt +Xt  (t; z)] lnXt  
1
 Xt
 Xt  (t; z)
  
 (dt; dz)
 +
 Z
 R
 fln [Xt +Xt  (t; z)] lnXt g  (dt; dz)
 =  (t)dt+  (t)dWt  
1
 2
  2(t)dt+
 Z
 jzj<R
 fln [1 +  (t; z)]  (t; z)g (dt; dz)
 +
 Z
 R
 ln [1 +  (t; z)]  (dt; dz)
 since X is c adl ag. Thus
 dYt =
  
 (t) 
1
 2
  2(t)
  
dt+
 Z
 jzj<R
 fln [1 +  (t; z)]  (t; z)g (dt; dz)
 + (t)dWt +
 Z
 R
 ln [1 +  (t; z)]  (dt; dz):
 Next follows a generalisation of Ito^?s formula which allows the derivative function to depend on
 multiple processes.
 Theorem 2.11 (The multi-dimensional Ito^ formula). Let X = (Xt)t 0  R
 n be an n-dimensional
 Ito^-L evy process of the form:
 dXt =  (t; !)dt+  (t; !)dWt +
 Z
 Rl
  (t; z; !) (dt; dz);
 where  : [0; T ]   ! Rn,  : [0; T ]   ! Rn m and  : [0; T ]  Rl   ! Rn l are adapted
 processes such that the integrals exist. Here Wt is an m-dimensional Brownian motion and
  (dt; dz) = [ 1(dt; dz1);    ;  l(dt; dzl)]
 =
  
 1(dt; dz1)  jz1j<R1 1(dt; dz1);    ;  l(dt; dzl)  jzlj<Rl l(dt; dzl)
  
where ( j)1 j l are independent Poisson random measures with respective compensator ( j)1 j l
 coming from l independent (1 dimensional) L evy processes and ( ) denotes the transpose.
2.2 Ito^ Formula for L evy Processes 28
 Note that each column  (k) of the n l matrix  = [ ij ] depends on z only through the kth coordinate
 zk, i.e
  (k)(t; z) =  (k)(t; zk); with z = (z1;    ; zl) 2 Rl:
 In particular, for i = 1; :::; n,
 dXi(t) =  i(t)dt+
 mX
 j=1
  ij(t)dW
 j
 t +
 lX
 j=1
 Z
 R
  ij(t; zj) j(dt; dzj):
 Let f 2 C1;2 ([0; T ];Rn) and Y = (Yt)t 0 such that Yt = f(t;Xt), then
 dYt =
 @f
 @t
 (t;Xt) dt+
 nX
 i=1
 @f
 @xi
 (t;Xt) [ i(t)dt+  i(t)dWt] +
 1
 2
 nX
 i;j=1
 (   )ij
 @2f
 @xi@xj
 (t;Xt) dt
 +
 lX
 k=1
 Z
 jzkj<Rk
 (
 f
  
t;Xt +  
(k)(t; z)
  
 f (t;Xt ) 
nX
 i=1
 @f
 @xi
 (t;Xt )  
(k)
 i (t; z)
 )
  k(dt; dzk)
 +
 lX
 k=1
 Z
 R
 n
 f
  
t;Xt +  
(k)(t; z)
  
 f (t;Xt )
 o
  (dt; dz)
 where  (k)i =  ik is the i
 th component of  (k) 2 Rn which is the kth column of the n  l matrix
  = [ ij ].
 The next corollaries apply the multidimensional Ito^ formula to some particularly useful functions.
 Corollary 2.12. Let
 dXi(t) =  i(t)dt+  i(t)dWt +
 Z
 R
  i(t; z) (dt; dz); i = 1; 2
 be two Ito^-L evy processes. Then the process (Zt)t 0 de ned by Zt = X1(t)X2(t) has P-dynamics:
 dZt =
  
 1(t)X2(t) +  2(t)X1(t) +  1(t) 2(t) +
 Z
 R
  1(t; z) 2(t; z) (dz)
  
dt
 + [ 1(t)X2(t) +  2(t)X1(t)] dWt
 +
 Z
 R
 [ 2(t; z)X1(t ) +  1(t; z)X2(t ) +  1(t; z) 2(t; z)]  (dt; dz):
2.2 Ito^ Formula for L evy Processes 29
 Proof. Applying Theorem 2.11 with f(t;X1; X2) = X1(t)X2(t)
 dZt = X2(t) [ 1(t)dt+  1(t)dWt] +X1(t) [ 2(t)dt+  2(t)dWt] +
 1
 2
 [ 1(t) 2(t) +  2(t) 1(t)]dt
 +
 Z
 R
 f[X1(t ) +  1(t; z)] [X2(t ) +  2(t; z)] X1(t )X2(t )
  X2(t ) 1(t; z) X1(t ) 2(t; z)g (dt; dz)
 +
 Z
 R
 f[X1(t ) +  1(t; z)] [X2(t ) +  2(t; z)] X1(t )X2(t )g  (dt; dz)
 =
  
 1(t)X2(t) +  2(t)X1(t) +  1(t) 2(t) +
 Z
 R
  1(t; z) 2(t; z) (dz)
  
dt
 + [ 1(t)X2(t) +  2(t)X1(t)] dWt
 +
 Z
 R
 [ 2(t; z)X1(t ) +  1(t; z)X2(t ) +  1(t; z) 2(t; z)]  (dt; dz):
 Corollary 2.13. Let
 dXi(t)
 Xi(t )
 =  i(t)dt+  i(t)dWt +
 Z
 R
  i(t; z) (dt; dz); i = 1; 2
 be two Ito^-L evy processes under the statistical probability measure P. Now consider the process
 Z = (Zt)t 0 de ned by Zt = X1(t)X2(t), then its P-dynamics are:
 dZt
 Zt 
= [ 1(t) +  2(t) +  1(t) 2(t)] dt+
 Z
 R
  1(t; z) 2(t; z) (dt; dz) + [ 1(t) +  2(t)] dWt
 +
 Z
 R
 [ 1(t; z) +  2(t; z) +  1(t; z) 2(t; z)] (   )(dt; dz):
 Proof. Applying Theorem 2.11 with f(t;X1; X2) = X1(t)X2(t)
 dZt = X2(t) [X1(t) 1(t)dt+X1(t) 1(t)dWt] +X1(t) [X2(t) 2(t)dt+X2(t) 2(t)dWt]
 +
 1
 2
 [ 1(t)X1(t) 2(t)X2(t) +  2(t)X2(t) 1(t)X1(t)]dt
 +
 Z
 R
 f[X1(t ) +X1(t) 1(t; z)] [X2(t ) +X2(t) 2(t; z)] X1(t )X2(t )
  X2(t )X1(t) 1(t; z) X1(t )X2(t) 2(t; z)g (dt; dz)
 +
 Z
 R
 f[X1(t ) +X1(t) 1(t; z)] [X2(t ) +X2(t) 2(t; z)] X1(t )X2(t )g  (dt; dz):
 Thus
 dZt
 Zt 
= [ 1(t) +  2(t) +  1(t) 2(t)] dt+ [ 1(t) +  2(t)] dWt +
 Z
 R
  1(t; z) 2(t; z) (dt; dz)
 +
 Z
 R
 [ 1(t; z) +  2(t; z) +  1(t; z) 2(t; z)]  (dt; dz):
2.2 Ito^ Formula for L evy Processes 30
 Corollary 2.14. Let
 dXi(t) =  i(t)dt+  i(t)dWt +
 Z
 R
  i(t; z) (dt; dz); i = 1; 2
 de ne two Ito^-L evy processes under the probability measure P with X2 strictly positive. Then the
 process (Zt)t 0 de ned by Zt =
 X1(t)
 X2(t)
 has P-dynamics:
 dZt =
  
 1(t)
 X2(t)
 +  Y (t)X1(t) 
 1(t) 2(t)
 X22 (t)
 +
 Z
 R
  1(t; z) Y (t; z) (dz)
  
dt
 +
  
 1(t)
 X2(t)
   2(t)
 X1(t)
 X22 (t)
  
dWt +
 Z
 R
  
 Y (t; z)X1(t ) +
  1(t; z)
 X2(t )
 +  1(t; z) Y (t; z)
  
 (dt; dz);
 where
  Y (t) =  
1
 X22 (t)
  
 2(t)  
2
 2(t)
 1
 X2(t)
  
+
 Z
 jzj<R
  
1
 X2(t ) +  2(t; z)
  
1
 X2(t )
 +
 1
 X22 (t )
  2(t; z)
  
 (dz);
  Y (t; z) =
 1
 X2(t ) +  2(t; z)
  
1
 X2(t )
 :
 Proof. By Corollary 2.8, considering Yt =
 1
 X2(t)
 dYt =
  
 
1
 X22 (t)
  
 2(t)  
2
 2(t)
 1
 X2(t)
  
+
 Z
 jzj<R
  
1
 X2(t ) +  2(t; z)
  
1
 X2(t )
 +
 1
 X22 (t )
  2(t; z)
  
 (dz)
 )
 dt
  
1
 X22 (t)
  2(t)dWt +
 Z
 R
  
1
 X2(t ) +  2(t; z)
  
1
 X2(t )
  
 (dt; dz):
 Applying Corollary 2.12 with Zt = X1(t)Yt
 dZt =
  
 1(t)
 X2(t)
 +  Y (t)X1(t) 
 1(t) 2(t)
 X22 (t)
 +
 Z
 R
  1(t; z) Y (t; z) (dz)
  
dt
 +
  
 1(t)
 X2(t)
   2(t)
 X1(t)
 X22 (t)
  
dWt
 +
 Z
 R
  
 Y (t; z)X1(t ) +
  1(t; z)
 X2(t )
 +  1(t; z) Y (t; z)
  
 (dt; dz);
 where
  Y (t) =  
1
 X22 (t)
  
 2(t)  
2
 2(t)
 1
 X2(t)
  
+
 Z
 jzj<R
  
1
 X2(t ) +  2(t; z)
  
1
 X2(t )
 +
 1
 X22 (t )
  2(t; z)
  
 (dz);
  Y (t; z) =
 1
 X2(t ) +  2(t; z)
  
1
 X2(t )
 :
2.2 Ito^ Formula for L evy Processes 31
 Corollary 2.15. Let
 dXi(t)
 Xi(t )
 =  i(t)dt+  i(t)dWt +
 Z
 R
  i(t; z) (dt; dz); i = 1; 2
 be two Ito^-L evy processes. Then the process (Zt)t 0 de ned by Zt =
 X1(t)
 X2(t)
 has P-dynamics:
 dZt
 Zt 
=
 "
  1(t)  2(t) +  
2
 2(t) +
 Z
 jzj<R
  22(t; z)
 1 +  2(t; z)
  (dz)  1(t) 2(t)
 #
 dt
 +
 Z
 R
  1(t; z)
  2(t; z)
 1 +  2(t; z)
  (dt; dz) + [ 1(t)  2(t)] dWt
 +
 Z
 R
  
 1(t; z) +
  2(t; z)
 1 +  2(t; z)
 +  1(t; z)
  2(t; z)
 1 +  2(t; z)
  
 (dt; dz):
 Proof. By Corollary 2.9, considering Yt =
 1
 X2(t)
 dYt
 Yt
 =
  
  2(t) +  
2
 2(t)
  
dt+
 Z
 jzj<R
  22(t; z)
 1 +  2(t; z)
  (dt; dz)  2(t)dWt  
Z
 R
  2(t; z)
 1 +  2(t; z)
  (dt; dz)
 =
 "
   2(t) +  
2
 2(t) +
 Z
 jzj<R
  22(t; z)
 1 +  2(t; z)
  (dz)
 #
 dt  2(t)dWt  
Z
 R
  2(t; z)
 1 +  2(t; z)
  (dt; dz):
 Applying Corollary 2.13 with Z(t) = X1(t)Y (t)
 dZt
 Zt 
=
 "
  1(t)  2(t) +  
2
 2(t) +
 Z
 jzj<R
  22(t; z)
 1 +  2(t; z)
  (dz)  1(t) 2(t)
 #
 dt
 +
 Z
 R
  1(t; z)
  2(t; z)
 1 +  2(t; z)
  (dt; dz) + [ 1(t)  2(t)] dWt
 +
 Z
 R
  
 1(t; z) +
  2(t; z)
 1 +  2(t; z)
 +  1(t; z)
  2(t; z)
 1 +  2(t; z)
  
 (dt; dz):
 Derivative securities are priced in the risk neutral probability measure, however the market prices
 are valued in the statistical probability measure. The following Girsanov Theorem is used to move
 from one measure to another.
 Theorem 2.16 (Girsanov change of measure). Let ( ;F ;F;P) be a  ltered probability space satis-
 fying the usual conditions. Let X1 and X2 be two real-valued adapted processes under the probability
 measure P, with dynamics
 dXi(t)
 Xi(t )
 =  i(t)dWt +
 Z
 R
  i(t; z) (dt; dz); for i = 1; 2;
 where Wt is a P-Brownian motion,  (dt; dz) is the compensator of the random measure of jumps
  and the standard integrability conditions are veri ed. If  i(t; z) is deterministic for i = 1; 2 and
2.2 Ito^ Formula for L evy Processes 32
  2(t; z) >  1, then the process Y =
 X1
 X2
 is a local martingale and has dynamics
 dYt
 Yt 
=  [ 2(t)  1(t)]dfWt  
Z
 R
  2(t; z)  1(t; z)
 1 +  2(t; z)
 (  e )(dt; dz)
 under an equivalent measure eP ? P, where fWt is a eP-Brownian motion given by
 dfWt = dWt   2(t)dt;
 and e (dt; dz) is the eP-compensator of  given by
 e (dt; dz) =  (t; z) (dt; dz)
 with  (t; z) = 1 +  2(t; z).
 The function  (t; z) is a non-negative solution to the equation
 Z
 R
  22(t; z)  1(t; z) 2(t; z)  1(t; z) +  2(t; z)
 1 +  2(t; z)
  (dt; dz) +
 Z
 R
  1(t; z)  2(t; z)
 1 +  2(t; z)
 f(t; z) (dt; dz) = 0:
 Proof. By applying Corollaries 2.9 and 2.13
 dYt
 Yt 
= [ 22(t)  1(t) 2(t)]dt+
 Z
 R
  22(t; z)
 1 +  2(t; z)
  (dt; dz) 
Z
 R
  1(t; z) 2(t; z)
 1 +  2(t; z)
  (dt; dz)
  [ 2(t)  1(t)]dWt  
Z
 R
  2(t; z)  1(t; z)
 1 +  2(t; z)
 (   )(dt; dz): (2.4)
 Since the continuous and discontinuous (jump) parts of a semimartingale do not interact [70], they
 can be studied independently. The continuous part of the process Y is
 [ 22(t)  1(t) 2(t)]dt [ 2(t)  1(t)]dWt =  [ 2(t)  1(t)][dWt   2(t)dt]:
 This implies that dfWt = dWt   2(t)dt [87, 15].
 The discontinuous part of Y is
 Z
 R
  22(t; z)
 1 +  2(t; z)
  (dt; dz) 
Z
 R
  1(t; z) 2(t; z)
 1 +  2(t; z)
  (dt; dz) 
Z
 R
  2(t; z)  1(t; z)
 1 +  2(t; z)
 (   )(dt; dz):
 Following ;ksendal and Sulem [88], the last expression can be written as:
 Z
 R
  1(t; z)  2(t; z)
 1 +  2(t; z)
 (  ~ ) (dt; dz) +
 Z
 R
  22(t; z)  1(t; z) 2(t; z)  1(t; z) +  2(t; z)
 1 +  2(t; z)
  (dt; dz)
 +
 Z
 R
  1(t; z)  2(t; z)
 1 +  2(t; z)
 f(t; z) (dt; dz)
 where e (dt; dz) =  (t; z) (t; z) for some   0 is the compensator of  under some new probability
 measure eP ? P. It is obvious from the last equation that for the process Y to be a martingale under
 the new measure, it is required that
 Z
 R
  22(t; z)  1(t; z) 2(t; z)  1(t; z) +  2(t; z)
 1 +  2(t; z)
  (dt; dz) +
 Z
 R
  1(t; z)  2(t; z)
 1 +  2(t; z)
 f(t; z) (dt; dz) = 0:
 The solution to this equation is  (t; z) = 1 +  2(t; z) [87].
2.3 Some Useful Results 33
 2.3 Some Useful Results
 In the 1930s, de Finetti and Kolmogorov obtained an explicit and simple expression which describes
 an in nitely divisible distribution in terms of its characteristic function. This formula is known as
 the L evy-Khinchin formula.
 Theorem 2.17 (L evy-Khinchin Formula). (i) Let (Xt)t>0 be a L evy process on Rd with charac-
 teristic triplet (c;  ;  ). Then
 Z
 Rd
 (jzj2 ^ 1) (dz) <1 and
 E[eiz Xt ] = et (z); z 2 Rd
 where
  (z) = i  z  
1
 2
 z  cz +
 Z
 Rd
  
eiz x  1 iz  x jxj 1
  
 (dx): (2.5)
 (ii) Conversely, if c is a symmetric positive matrix,  2 Rd and  is a positive measure on Rd nf0g
 that satis es
 Z
 Rd
 (jzj2 ^ 1) (dz) < 1, then there exists a L evy process on Rd (unique in law)
 whose characteristic function is given by Equation 2.5.
 Note that the characteristic function is the Fourier transform of the probability density function
 f(x).
 De nition 2.18. The Fourier transform of a function f 2 L1(R) is the function f^ = F(f) de ned
 by
 f^(z) =
 Z
 R
 eixzf(x)dx:
 Example 3. Let us consider two simple cases:
 1. X follows a Brownian motion; its dynamics are given by dXt =  dt +  dWt and its L evy
 characteristics are ( 2; 0;  ). Therefore
  X(z) = i z  
1
 2
  2z2
 E[eizXt ] = exp
  
t
  
i z  
1
 2
  2z2
   
:
 2. X is a Poisson process of intensity  ; its L evy characteristics are (0;  ; 0). Therefore
  X(z) =
 Z
 R
  
eiz x  1
  
 (dx)
 =  
 
eiz x  1
  
:
2.3 Some Useful Results 34
 Since  counts the average number of jumps which is  . Hence,
 E[eizXt ] = exp
  
 
 
eiz x  1
   
:
 Theorem 2.19 (Exponential moments [35]). Let (Xt)t>0 be a L evy process on R with characteristic
 triplet (c;  ;  ) and let u 2 R. The exponential moment E[euXt ] is  nite for some t or, equivalently,
 for all t > 0 if and only if
 Z
 jxj>1
 eux (dx) <1. In this case
 E[euXt ] = et X( iu):
 where  X is the characteristic exponent of X given by Equation 2.5.
 For a purely jump process, its exponential moment can be computed analytically. The next theorems
 give these moments for a Poisson integral in the cases when it is compensated or not.
 Theorem 2.20. Let A be bounded from below. Then,
 (i) for each t  0,
 Z
 A
 f(x) (t; dx) has compound Poisson distribution such that, for each u 2 Rd,
 E
  
exp
  
i
  
u;
 Z
 A
 f(x) (t; dx)
    
= exp
  
t
 Z
 A
 (ei(u;x)  1) f (dx)
  
(ii)
 E
  
exp
  
i
  
u;
 Z
 A
 f(x)(   )(t; dx)
    
= exp
  
t
 Z
 A
 h
 ei(u;x)  1 i(u; x)
 i
  f (dx)
  
where  f =   f 1 and for x; y 2 Rd such as x = (x1; x2;    ; xd) and y = (y1; y2;    ; yd),
 hx; yi =
 dX
 i=1
 xiyi.
 Proof. See Theorem 2:3:8(1) and Equation (2:9) in Applebaum?s book [7].
 Corollary 2.21. Working in R and taking u =  i, we have
 E
  
exp
  Z
 A
 f(x) (t; dx)
   
= exp
  
t
 Z
 A
 (ex  1) f (dx)
  
E
  
exp
  Z
 A
 f(x)(   )(t; dx)
   
= exp
  
t
 Z
 A
 [ex  1 x] f (dx)
  
where  f =   f 1.
 The previous expectations can be simpli ed further when  (t; z) is deterministic as shown in the
 next theorem.
2.3 Some Useful Results 35
 Theorem 2.22. If  (t; z) is deterministic, the exponential moment of a Poisson integral and a
 compensated Poisson integral are respectively
 E
  
exp
  Z t
 0
 Z
 R
  (t; z)(   )(dt; dz)
   
= exp
  Z t
 0
 Z
 R
 h
 e (t;z)  1  (t; z)
 i
  (dz)dt
  
;
 E
  
exp
  Z t
 0
 Z
 R
  (t; z) (dt; dz)
   
= exp
  Z t
 0
 Z
 R
 h
 e (t;z)  1
 i
  (dz)dt
  
:
 Proof. This proof is based on Exercice 1:6 in ;ksendal and Sulem?s book [88].
 Using Corollary 2.10, the equation
 dXt
 Xt 
=
 Z
 R
 (e (t;z)  1)(   )(dt; dz); X0 = 1 (2.6)
 has solution
 XT = exp
 (Z T
 0
 Z
 R
  (t; z) (dt; dz) 
Z T
 0
 Z
 R
 (e (t;z)  1) (dz)dt
 )
 = exp
 (Z T
 0
 Z
 R
  (t; z)(   )(dt; dz) 
Z T
 0
 Z
 R
 h
 e (t;z)  1  (t; z)
 i
  (dz)dt
 )
 :
 Assuming that
 Z T
 0
 Z
 R
 (e (t;z)  1)2 (dz)dt < 1;
 from Equation (2.6), X is a martingale, i.e. E[Xt] = 1, hence
 E
 "
 exp
 (Z T
 0
 Z
 R
  (t; z)(   )(dt; dz)
 )#
 = exp
 (Z T
 0
 Z
 R
 h
 e (t;z)  1  (t; z)
 i
  (dz)dt
 )
 :
 Note that
 E
 "
 exp
 (Z T
 0
 Z
 R
  (t; z)(   )(dt; dz)
 )#
 = E
 "
 exp
 (Z T
 0
 Z
 R
  (t; z) (dt; dz)
 )#
 E
 "
 exp
 (
  
Z T
 0
 Z
 R
  (t; z) (dz)dt
 )#
 = E
 "
 exp
 (Z T
 0
 Z
 R
  (t; z) (dt; dz)
 )#
 exp
 (
  
Z T
 0
 Z
 R
  (t; z) (dz)dt
 )
 ;
 thus
 E
 "
 exp
 (Z T
 0
 Z
 R
  (t; z) (dt; dz)
 )#
 = exp
 (Z T
 0
 Z
 R
 h
 e (t;z)  1
 i
  (dz)dt
 )
 The next two results will be used when pricing IL securities using the martingale method in Chapter
 3. The convexity correction allows to compute the expected value of the product of two martingales.
2.3 Some Useful Results 36
 Theorem 2.23. (Convexity correction) Let X1 and X2 be two P-martingales with dynamics given
 by:
 dXi(t)
 Xi(t )
 =  i(t)dW (t) +
 Z
 R
  i(t; z)(   )(dt; dz); for i = 1; 2;
 where  i(t) and  i(t) are deterministic and
 Z T
 0
 Z
 R
  2i (t; z) (dt; dz) < 1 for i = 1; 2. Then for all
 t 2 [0; T ]
 EPt [X1(T )X2(T )] = X1(t)X2(t) exp
 "Z T
 t
  1(u) 
 
2 (u)du+
 Z
 R
  1(t; z) 2(t; z) (dt; dz)
 #
 (2.7)
 where ( ) denotes the transpose.
 Remark The exponential factor in Equation (2.7) is sometimes referred to the as convexity correc-
 tion term.
 Proof. This proof is from [63] where it was proved in the case of  nite jumps. The proof when
 considering in nite jumps is similar.
 Considering Zt = X1(t)X2(t), by Corollary 2.13
 dZt
 Zt 
=  1(t) 2(t)dt+
 Z
 R
  1(t; z) 2(t; z) (dt; dz) + [ 1(t) +  2(t)] dWt
 +
 Z
 R
 [ 1(t; z) +  2(t; z) +  1(t; z) 2(t; z)] (   )(dt; dz):
 Since dW and (   ) are P-martingales, for every t  T ,
 EPt [ZT ] = Zt + E
 P
 t
 "Z T
 t
 ZuAudu
 #
 where
 Au =  1(u) +  2(u) +  1(u) 2(u) +
 Z
 R
  1(u; z) 2(u; z) (dz)
 is an integrable function of u.
 Since A is a non stochastic process which is integrable, the expectation can be moved within the
 integral sign. Assuming that Cs = EPt [Zs] with s  t yields
 Cs = Zt +
 Z s
 t
 AuCudu:
 Taking the derivative of this equation gives the the following ordinary di erential equation
 8
 <
 :
 _Cs = CsAs
 Ct = Yt
2.4 Examples of L evy Processes 37
 The solution to this di erential equation is given by
 CT = Yt exp
 "Z T
 t
 Audu
 #
 :
 The Bayes theorem relates the conditional and marginal probabilities of two random events.
 Theorem 2.24 (Bayes theorem). Let X = (Xt)t 0 be a stochastic process on ( ;F ;P), let Q be
 another probability measure on ( ;F), absolutely continuous with respect to P and with Radon-
 Nikodym derivative
  =
 dQ
 dP
 on F :
 Let G be a  -algebra with G  F , then
 EQ[X j G] =
 EP[  X j G]
 EP[ j G]
 ; Q a:s:
 2.4 Examples of L evy Processes
 In nitely divisible distributions have been used for modelling  nancial data as early as 1980 [95] to
 incorporate skewness and excess kurtosis. Examples of such distributions are the Variance Gamma
 (VG), the Normal Inverse Gausssian (NIG), the Generalized Hyperbolic (GH) model, the GH skew
 Student?s t distribution and the Hyperbolic model. The VG distribution was introduced by Madan
 and Seneta [77, 78] to model stock returns in the late 1980s. In 1995, Eberlein and Keller [41]
 used the Hyperbolic distribution and Barndor -Nielsen [11] proposed the NIG L evy process. All
 the previous models were brought together as special cases of the GH model, which was developed
 by Eberlein and co-workers in a series of papers [47, 42, 91]. This section presents these selected
 L evy processes and other useful results. A more detailed coverage of L evy processes can be found
 in [91, 95].
 Note that throughout this section, the GH model refers to the univariable Generalized Hyperbolic
 distribution.
 2.4.1 The Generalized Hyperbolic Distribution
 The name \Generalized Hyperbolic" is due to the fact that the GH distribution log-density is
 hyperbolic while the Gaussian distribution log-density is a parabola. In 1977, Barndor -Nielsen [10]
2.4 Examples of L evy Processes 38
 introduced the Generalized Hyperbolic distribution to model grain size distributions of wind blown
 sand. He was looking for a satisfactory explanation to some empirical laws in geology; later on, this
 distribution was used in  nancial Mathematics [41, 91].
 De nition 2.25 (Generalized Hyperbolic distribution). A univariate GH distribution is de ned by
 the following Lebesgue density
 gh(x; ;  ;  ;  ;  ) = a( ;  ;  ;  )[ 2 + (x  )2]
 1
 2 (  
1
 2 )K  12
  
 
p
  2 + (x  )2
  
e (x  )(2.8)
 a( ;  ;  ;  ) =
 ( 2   2)
  
2
 p
 2    
1
 2   K ( 
p
  2   2)
 (2.9)
 where x 2 R and K is a modi ed Bessel function of the third kind (See Figure 2.1)
 K (z) =
 1
 2
 Z 1
 0
 y  1 exp
  
 
z
 2
  
y +
 1
 y
   
dy for z > 0: (2.10)
 Figure 2.1 Modi ed Bessel function of the third kind.
2.4 Examples of L evy Processes 39
 The domain of variation of the parameters is  2 R and
   0; j j <  if  > 0;
  > 0; j j <  if  = 0;
  > 0; j j   if  < 0:
 The parameters  ,  ,  and  a ect respectively the location, the scale, the skewness and the
 kurtosis.
 Proposition 2.26 (Mean and Variance). The mean and variance of a generalized hyperbolic dis-
 tributed random variate X are given by [91]
 E[X] =  +
   
p
  2   2
 K +1( )
 K ( )
 ;
 V ar[X] =  2
 (
 K +1( )
  K ( )
 +
  2
  2   2
 "
 K +2( )
 K ( )
  
 
K +1( )
 K ( )
  2
 #)
 ;
 where  =  
p
  2   2.
 Proposition 2.27. The characteristic function of the generalized hyperbolic distribution is given by
 ?GH(u) = e
 i u
  
 2   2
  2  ( + iu)2
   
2 K ( 
p
  2  ( + iu)2)
 K ( 
p
  2   2)
 The GH can also be seen as a normal variance-mean mixture in the form
 gh(x; ;  ;  ;  ;  ) =
 Z 1
 0
 N (x; +  w;w)  gig(w; ;  2;  2   2)dw
 where N ( ) is the normal density function and gig( ) the density function of a generalized inverse
 Gaussian(GIG).
 De nition 2.28 (Generalized Inverse Gaussian distribution). A univariate GIG distribution is
 de ned by the following Lebesgue density
 gig(x; ;  ;  ) =
  
 
 
  
2
 2K 
 p
   
 x  1 exp
  
 
1
 2
   
x
 +  x
   
, for x > 0;
 where K is a modi ed Bessel function of the third kind and  2 R and  ;  2 R+.
 Remark The normal distribution is obtained from the GH distribution by considering the following
 limit case:  !1 and
  
 
!  2.
2.4 Examples of L evy Processes 40
 Although the GH distribution is highly  exible, it is seldom used in practical applications. This
 might be due to the fact that even for very large sample, it is hard to determine which subclass is
 the most appropriate [91]. Instead, speci c subclasses have been applied in various situations for
 parameter estimation. The following subsections review some of these subclasses.
 2.4.2 The Hyperbolic Distribution
 A univariate hyperbolic (HYP) distribution is obtained from a GH distribution for  = 1.
 De nition 2.29 (Hyperbolic distribution). A univariate HYP distribution is de ned by the follow-
 ing Lebesgue density
 hyp(x; ;  ;  ;  ) =
 p
  2   2
 2  K1( 
p
  2   2)
 exp
 h
   
p
  2 + (x  )2 +  (x  )
 i
 ;
 where x;  2 R; 0   and j j <  .
 The mean and variance of an HYP distribution can easily be computed from that of the GH distri-
 bution.
 2.4.3 The Normal Inverse Gaussian Distribution
 The name \Normal Inverse Gaussian" (NIG) stems from the fact that the NIG distribution can be
 represented as a mixture of a Generalized Inverse Gaussian with a Normal distribution. A univariate
 NIG distribution is obtained from a GH distribution for  =  
1
 2
 .
 De nition 2.30 (Normal Inverse Gaussian distribution). A univariate NIG distribution is de ned
 by the following Lebesgue density
 nig(x; ;  ;  ;  ) =
   
 
exp
 h
  
p
  2   2 +  (x  )
 i K1[ 
p
  2 + (x  )2]
 p
  2 + (x  )2
 ;
 where x;  2 R; 0   and 0  j j   .
 The NIG distribution is a lot easier to handle than the HYP distribution because it has a parameter
 additivity property similar to that of the the normal distribution [92]. If (Xi)1 i n are independent
 NIG random variables with common parameters  and  but having individual parameters  i and
  i, then
 nX
 i=1
 Xi is NIG distributed with parameters
  
 ;  ;
 nX
 i=1
  i;
 nX
 i=1
  i
 !
 . Furthermore, if X  
nig( ;  ;  ;  ) and Y = aX + b, then
 Y  nig
  
 
jaj
 ;
  
a
 ; jaj ; a + b
  
:
2.4 Examples of L evy Processes 41
 The characteristic function of the NIG distribution is given by
 ?NIG(u) = exp
 n
  [
 p
  2   2  
p
  2  ( + iu)2] + iu 
o
 :
 (a) Probability density function (b) Sample paths for  = 100,  = 1,  = 0 and
  = 0:01
 Figure 2.2 Probability density function and sample path for NIG.
 2.4.4 The Variance Gamma Distribution
 The Variance Gamma (VG) process can be expressed as the di erence between two independent
 Gamma processes [95]. The Gamma process X(Gamma) =
 n
 X(Gamma)t
 o
 t 0
 starts at zero and has
 independent and stationary increments. The increments are Gamma distributed, i.e. X(Gamma)t is
 Gamma(at; b) distributed. So if X = (Xt)t 0 and Y = (Yt)t 0 are two Gamma processes, a VG
 density function can be expressed in the following way,
 fV G(x) = fX+( Y )(x) =
 Z 1
  1
 fX(x+ s)fY (s)ds
 where fX and fY are Gamma density functions. The Gamma density function is given by
 fG(x; a; b) =
 ba
  (a)
 xa 1 exp( xb); x > 0:
 The previous method is commonly used when simulating VG paths (See Figures 2.4(a) and 2.4(b)).
 Alternatively, the representation of a VG process as a Brownian motion subordinated by a Gamma
 process can also be used. A subordinated L evy process is a time changing process for which the
 time changes according to another \increasing" L evy process. The latter process is referred to as
2.4 Examples of L evy Processes 42
 the subordinator. In this situation, a VG process has three parameters:  ,  and  which are
 respectively the volatility of the underlying Brownian motion, the drift of the Brownian motion and
 the variance of the subordinator.
 Figure 2.3 Probability density function of some VG processes.
 Senata [96] introduced another approach whereby the probability density function of a VG distri-
 bution with parameters ( ;  ;  ;  ) is
 vg(x;  ;  ;  ;  ) =
 2 exp
  
(x  )  
 2
  
 
p
 2  
1
   ( 1 )
 "
 (x  )2
  2 + 2 2= 
# 1
 2  
1
 4
  K 1
   
1
 2
  
jx  j
  2
 p
  2 + 2 2= 
 
;
 and its characteristic function is
 ?V G(x;  ;  ;  ;  ) = e
 i x
  
1 i  x+
  2 
2
 x2
   1= 
:
 2.4.5 The GH Skew Student?s t Distribution
 The GH skew Student?s t-distribution is ideal for  nancial modelling. It is not only almost as
 analytically tractable as the NIG distribution, but its parameter estimation using the maximum
2.5 Option Pricing Using the Fast Fourier Transform 43
 (a) Subordinated VG (b) Di erence of 2 Gamma
 Figure 2.4 Sample paths of a VG process with  = 0:2,  = 0:5 and  = 0:25.
 likelihood method is quite straightforward [39]. Moreover, the GH skew Student?s t-distribution is
 the only subclass of the GH distribution for which one tail has polynomial behaviour while the other
 has exponential behaviour. This generalisation of the usual Student?s t distribution is obtained from
 Equation (2.8) by letting  =   2 ;  > 0 and  ! j j > 0. Its probability density function is [1]
 fSt(x; ;  ;  ;  ) =
 2
 1  
2   j j
  +1
 2 exp [ (x  )]
  
 
 
2
  p
  
hp
  2 + (x  )2
 i  +1
 2
 K  +1
 2
 h
  2
 p
  2 + (x  )2
 i
 ;  6= 0;
 fSt(x; ;  ;  ;  ) =
  
 
 +1
 2
  
p
    
 
 
2
  
 
1 +
 (x  )2
  2
    +12
 ;  = 0:
 The mean and the variance of the skewed Student?s t distributed random variate X are
 E(X) =  +
   2
   2
 ;
 V ar(X) =
 2 2 4
 (  2)2(  4)
 +
  2
   2
 :
 The mean is  nite only when  > 2 and the same is true for the variance when1  > 4.
 2.5 Option Pricing Using the Fast Fourier Transform
 Under the assumption that prices follow a L evy distribution or an exponential L evy distribution,
 option pricing using the fast Fourier transform is performed in two steps. First, the Fourier transform
 of the contingent claim is computed, subsequently the Fourier inverse method gives the option price.
 1See [21] for a detailed derivation and coverage of the mean and variance for all possible values of  .
2.5 Option Pricing Using the Fast Fourier Transform 44
 This methodology was  rst proposed by Carr and Madan [26] to price equity derivatives driven
 by variance gamma processes, but it has general applicability. The current section presents the
 valuation of European call-like option since IL caplets,  oorlets and swaptions (priced later) can be
 viewed as particular European call options.
 Let rt denote the market interest rate and Rt = ln rt. This section values an European call with
 underlying rt and strike k. Throughout this section, the characteristic function of rt
  T (u) = E[exp(iurt)]
 is considered known analytically. Since the returns? distribution is easily deduced from the market?s
 observation (See Chapter 6), this is not too far fetched. The previous characteristic function is also
 de ned by
  T (u) =
 Z 1
  1
 eiuRqT (R)dR; 8 u 2 R
 where qT ( ) is the density of Rt under the risk neutral probability. The European call?s value is
 cT (k) = pn(0; T )EQ[(rt  k)+]
 = pn(0; T )
 Z 1
 k
  
eR  eK
  
qT (R)dR:
 However, the cT function is not square integrable in K, i.e. cT does not decay as K !  1 (or, i.e.
 k ! 0), thus its Fourier transform does not exist. Following Carr and Madan [26], the modi ed call
 price is
 CT (K) = exp( k)cT (k);
 with  > 0 chosen such that CT (K) is integrable in  1.
 The Fourier transform of CT (K) is
  T (v) =
 Z +1
  1
 eivKCT (K)dK: (2.11)
 The fact that
 CT (K)  
K! 1
 r0 exp( K);
 ensures the integrability of the square of CT (K) at  1. However, this might accentuate the problem
 at +1. For the moment, the assumption is made that  (0) is de ned and CT (K) is integrable at
 +1. The latter point will be taken care of in the paragraph containing Equation 2.12.
2.5 Option Pricing Using the Fast Fourier Transform 45
 The Fourier inversion formula gives
 CT (K) =
 1
 2 
Z +1
  1
 e ivK T (v)dv;
 cT (K) =
 1
 2 
exp(  K)
 Z +1
  1
 e ivK T (v)dv:
 The price cT (K) is real, therefore 8K 2 R,
 =
  Z +1
  1
 e ivK T (v)dv
  
= 0:
 Let a(v) and b(v) denote respectively the real and imaginary parts of  T (v). They are de ned by
 a : v  !
 Z +1
  1
 cos(vK)cT (K)dK
 b : v  !
 Z +1
  1
 sin(vK)cT (K)dK
 a is even and b is odd. Thus, 8v 2 R,
  ( v) = a(v) ib(v):
 Let A and B be the functions de ned for all K 2 R by:
 A(K) =
 Z 0
  1
 e ivK T (v)dv
 B(K) = 2 exp( K)cT (K) A(K)
 =
 Z +1
 0
 e ivK T (v)dv:
 With the change of variable v !  v,
 A(K) =
 Z 0
 +1
  eivK T ( v)dv
 =
 Z +1
 0
 fcos(vK)a(v) + sin(vK)b(v) + i [sin(vK)a(v) cos(vK)b(v)]g dv:
 Comparing the last equation with:
 B(K) =
 Z +1
 0
 e ivK T (v)dv
 =
 Z +1
 0
 fcos(vK)a(v) + sin(vK)b(v) i [sin(vK)a(v) cos(vK)b(v)]g dv;
 notice that
 <[A(K)] = <[B(K)];
 =[A(K)] =  =[B(K)]:
2.5 Option Pricing Using the Fast Fourier Transform 46
 Hence
 2 exp( K)cT (K) = 2<[B(K)];
 and
 cT (K) =
 exp(  K)
  
<
  Z +1
 0
 e ivK T (v)dv
  
:
 To get the call price as a function of the characteristic function  T , the  rst step is expressing  T
 as function of  T . From Equation (2.11),
  T (v) = pn(0; T )
 Z +1
  1
 Z +1
 K
 e KeivK
  
eR  eK
  
qT (R)dRdK:
 The integration domain is de ned by the upper half plane de ned by R = K. With the use of the
 Fubini theorem,
  T (v) = pn(0; T )
 Z +1
  1
  Z R
  1
 e K+ivK+R  e K+ivK+K
 !
 qT (R)dRdK
 = pn(0; T )
 Z +1
  1
 qT (R)
  
e K+ivK+R
  + iv
  
e K+ivK+K
  + iv + 1
  R
  1
 dR
 = pn(0; T )
 Z +1
  1
 qT (R)
  
e K+ivK+R
  + iv
  
e K+ivK+K
  + iv + 1
  
dR
 = pn(0; T )
 Z +1
  1
 qT (R)
  
e( +iv+1)R
 ( + iv)( + iv + 1)
  
dR
 =
 pn(0; T ) T [v  i(1 +  )]
  2 +   v2 + iv(2 + 1)
 :
 The integrability condition at +1 on  which was  T (0) <1 becomes  T [0 i(1 + )] <1, then:
 Z 1
  1
 qT (R)e
 (1+ )RdR < +1; (2.12)
 i.e. EQ
  
r +1T
  
< +1.
 Hence
 cT (K) =
 pn(0; T )e  K
  
<
  Z +1
 0
 e ivK T [v  i(1 +  )]
  2 +   v2 + iv(2 + 1)
  
:
 Notice that for  = 0, i.e. non modi ed price of the call, there is a valuation problem under the
 integral sign in zero. The choice of a value of  is important for the convergence speed. Carr and
 Madan [26] suggest close to 0:25 and Schoutens [95] 0:75 to price stock options, while Wu [110]
 proposes 1 for currency and interest rate options.
 Section 6.2 describes how to use Fast Fourrier Transform (FFT) to discretise and implement this
 pricing scheme.
Chapter 3
 Heath-Jarrow-Morton Model
 In order to improve the match between model generated and market observed in ation linked (IL)
 securities prices, this chapter assumes that the consumer price index?s log return, nominal and real
 forward rates follow L evy processes. This is an extension of the work of Hinnerich [63] where the
 probability measure had only  nite jump processes contrary to in nite jump processes that are used
 in this chapter. Pricing formulas for swaps, swaptions, caps and  oors are derived. Finally, an
 example of calibration to market data with numerical details is performed.
 Here is a summary of the content of this chapter. The  rst section is related to Bj ork, Di Masi,
 Kabanov and Runggaldier [16], where a general semimartingale approach is used for modelling of the
 in ation linked market in the L evy setting. Having introduced some basic assumptions, the models
 for nominal and IL bond prices are speci ed through the dynamics of in ation, domestic and real
 instantaneous forward rates. We derive expressions for the real spot and forward in ation rates and
 consider the problems of existence of nominal risk neutral martingale measures respectively. As a by
 product we obtain HJM-type conditions on the coe cients for the IL market. Finally, we investigate
 the question of absence of arbitrage in the international bond market.
 The second section is motivated by Eberlein and  Ozkan [46], where a L evy Libor model based on
 a time-inhomogeneous L evy processes has been introduced. After presentation of several technical
 results concerning the properties of the driving time-inhomogeneous L evy process, we translate a
 few models from the semimartingale setting in the  rst section to the current L evy setting. In
 particular, we specify the models for domestic and foreign instantaneous forward rates, bond prices
 and foreign spot and forward exchange rates. This allows us to proceed with the speci cation of
 47
3.1 The Extended HJM Model 48
 the dynamics for domestic and foreign forward processes, followed by the models for domestic and
 foreign forward Libor rates. Finally, we consider the relationship between domestic and foreign  xed
 income markets in the discrete-tenor framework.
 3.1 The Extended HJM Model
 This section extends the HJM model to Ito^-L evy processes. Using the martingale approach, dynamics
 are derived for nominal bonds, in ation linked bonds, real bonds and in ation under the risk neutral
 measure. Contrary to previous work [71, 81, 82, 14], no initial assumption is made that the foreign
 currency analogy holds. Instead, the foreign currency analogy is a result.
 Assumption 1. The probability space carries both an n-dimensional Wiener process W P and a
 Poisson random measure  (dt; dz) over R+  R with compensator  P(dt; dz) =  P(dz)dt. The
 probability space?s  ltration F = (Ft)t 0 is generated both by W P and  (i.e. Ft = FW
 P
 t _F
  
t ) which
 are independent. The L evy measure  P is on R and satis es:
 (i)  P(0) = 0;
 (ii)
 Z T
 0
 Z
 R
 (z2 ^ 1) P(dz)dt <1.
 For a \smooth" yield curve to be deductible from the market bonds? prices, the next initial assump-
 tion is needed where IP stands for in ation protected.
 Assumption 2. There exists a (nominal) market for T -bonds and T -IP-bonds for all maturities
 T > 0. Furthermore, for every  xed t, the nominal bond pn(t; T ) and the in ation linked bond
 pIP (t; T ) are di erentiable with respect to the maturity T .
 The corresponding real bond is de ned by
 pn(t; T ) =
 pIL(t; T )
 I(t)
 :
 Instantaneous forward rates, contracted at time t are de ned by
 fi(t; T ) =  
@ ln pi(t; T )
 @T
 for i = r; n:
 For i = n (resp. i = r), the forward rate is a nominal (resp. real) instantaneous forward rate. From
 these forward rates, the instantaneous interest rates are deduced by
 ri(t) = fi(t; t) for i = r; n:
3.1 The Extended HJM Model 49
 The money market accounts are given by
 Bi(t) = e
 R t
 0 r
 i(s)ds for i = r; n:
 For i = n, Bn(t) is the nominal money market account at time t measured in dollars; while Br(t) is
 the real money market account at time t measured in CPI basket.
 Similarly to Jarrow and Yildirim, the next assumption  rst gives speci cations for the dynamics of
 the consumer price index, the nominal and real forward rates in the statistical probability measure.
 Assumption 3. Under the objective probability measure P, the dynamics of fr and fn for every
  xed T > 0 and the dynamics of I are given by:
 dfi(t; T ) =  
i(t; T )dt+  i(t; T )dW Pt +
 Z
 R
  i(t; z; T ) (dt; dz) i = r; n (3.1)
 dI(t) = I(t )aI(t)dt+ I(t )bI(t)dW Pt + I(t )
 Z
 R
 cI(t; z) (dt; dz);
 with
  (dt; dz) =
 8
 <
 :
  (dt; dz)  (dz)dt; jzj < R
  (dt; dz); jzj  R
 where  i(t; T ),  i(t; T ),  i(t; z; T ), aI(t), bI(t) and cI(t; z) are adapted processes with
 Z T
 0
 Z T
 t
 j i(u; s)jdsdu <1;
 Z T
 0
 Z T
 t
 j i(u; s)j2dsdu <1;
 for all  nite t and T  t;  i(t; z; T ) :   R+  R R+ is a real valued function satisfying
 Z T
 0
 Z
 R
 Z T
 t
 j i(u; z; s)j2ds (du; dz) <1;
 for  nite t and T  t. These conditions guarantee integrability of the coe cients and are satis ed
 if the coe cients are bounded for t and T from a bounded set and  ([0; t]  R) < 1 for  nite t.
 Additionally,  (t; T ),  (t; T ) and  (t; x; T ) equal zero for T < t.
 The real world Brownian motion W P will sometime be noted W in short form.
 Assumption 4. The market is arbitrage free.
 In the current economy, the investor maintains his real value holdings in the form of real bonds
 and the real money market account. In nominal currency these are respectively represented by
 PIP (t; T ) = I(t)Pr(t; T ) and I(t)Br(t). Let BIP (t) denote the nominal value of the real money bank
 account, i.e. BIP (t) = I(t)Br(t). Assumption 4 is equivalent to the existence of a (not necessary
3.1 The Extended HJM Model 50
 unique) nominal risk neutral probability measure Qn. The probability measure Qn is such that
 Pn(t; T )
 Bn(t)
 ,
 PIP (t; T )
 Bn(t)
 and
 I(t)Br(t)
 Bn(t)
 are Qn-martingales [3, 71].
 Proposition 3.1. If fn(t; T ), fr(t; T ) and I(t) satisfy the Assumption 3 then I(t), pn(t; T ), pIP (t; T )
 and pr(t; T ) will under the nominal martingale measure Qn satisfy:
 dI(t)
 I(t )
 =
 "
 rn(t) rr(t) +
 Z
 jzj R
 cI(t; z) (dz)
 #
 dt+ bI(t)dWt +
 Z
 R
 cI(t; z)~ (dt; dz);(3.2)
 dpn(t; T )
 pn(t; T )
 = rn(t)dt+  
n(t; T )dWt +
 Z
 R
  n(t; z; T )~ (dt; dz); (3.3)
 dpIP (t; T )
 pIP (t ; T )
 = rn(t)dt+  
IP (t; T )dWt +
 Z
 R
  IP (t; z; T )~ (dt; dz); (3.4)
 dpr(t; T )
 pr(t ; T )
 = ar(t; T )dt+  r(t; T )dWt +
 Z
 R
  r(t; z; T )~ (dt; dz); (3.5)
 where
  i(t; T ) =  
Z T
 t
  i(t; u)du; for i = n; r;
  i(t; z; T ) = exp
  
Di(t; z; T )
  
 1; for i = n; r;
  IP (t; T ) =
  
Sr(t; T ) + bI(t)
  
;
 ar(t; T ) = rr(t) S
 r(t; T )bI(t) 
Z
 jzj R
 cI(t; z) (dz) +
 Z
 jzj<R
  
cI(t; z)
  2
 1 + cI(t; z)
  (dz)
  
Z
 R
  
exp [Dr(t; z; T )]cI(t; z) 
cI(t; z)
 1 + cI(t; z)
  
 (dz);
  IP (t; z; T ) = exp [Dr(t; z; T )][1 + cI(t; z)] 1;
 Si(t; T ) =  
Z T
 t
  i(t; u)du;
 Di(t; z; T ) =  
Z T
 t
  i(t; z; u)du:
 Proof. The process pi for i = r; n is de ned by
 pi(t; T ) = exp [X(t; T )] ; i:e: ln pi(t; T ) = X(t; T );
 with X(t; T ) =  
Z T
 t
 fi(t; u)du.
 Integrating Equation 3.1 with respect to t on [0; t] gives
 fi(t; u) = fi(0; u) +
 Z t
 0
  i(s; u)ds+
 Z t
 0
  i(s; u)dWs +
 Z t
 0
 Z
 R
  i(s; z; u) (ds; dz):
 In particular, for the instantaneous interest rate ri(t) = fi(t; t)
 ri(t) = fi(0; t) +
 Z t
 0
  i(s; t)ds+
 Z t
 0
  i(s; t)dWs +
 Z t
 0
 Z
 R
  i(s; z; t) (ds; dz):
3.1 The Extended HJM Model 51
 Splitting the integrals, then using Fubini theorem, we get
 X(t; T ) =  
Z T
 t
 fi(0; u)du 
Z t
 0
 Z T
 t
  i(s; u)duds 
Z t
 0
 Z T
 t
  i(s; u)dudWs
  
Z t
 0
 Z T
 t
 Z
 R
  i(s; z; u)du (ds; dz)
 =  
Z T
 0
 fi(0; u)du 
Z t
 0
 Z T
 s
  i(s; u)duds 
Z t
 0
 Z T
 s
  i(s; u)dudWs
  
Z t
 0
 Z T
 s
 Z
 R
  i(s; z; u)du (ds; dz) +
 Z t
 0
 fi(0; u)du+
 Z t
 0
 Z t
 s
  i(s; u)duds
 +
 Z t
 0
 Z t
 s
  i(s; u)dudWs +
 Z t
 0
 Z t
 s
 Z
 R
  i(s; z; u)du (ds; dz)
 = X(0; T ) 
Z t
 0
 Z T
 s
  i(s; u)duds 
Z t
 0
 Z T
 s
  i(s; u)dudWs
  
Z t
 0
 Z T
 s
 Z
 R
  i(s; z; u)du (ds; dz) +
 Z t
 0
 fi(0; u)du+
 Z t
 0
 Z u
 0
  i(s; u)dsdu
 +
 Z t
 0
 Z u
 0
  i(s; u)dWsdu+
 Z t
 0
 Z u
 0
 Z
 R
  i(s; z; u) (ds; dz)du
 Noticing that the four last terms are equal to
 Z t
 0
 ri(s)ds, we end up with
 X(t; T ) = X(0; T ) +
 Z t
 0
 ri(s)ds 
Z t
 0
 Z T
 s
  i(s; u)duds 
Z t
 0
 Z T
 s
  i(s; u)dudWs
  
Z t
 0
 Z T
 s
 Z
 R
  i(s; z; u)du (ds; dz)
 Let us de neAi(t; T ) =  
Z T
 t
  i(t; u)du, Si(t; T ) =  
Z T
 t
  i(t; u)du andDi(t; z; T ) =  
Z T
 t
  i(t; z; u)du,
 if di erentiating the previous equation, we have
 dX(t; T ) =
  
ri(t) +A
 i(t; T )
  
dt+ Si(t; T )dWt +
 Z
 R
 Di(t; z; T ) (dt; dz):
 Using the one-dimensional Ito^ formula with pi(t; T ) = f [X(t; T )] = exp [X(t; T )],  (t; T ) = ri(t) +
 Ai(t; T ),  (t; T ) = Si(t; T ) and  (t; z; T ) = Di(t; z; T )
 dpi(t; T ) = pi(t; T ) [ (t; T )dt+  (t; T )dWt] +
 1
 2
  2(t; T )pi(t; T )dt
 +
 Z
 jzj<R
 fpi(t ; T ) exp [ (t; z; T )] pi(t ; T ) pi(t ; T ) (t; z; T )g (dt; dz)
 +
 Z
 R
 fpi(t ; T ) exp [ (t; z; T )] pi(t ; T )g  (dt; dz):
3.1 The Extended HJM Model 52
 Thus
 dpi(t; T )
 pi(t ; T )
 =
  
 (t; T ) +
 1
 2
  2(t; T )
  
dt+  (t; T )dWt +
 Z
 jzj<R
 fexp [ (t; z; T )]
   (t; z; T ) 1g (dt; dz) +
 Z
 R
 fexp [ (t; z; T )] 1g  (dt; dz)
 =
  
ri(t) +A
 i(t; T ) +
 1
 2
  
 Si(t; T )
  
 2
  
dt+ Si(t; T )dWt +
 Z
 jzj<R
  
exp
  
Di(t; z; T )
  
 Di(t; z; T ) 1
  
 (dt; dz) +
 Z
 R
  
exp
  
Di(t; z; T )
  
 1
  
 (dt; dz)
 =
  
ri(t) +A
 i(t; T ) +
 1
 2
  
 Si(t; T )
  
 2
  
dt+ Si(t; T )dWt +
 Z
 jzj<R
  
exp
  
Di(t; z; T )
  
 Di(t; z; T ) 1
  
 (dz)dt+
 Z
 R
  
exp
  
Di(t; z; T )
  
 1
  
 (dt; dz) (3.6)
 with  (dt; dz) =  (dz)dt.
 Combining the P-dynamics of pr(t; T ) and I(t), the Corollary 2.13 applied to pIP (t; T ) = pr(t; T )I(t)
 with
  1(t; T ) =
  
rr(t) +A
 r(t; T ) +
 1
 2
 kSr(t; T )k2
  
+
 Z
 jzj<R
 fexp [Dr(t; z; T )] Dr(t; z; T ) 1g  (dz);
  1(t; T ) = S
 r(t; T );
  1(t; z; T ) = fexp [D
 r(t; z; T )] 1g ;
  2(t) = a
 I(t);  2(t) = b
 I(t);  2(t; z) = c
 I(t; z);
 gives
 dpIP (t; T )
 pIP (t; T )
 = [ 1(t; T ) +  2(t) +  1(t; T ) 2(t)] dt+
 Z
 R
  1(t; z; T ) 2(t; z) (dt; dz)
 + [ 1(t; T ) +  2(t)] dWt +
 Z
 R
 [ 1(t; z; T ) +  2(t; z) +  1(t; z; T ) 2(t; z)] (   )(dt; dz)
 =
 ( 
rr(t) +A
 r(t; T ) +
 1
 2
 kSr(t; T )k2
  
+
 Z
 jzj<R
 fexp [Dr(t; z; T )] Dr(t; z; T ) 1g  (dz)
 +aI(t) + Sr(t; T )bI(t)
  
dt+
 Z
 R
 fexp [Dr(t; z; T )] 1g cI(t; z) (dt; dz)
 +
  
Sr(t; T ) + bI(t; T )
  
dWt + +
 Z
 R
  
[1 + cI(t; z)] exp [Dr(t; z; T )] 1
  
(   )(dt; dz):
 Next, we would like to change measure from P to the equivalent (nominal) martingale measure Qn.
 By the Girsanov change of measure, we know there is a P-adapted process ht and a P-predictable
 process  (t; z)   1 8z 2 R such that dLt = htLtdW P +
 Z
 R
  (t; z)~ P(dt; dz) where LT =
 dQn
 dP
 on
 FT so that dW Pt = htdt+dWt and  t(dz) =  
P
 t (1 + (t; z)). Here W denotes a Q
 n-Brownian motion
3.1 The Extended HJM Model 53
 and  t(dt; dz) it the compensator of  under the Qn-measure. Furthermore ~ (dt; dz) denotes the
 compensated Poisson random measure under Qn, i.e.
 ~ (dt; dz) =  (dt; dz)  (dt; dz)
 =  (dt; dz)  Pt (dz) (t; z):
 Hence the dynamics of I(t), pn(t; T ), pr(t; T ) and pIP (t; T ) under Qn are given by:
 dI(t)
 I(t )
 =
  
aI(t) + bI(t)ht +
 Z
 R
 cI(t; z) (t; z) P(dz)
  
dt+ bI(t)dWt +
 Z
 R
 cI(t; z)~ (dt; dz);(3.7)
 dpi(t; T )
 pi(t; T )
 = ai(t; T )dt+  i(t; T )dWt +
 Z
 R
  i(t; z; T )~ (dt; dz); i = n; r; (3.8)
 dpIP (t; T )
 pIP (t; T )
 = aIP (t; T )dt+  IP (t; T )dWt +
 Z
 R
  IP (t; z; T )~ (dt; dz); (3.9)
 where
 ai(t; T ) = ri(t) +A
 i(t; T ) +
 1
 2
  
 Si(t; T )
  
 2 + htS
 i(t; T ) +
 Z
 jzj<R
  
exp
  
Di(t; z; T )
  
 Di(t; z; T ) 1
  
 (dz) +
 Z
 R
  
exp
  
Di(t; z; T )
  
 1
  
 (t; z) P(dz); (3.10)
  i(t; T ) = Si(t; T );
  i(t; z; T ) = exp
  
Di(t; z; T )
  
 1;
 aIP (t; T ) = rr(t) +A
 r(t; T ) +
 1
 2
 kSr(t; T )k2 +
 Z
 jzj<R
 fexp [Dr(t; z; T )] Dr(t; z; T ) 1g  (dz)
 +aI(t) + Sr(t; T )bI(t) +
 Z
 R
 fexp [Dr(t; z; T )] 1g cI(t; z) (dz)
 +
 Z
 R
  
[1 + cI(t; z)] exp [Dr(t; z; T )] 1
  
 (t; z) P(dz) + ht
  
Sr(t; T ) + bI(t)
  
;(3.11)
  IP (t; T ) =
  
Sr(t; T ) + bI(t)
  
;
  IP (t; z; T ) = exp [Dr(t; z; T )][1 + cI(t; z)] 1:
 From Assumption 4,
 Pn(t; T )
 Bn(t)
 and
 PIP (t; T )
 Bn(t)
 are Qn-martingales, hence the drift of pn(t; T ) and
 pIP (t; T ) must equal the nominal short rate, that is an(t; T ) = aIP (t) = rn(t). This, together with
 Equations (3.8) and (3.9), implies Equations (3.3) and (3.4), respectively.
3.1 The Extended HJM Model 54
 If we insert the condition an(t; T ) = aIP (t; T ) = rn(t) into the drift Equations (3.10) and (3.11)
 rn(t) = rn(t) +A
 n(t; T ) +
 1
 2
 kSn(t; T )k2 + htS
 n(t; T ) +
 Z
 jzj<R
 fexp [Dn(t; z; T )]
  Dn(t; z; T ) 1g  (dz) +
 Z
 R
 fexp [Dn(t; z; T )] 1g  (t; z) P(dz)
 = rr(t) +A
 r(t; T ) +
 1
 2
 kSr(t; T )k2 +
 Z
 jzj<R
 fexp [Dr(t; z; T )]
  Dr(t; z; T ) 1g  (dz) + aI(t) + Sr(t; T )bI(t) +
 Z
 R
 fexp [Dr(t; z; T )] 1g cI(t; z) (dz)
 +ht
  
Sr(t; T ) + bI(t)
  
+
 Z
 R
  
[1 + cI(t; z)] exp [Dr(t; z; T )] 1
  
 (t; z) P(dz):
 Since these equations must hold for all T , we get Equations (3.12), (3.13), (3.14) and (3.16).
 An(t; T ) =  
1
 2
 kSn(t; T )k2  htS
 n(t; T ) 
Z
 jzj<R
 fexp [Dn(t; z; T )] Dn(t; z; T )g  (dz)
  
Z
 R
 exp [Dn(t; z; T )] (t; z) P(dz); (3.12)
 Z
 jzj<R
  (dz) =  
Z
 R
  (t; z) P(dz) (3.13)
 rn(t) = rr(t) 
Z
 jzj<R
  (dz) + aI(t) 
Z
 R
 cI(t; z) (dz) + htb
 I(t) 
Z
 R
  (t; z) P(dz);
 = rr(t) + a
 I(t) 
Z
 R
 cI(t; z) (dz) + htb
 I(t) (3.14)
 from Equation (3.13);
 aI(t) = rn(t) rr(t) +
 Z
 R
 cI(t; z) (dz) htb
 I(t); (3.15)
 Ar(t; T ) =  
1
 2
 kSr(t; T )k2  
Z
 jzj<R
 fexp [Dr(t; z; T )] Dr(t; z; T )g  (dz) Sr(t; T )bI(t) htS
 r(t; T )
  
Z
 R
 exp [Dr(t; z; T )]cI(t; z) (dz) 
Z
 R
 [1 + cI(t; z)] exp [Dr(t; z; T )] (t; z) P(dz): (3.16)
 Inserting Equation (3.15) into Equation (3.7)
 dI(t)
 I(t )
 =
  
rn(t) rr(t) +
 Z
 R
 cI(t; z) (dz) +
 Z
 R
 cI(t; z) (t; z) P(dz)
  
dt+ bI(t)dWt +
 Z
 R
 cI(t; z)~ (dt; dz)
 =
 "
 rn(t) rr(t) +
 Z
 jzj R
 cI(t; z) (dz)
 #
 dt+ bI(t)dWt +
 Z
 R
 cI(t; z)~ (dt; dz) (3.17)
 because of Equation (3.13). Furthermore,
 Z
 jzj R
 cI(t; z) (dz) is  nite by de nition. Hence Equation
 (3.2) is proved.
3.1 The Extended HJM Model 55
 Let us now  nd the dynamics of pr under Qn; by de nition pr(t; T ) =
 pIP (t; T )
 I(t)
 . Considering
 Yt =
 1
 I(t)
 , Equation (3.17) and Corollary 2.9 give
 dYt
 Yt
 =
 (
  rn(t) + rr(t) 
Z
 jzj R
 cI(t; z) (dz) +
  
bI(t)
  2
 )
 dt+
 Z
 jzj<R
  
cI(t; z)
  2
 1 + cI(t; z)
  (dt; dz)
  bI(t)dWt  
Z
 R
 cI(t; z)
 1 + cI(t; z)
 ~ (dt; dz)
 =
 (
  rn(t) + rr(t) 
Z
 jzj R
 cI(t; z) (dz) +
  
bI(t)
  2
 +
 Z
 jzj<R
  
cI(t; z)
  2
 1 + cI(t; z)
  (dz)
 )
 dt
  bI(t)dWt  
Z
 R
 cI(t; z)
 1 + cI(t; z)
 ~ (dt; dz)
 Applying Corollary 2.13 to pr(t; T ) = pIP (t; T )Yt
 dpr(t; T )
 pr(t ; T )
 =
 "
 rn(t) rn(t) + rr(t) 
Z
 jzj R
 cI(t; z) (dz) +
  
bI(t)
  2
 +
 Z
 jzj<R
  
cI(t; z)
  2
 1 + cI(t; z)
  (dz)
   IP (t; T )bI(t)
  
dt 
Z
 R
  IP (t; z; T )
 cI(t; z)
 1 + cI(t; z)
  (dt; dz) +
  
 IP (t; T )t bI(t)
  
dWt
 +
 Z
 R
  
 IP (t; z; T ) 
cI(t; z)
 1 + cI(t; z)
   IP (t; z; T )
 cI(t; z)
 1 + cI(t; z)
  
~ (dt; dz)
 =
 "
 rr(t) +
  
bI(t)
  2
   IP (t; T )bI(t) 
Z
 jzj R
 cI(t; z) (dz) +
 Z
 jzj<R
  
cI(t; z)
  2
 1 + cI(t; z)
  (dz)
  
Z
 R
  IP (t; z; T )cI(t; z)
 1 + cI(t; z)
  (dz)
  
dt+
  
 IP (t; T ) bI(t)
  
dWt
 +
 Z
 R
  
 IP (t; z; T ) 
cI(t; z)
 1 + cI(t; z)
  
 IP (t; z; T )cI(t; z)
 1 + cI(t; z)
  
~ (dt; dz)
 dpr(t; T )
 pr(t ; T )
 =
 "
 rr(t) +
  
bI(t)
  2
  
 
Sr(t; T ) + bI(t)
  
bI(t) 
Z
 jzj R
 cI(t; z) (dz) +
 Z
 jzj<R
  
cI(t; z)
  2
 1 + cI(t; z)
  (dz)
  
Z
 R
  
exp [Dr(t; z; T )][1 + cI(t; z)] 1
  cI(t; z)
 1 + cI(t; z)
  (dz)
  
dt+
  
Sr(t; T ) + bI(t) bI(t)
  
dWt
 +
 Z
 R
  
exp [Dr(t; z; T )][1 + cI(t; z)] 1 
cI(t; z)
 1 + cI(t; z)
  
 
exp [Dr(t; z; T )][1 + cI(t; z)] 1
  cI(t; z)
 1 + cI(t; z)
  
~ (dt; dz)
 dpr(t; T )
 pr(t ; T )
 =
 "
 rr(t) S
 r(t; T )bI(t) 
Z
 jzj R
 cI(t; z) (dz) +
 Z
 jzj<R
  
cI(t; z)
  2
 1 + cI(t; z)
  (dz)
  
Z
 R
  
exp [Dr(t; z; T )]cI(t; z) 
cI(t; z)
 1 + cI(t; z)
  
 (dz)
  
dt
 +Sr(t; T )dWt +
 Z
 R
 fexp [Dr(t; z; T )] 1g ~ (dt; dz)
 which proves Equation (3.5).
3.1 The Extended HJM Model 56
 By construction in the previous proof,
 Pn(t; T )
 Bn(t)
 and
 PIP (t; T )
 Bn(t)
 are Qn-martingales. For Qn to be
 the sought nominal risk neutral probability measure,
 I(t)Br(t)
 Bn(t)
 also has to be a Qn-martingale. The
 next Lemma states a necessary condition for such a probability measure to exist.
 Lemma 3.2. For the probability measure Qn to be a nominal risk neutral probability measure, the
 integral
 Z
 jzj R
 cI(t; z) (dz) has to be zero.
 Proof. The de nition of Br(t) and the Qn-dynamics of I(t) given in Equation (3.2) yield
 dBIP (t)
 BIP (t )
 =
 "
 rn(t) +
 Z
 jzj R
 cI(t; z) (dz)
 #
 dt+ bI(t)dWt +
 Z
 R
 cI(t; z)~ (dt; dz):
 However BIP if a Qn-martingale is and only if the drift term in the previous equation is rn(t).
 This necessary condition for the existence of a nominal risk neutral probability measure is not part
 of the generalisation of the previously obtained conditions in more speci c settings (See Hinnerich
 [63], Corollary 2:1 and Jarrow and Yildrim [71], Proposition 2). This additional condition is a
 restriction only on the in ation?s jumps. The fact that this condition is on a jump component was
 predictable from the work of Jarrow and Yildrim [71], under some drift and volatility restrictions, a
 unique equivalent risk neutral measure always exists under the normality assumption. However, after
 introducing  nite jumps in the probability measure, Hinnerich did not get this additional condition.
 A su cient condition for the previous condition to be met is the next assumption which is commonly
 used with L evy processes in Finance in di erent versions [43, 100, 45]. Before stating the assumption,
 the general work setting needs to be rede ned.
 The nominal probability measure is endowed with a canonical  ltration F = (Ft)t 0, where the
 driving process L = (Lt)t 0 is a time inhomogeneous L evy process, i.e. a process with indepen-
 dent increments and absolutely continuous characteristics (PIIAC). The law of the process Lt with
 characteristic triplet (c;  ;  ) is given by the L evy-Khinchin Formula (See Equation (2.5)) with the
 standard integrability conditions, where
 b =
 Z t
 0
 bsds; c =
 Z t
 0
 csds;  (z) =
 Z t
 0
  sds;
 with the integrals being componentwise. Further assumption is made that
 Z T
 0
  
j sj+ kcsk+
 Z
 Rd
 (jz2j ^ 1) (dz)
  
ds <1 , for T > 0;
 where j  j is the norm corresponding to the Euclidian scalar product on Rd and k k denotes any norm
 on the d d matrices. The following additional moment assumption is made
3.1 The Extended HJM Model 57
 Assumption 5. There are constant M , " > 0, such that for every u 2 [ (1 + ")M; (1 + ")M ]d
 Z T
 0
 Z
 jzj>R
 exp hu; zi  (dz)ds <1 , for T > 0;
 where h ;  i is the Euclidian scalar product on Rd and R is a positive constant generally taken equal
 to one.
 The previous assumption is equivalent to E[exp hu; Lti] < 1 for t 2 [0; T ] and u 2 Rd. This is a
 natural assumption especially when using the Fast Fourier transform or Laplace transform for option
 pricing. Recall that for these methods, the characteristic function is supposedly initially known and
 needs to be  nite for obvious reasons. Furthermore, in the HJM framework, the underlying processes
 are always exponentials of stochastic integrals with respect to the driving processes L. In order to
 allow the pricing of derivatives these underlying processes have to be martingales under the nominal
 risk neutral measure and, therefore, a priori have to have  nite expectations, which is exactly the
 previous assumption.
 In particular, under the previous assumption, the variable Lt itself has  nite expectation and con-
 sequently a truncation is not needed. In fact, now L is not only a semimartingale, but a special
 semimartingale and Assumption 3 can be relaxed. Under the objective probability measure P, the
 dynamics of fi (for every  xed T > 0 and i = n; r) and the dynamics of I are given by:
 dfi(t; T ) =  
i(t; T )dt+  i(t; T )dW Pt +
 Z
 R
  i(t; z; T )( (dt; dz)  (dz)dt) i = r; n
 dI(t) = I(t )aI(t)dt+ I(t)bI(t )dW Pt + I(t )
 Z
 R
 cI(t; z)[ (dt; dz)  (dz)dt];
 with the standard integrability conditions. The canonical representation of the process L is
 Lt =
 Z t
 0
  sds+
 Z t
 0
 p
 csdWs +
 Z t
 0
 Z
 Rd
 z(   )(ds; dz);
 where
 p
 cs is a measurable version of the square root of cs.
 Henceforth, Assumption 5 is supposed veri ed.
 For a nominal risk-free probability measure to exist, additional restrictions on the drift terms in
 Assumption 3 and some non-degeneracy conditions upon the volatilities are needed.
 Corollary 3.3. The drift conditions that have to be satis ed in order for the market to be free of
 arbitrage are:
  n(t; T ) =  n(t; T )
 "Z T
 t
  (t; u)du ht
 #
  
Z
 jzj<R
  n(t; z; T ) n(t; z; T ) (dz)
  
Z
 R
 [ n(t; z; T ) + 1]  n(t; z; T ) (t; z) P(dz);
3.1 The Extended HJM Model 58
  r(t; T ) =  r(t; T )
 "Z T
 t
  r(t; u)du ht  b
 I(t)
 #
  
Z
 jzj<R
  r(t; z; T ) r(t; z; T ) (dz)
  
Z
 R
 [ r(t; z; T ) + 1]  r(t; z; )cI(t; z) (dz)
  
Z
 R
 [1 + cI(t; z)] [ r(t; z; T ) + 1]  r(t; z; ) (t; z) P(dz);
 Z
 jzj<R
  (dz) =  
Z
 R
  (t; z) P(dz);
 aI(t) = rn(t) rr(t) +
 Z
 R
 cI(t; z) (dz) htb
 I(t):
 Hinnerich?s Corollary 2:1 in [63] and Jarrow and Yildrim?s Proposition 2 in [71] are just particular
 cases of this corollary.
 Proof. We use Equations (3.12), (3.14), (3.13) and (3.16) and take the T -derivative of the  rst
 two equations.
 An(t; T ) =  
1
 2
 kSn(t; T )k2  htS
 n(t; T ) 
Z
 jzj<R
 fexp [Dn(t; z; T )]
  Dn(t; z; T )g  (dz) 
Z
 R
 exp [Dn(t; z; T )] (t; z) P(dz);
   n(t; T ) =   n(t; T )
 Z T
 t
  (t; u)du+ ht 
n(t; T ) 
Z
 jzj<R
 f  n(t; z; T ) exp [Dn(t; z; T )]
 + n(t; z; T )g  (dz) +
 Z
 R
 exp [Dn(t; z; T )] n(t; z; T ) (t; z) P(dz);
  n(t; T ) =  n(t; T )
 "Z T
 t
  (t; u)du ht
 #
 +
 Z
 jzj<R
 f1 exp [Dn(t; z; T )]g  n(t; z; T ) (dz)
  
Z
 R
 exp [Dn(t; z; T )] n(t; z; T ) (t; z) P(dz)
 =  n(t; T )
 "Z T
 t
  (t; u)du ht
 #
  
Z
 jzj<R
  n(t; z; T ) n(t; z; T ) (dz)
  
Z
 R
 [ n(t; z; T ) + 1]  n(t; z; T ) (t; z) P(dz);
 Ar(t; T ) =  
1
 2
 kSr(t; T )k2  Sr(t; T )bI(t) htS
 r(t; T ) 
Z
 jzj<R
 fexp [Dr(t; z; T )]
  Dr(t; z; T )g  (dz) 
Z
 R
 exp [Dr(t; z; T )]cI(t; z) (dz)
  
Z
 R
 [1 + cI(t; z)] exp [Dr(t; z; T )] (t; z) P(dz);
   r(t; T ) =   r(t; T )
 "Z T
 t
  r(t; u)du ht  b
 I(t)
 #
  
Z
 jzj<R
 f  r(t; z; T ) exp [Dr(t; z; T )]
 + r(t; z; T )g  (dz) +
 Z
 R
 exp [Dr(t; z; T )] r(t; z; )cI(t; z) (dz)
 +
 Z
 R
 [1 + cI(t; z)] exp [Dr(t; z; T )] r(t; z; ) (t; z) P(dz);
3.1 The Extended HJM Model 59
  r(t; T ) =  r(t; T )
 "Z T
 t
  r(t; u)du ht  b
 I(t)
 #
 +
 Z
 jzj<R
 f1 exp [Dr(t; z; T )]g  r(t; z; T ) (dz)
  
Z
 R
 exp [Dr(t; z; T )] r(t; z; )cI(t; z) (dz)
  
Z
 R
 [1 + cI(t; z)] exp [Dr(t; z; T )] r(t; z; ) (t; z) P(dz);
 =  r(t; T )
 "Z T
 t
  r(t; u)du ht  b
 I(t)
 #
  
Z
 jzj<R
  r(t; z; T ) r(t; z; T ) (dz)
  
Z
 R
 [ r(t; z; T ) + 1]  r(t; z; )cI(t; z) (dz)
  
Z
 R
 [1 + cI(t; z)] [ r(t; z; T ) + 1]  r(t; z; ) (t; z) P(dz);
 Z
 jzj<R
  (dz) =  
Z
 R
  (t; z) P(dz);
 aI(t) = rn(t) rr(t) +
 Z
 R
 cI(t; z) (dz) htb
 I(t):
 Under Assumption 5, the forward rates? dynamics can be rewritten in the form
 dfi(t; T ) =  
i(t; T )dt  i(t; T )dLt i = r; n (0  t  T ); (3.18)
 where  i and  i satisfy the usual integrability conditions (See [47, 44]).
 In this setting, the stochastic di erential equation (SDE) (3.6) can be rewritten [89] as
 dpi(t; T )
 pi(t ; T )
 = [ri(t) A
 i(t; T )]dt+  i(t; T )dLt;
 where Ai(t; T ) =
 Z T
 t^T
  i(t; u)du and  i(t; T ) =
 Z T
 t^T
  i(t; u)du.
 Therefore
 pi(t; T ) = pi(0; T ) exp
  Z t
 0
 [ri(s) A
 i(s; T )]ds+
 Z t
 0
  i(s; T )dLs
  
(3.19)
 = pi(0; T )Bi(t) exp
  
 
Z t
 0
 Ai(s; T )ds+
 Z t
 0
  i(s; T )dLs
  
By setting T = t, the risk free savings account can be written as
 Bi(t) =
 1
 pi(0; t)
 exp
  Z t
 0
 Ai(s; t)ds 
Z t
 0
  i(s; t)dLs
  
: (3.20)
 Using the two previous equations, the bond prices can be rewritten as
 pi(t; T ) =
 pi(0; T )
 pi(0; t)
 exp
  Z t
 0
  
Ai(s; t) Ai(s; T )
  
ds+
 Z t
 0
  
 i(s; T )  i(s; t)
  
dLs
  
=
 pi(0; T )
 pi(0; t)
 exp
  
 
Z t
 0
 Ai(s; t; T )ds+
 Z t
 0
  i(s; t; T )dLs
  
; (3.21)
3.1 The Extended HJM Model 60
 where
 Ai(s; t; T ) = Ai(s; T ) Ai(s; t);
  i(s; t; T ) =  i(s; T )  i(s; t):
 In the risk neutral measure, the nominal money market account is the numeraire, hence discounted
 bond prices are martingales. The bond prices take the form [44]
 pi(t; T ) = pi(0; T ) exp
  
ds
 Z t
 0
 [r(s)  is( 
i(s; T ))]ds+
 Z t
 0
  i(s; T )dLs
  
; (3.22)
 where  is the Laplace cumulant of L1, given by
  (z) = ?1( iz):
 Comparison with Equation 3.19, yields
 Ai(s; T ) =  is( 
i(s; T )): (3.23)
 Note that the latter drift condition is only \part" of the conditions in Corollary 3.3 for a nominal
 risk neutral probability measure to exist. The following sections assume that the latter conditions
 are in force.
 The next propositions and corollaries present some needed probability measures, change of proba-
 bility measure and associated properties that will be used later for IL options? pricing.
 Proposition 3.4. Let QIP denote the probability measure de ned by
 dQIP
 dQn
 = ZT on FT
 where
 Zt =
 BIP (t)
 Bn(t)
 Bn(0)
 BIP (0)
 ;
 then QIP is a martingale measure for the numeraire BIP .
 The expression BIP (t)Bn(t) is the discounted nominal value of the real money bank account.
 Proof. Let  be a stochastic process such that
  (t)
 Bn(t)
 is a Qn-martingale, i.e. so that  is an
 arbitrage free price process. We have to show that the process
  (t)
 BIP (t)
 is a QIP -martingale. Let
 s  t, then Bayes formula gives that
 EIPs
  
 (t)
 BIP (t)
  
=
 EQs
 h
 Zt
  (t)
 BIP (t)
 i
 Zs
 =
 EQs
 h
 BIP (t)
 Bn(t)
  (t)
 BIP (t)
 i
 Bn(0)
 BIP (0)
 Zs
 = EQs
  
 (t)
 Bn(t)
  
Bn(s)
 BIP (s)
 =
  (s)
 Bn(s)
 Bn(s)
 BIP (s)
 =
  (s)
 BIP (s)
3.1 The Extended HJM Model 61
 Corollary 3.5. De ne QT IP by
 dQT IP
 dQn
 = ZT on FT
 where
 Zt =
 pIP (t; T )
 Bn(t)
 Bn(0)
 pIP (0; T )
 then QT IP is a martingale measure for the numeraire pIP (t; T ).
 The expression pIP (t;T )Bn(t) is the price at time t of the discounted real zero-coupon bond denominated
 in nominal terms.
 If one exchanges BIP (t) for pIP (t; T ) in the proof of Proposition 3.4 the corollary follows.
 Proposition 3.6. Let  n denote an arbitrage free price in the nominal economy. De ne the process
  r by  r(t) =
  n(t)
 I(t)
 . De ne Qr by
 dQr
 dQn
 = ZT on FT
 where
 Zt =
 Br(t)I(t)
 Bn(t)
 Bn(0)
 Br(0)I(0)
 :
 Then Qr is a martingale measure for the numeraire Br(t) and
  r(t)
 Br(t)
 is a Qr-martingale:
 Proof. From Proposition 3.4 it follows that
  n(t)
 BIP (t)
 is a QIP -martingale and that Qr is equal
 to QIP . Since
  r(t)
 Br(t)
 =
  r(t)I(t)
 Br(t)I(t)
 =
  n(t)
 BIP (t)
 it follows that  r(t)Br(t) is a Q
 r-martingale.
 Corollary 3.7. De ne QT;r by
 dQT;r
 dQn
 = ZT on FT
 where
 Zt =
 pr(t; T )I(t)
 Bn(t)
 Bn(0)
 pr(0; T )I(0)
3.2 In ation Linked Swaps 62
 then QT;r is a martingale measure for the numeraire pr(t; T )and
  r(t)
 pr(t)
 is a QT;r-martingale:
 Corollary 3.8.
 pr(t; T )
 Br(t)
 is a Qr-martingale
 pr(t; S)
 pr(t; T )
 is a QT;r-martingale:
 Assumption 6. We assume that  n;  r;  n;  r; cI ;  P are deterministic. Under this assumption, for
 1-dimensionnal processes, there is a single risk neutral measure and thus a unique fair price for IL
 derivatives just as in the Black-Scholes pricing theory [47].
 3.2 In ation Linked Swaps
 This section focuses on the pricing of the most commonly traded in ation linked (IL) swaps.
 Zero coupons and year-on-year IL swaps are priced in the previously built HJM framework. Let
 T0; T1;    ; TM denote a  xed set of increasing times and  i be de ned by
  i = Ti  Ti 1; for i = 1;    ;M:
 A typical swap starts at time T0 with payments occurring at time T1; T2;    ; TM . On each payment
 date, Party A pays Party B the in ation rate over a prede ned period while Party B pays Party
 A a  xed rate. The in ation rate is computed as the percentage increase of the level of the price
 index over a period of time. In the previous description, Party A has entered a receiver swap (i.e.
 he receives a  xed amount) while Party B has entered a payer swap.
 In what follows to lighten the formulas,  [t;  ] is used to denote the price in nominal currency (e.g.
 rands, dollars), of the payo ( ).
 3.2.1 Zero Coupon In ation Indexed Swap
 Mercurio [81] showed that the fair price of a ZCIIS is model independent using martingale methods.
 This fact can also be proved using a replicating argument as proved by Hinnerich [63]. Both proofs
 are provided in this subsection, this result will be used afterwards to price year-on-year swaps.
 A ZCIIS over the time interval [T0; T ] has only one payment at time T without any intermediary
 payments. If Z0(T;K) denotes the corresponding payer ZCIIS with swap rate K and nominal N ,
 then a  xed amount of
3.2 In ation Linked Swaps 63
 N
  
(1 +K)T T0  1
  
is paid out at time T and a  oating amount of
 N
  
I(T )
 I(T0)
  1
  
is received at time T . Henceforth, the nominal will be taken to be equal to one for simpli cation.
 If Z0(t; T;K) denotes the price of Z0(T;K) at time t, then the its payo is
 Z0(T; T;K) =
 I(T )
 I(T0)
  (1 +K)T T0
 and
 Z0(t; T;K) = En
 (
 exp
  
 
Z T
 t
 rn(s)ds
 ! 
I(T )
 I(T0)
  (1 +K)T T0
    
 
 
 
Ft
 )
 for t 2 [T0; T ], where Ft is the corresponding  ltration. In particular
 Z0(T0; T;K) =  
 
T0;
 I(T )
 I(T0)
  (1 +K)T T0
  
=  
 
T0;
 I(T )
 I(T0)
  
  
 
T0; (1 +K)
 T T0
  
: (3.24)
 The in ation linked leg of the ZCIIS is
  
 
T0;
 I(T )
 I(T0)
  
=
 pn(T0; T )
 I(T0)
 ET;nT0 [I(T )]
 =
 pn(T0; T )
 I(T0)
 ET;nT0
  
I(T )pr(T; T )
 pn(T; T )
  
= pr(T0; T )
 since pr(T; T ) = pn(T; T ) = 1 and
 I(t)pr(t; T )
 pn(t; T )
 =
 pIP (t; T )
 pn(t; T )
 is a QT;n-martingale.
 The  xed leg of the ZCIIS is
  
 
T0; (1 +K)
 T T0
  
= pn(T0; T )(1 +K)
 T T0 :
 Hence fair price of the payer ZCIIS becomes
 Z0(T0; T;K) = pr(T0; T ) pn(T0; T )(1 +K)
 T T0 :
3.2 In ation Linked Swaps 64
 This price is not based on any assumption about the interest rate behaviour or assets? dynamics,
 but only on a no-arbitrage argument. The result was  rst stated in [81]. Its next simple replicating
 argument counterpart was stated in [63].
 To replicate the  oating leg of the swap
 (i) At time T0 buy
 1
 I(T0)
 IL bonds with maturity date T .
 (ii) At time T the dollar value of
 1
 I(T0)
 CPI units will be received, that is
 I(T )
 I(T0)
 .
 (iii) The price at time T0 of
 1
 I(T0)
 IL bonds is
 1
 I(T0)
 I(T0)pr(T0; T ) = pr(T0; T ).
 3.2.2 Year-on-Year In ation Indexed Swaps
 Contrary to the ZCIIS, the year-on-year IL swap?s fair price is model independent. In this subsection,
 the year-on-year IL swap (YYIIS) is priced in the L evy setting speci ed in Section 3.1.
 A YYIIS has multiple payments dates. Let YMm (K) denote a payer YYIIS that starts at time Tm
 with payment dates at Tm+1, Tm+2,    , TM . For each period [Ti; Ti+1] for i = m;    ;M  1 a  xed
 amount of
  i+1K
 is paid out at time Ti+1. For the same period a  oating amount of
  i+1[Xi+1  1]
 where
 Xi+1 =
 I(Ti+1)
 I(Ti)
 is received at time Ti+1.
 Let YMm (t;K) denote the price of a Y
 M
 m (K) at time t where t  Tm, then
 YMm (t;K) =
 M 1X
 i=m
  [t;  i+1(Xi+1  1)] 
M 1X
 i=m
  [t;  i+1K]
 =
 M 1X
 i=m
  [t;  i+1Xi+1] (K + 1)
 M 1X
 i=m
  i+1pn(t; Ti+1); (3.25)
 by standard no-arbitrage pricing theory. Therefore the pricing of YMm comes back to the computation
 of
 M 1X
 i=m
 [t;  i+1Xi+1], which is achieved through the forward swap rate. The forward swap rate of a
3.2 In ation Linked Swaps 65
 YYIIS is the value of swap rate for which the fair price of the swap is zero. Let RMm (t) denote the
 forward swap rate for the swap YMm (K). By de nition, Y
 M
 m [t; R
 M
 m (t)] = 0 and so R
 M
 m (t) is given by:
 RMm (t) =
 M 1X
 i=m
  [t;  i+1(Xi+1  1)] 
M 1X
 i=m
  i+1pn(t; Ti+1)
 M 1X
 i=m
  i+1pn(t; Ti+1)
 : (3.26)
 Now the computation of the model-dependent expression
 M 1X
 i=m
  [t;  i+1Xi+1] will provide explicit
 formulas for both the swap price and the forward swap rate.
 For m = 1 and t < T2, the  rst term of the summation is
  [t;  2X2] = pn(t; T2)E
 T2;n
 t
  
 2
 I(T2)
 I(T1)
  
= pn(t; T2) 2E
 T2;n
 t
  
1
 I(T1)
 ET2;nT1 [I(T 2)]
  
= pn(t; T2) 2E
 T2;n
 t
  
1
 I(T1)
 ET2;nT1
  
I(T2)pr(T2; T2)
 pn(T2; T2)
   
= pn(t; T2) 2E
 T2;n
 t
  
pr(T1; T2)
 pn(T1; T2)
  
: (3.27)
 Because the num eraire of the expectation in Equation (3.27) is pn(T1; T2), the QT1;n-forward measure
 is more appropriate for its valuation. The change of measure is done with the Bayes formula and
 the Radon-Nikod ym derivative ZT2;n=T1;nt that satis es for every t 2 [0; T2]
 ZT2;n=T1;nt =
 dQT2;n
 dQT1;n
  
 
 
 
t
 =
 pn(t; T2)
 pn(t; T1)
 pn(0; T1)
 pn(0; T2)
 :
 Hence
 ET2;nt
  
pr(T1; T2)
 pn(T1; T2)
  
=
 ET1;nt
 h
 pr(T1;T2)
 pn(T1;T2)
 ZT2;n=T1;nT1
 i
 ZT2;n=T1;nt
 = ET1;nt
  
pr(T1; T2)
 pn(T1; T2)
 pn(T1; T2)
 pn(T1; T1)
  
pn(t; T1)
 pn(t; T2)
 =
 pn(t; T1)
 pn(t; T2)
 ET1;nt [pr(T1; T2)] (3.28)
 The combination of Equations (3.27) and (3.28) yields
  [t;  2X2] =  2pn(t; T1)E
 T1;n
 t [pr(T1; T2)]: (3.29)
 So far, no model assumption has been used. However, the expectation in Equation (3.29) is model
 dependent. Mercurio use a di usion model to computed it in an environment without jumps [81]
3.2 In ation Linked Swaps 66
 while Hinnerich [63] used the martingale approach in a jump di usion setting. The latter approach
 will also be used in the L evy setting.
 Once more, the Bayes formula is used, but this time to change measure from QT1;n to QT1;r. The
 expected value in equation (3.29) can thus be rewritten as
 ET1;nt [pr(T1; T2)] =
 ET1;rt
 h
 pr(T1;T2)
 pr(T1;T1)
 ZT1;n=T1;rT1
 i
 ZT1;n=T1;rt
 ; (3.30)
 where
 ZT1;n=T1;rt =
 dQT1;n
 dQT1;r
  
 
 
 
t
 =
 pn(t; T1)
 pr(t; T1)I(t)
 pr(0; T1)I(0)
 pn(0; T1)
 :
 Under Qn we have the following dynamics
 dI(t)
 I(t )
 = [rn(t) rr(t)] dt+ b
 I(t)dWt +
 Z
 R
 cI(t; z)~ (dt; dz);
 dpn(t; T1)
 pn(t ; T1)
 = rn(t)dt+  
n(t; T1)dWt +
 Z
 R
  n(t; z; T1)~ (dt; dz);
 dpr(t; T1)
 pr(t ; T1)
 = ar(t; T )dt+  r(t; T1)dWt +
 Z
 R
  r(t; z; T1)~ (dt; dz)
 Using Corollary 2.13 with Z(t) = pr(t; T1)I(t)
 dZ(t)
 Z(t )
 =
  
ar(t; T ) + rn(t) rr(t) +  
r(t; T1)b
 I(t)
 +
 Z
 R
  r(t; z; T1)c
 I(t; z) (dz)
  
dt+
  
 r(t; T1) + b
 I(t)
  
dWt
 +
 Z
 R
  
 r(t; z; T1) + c
 I(t; z) +  r(t; z; T1)c
 I(t; z)
  
~ (dt; dz)
 By Corollary 2.9 with Y (t) =
 1
 Z(t)
 dY (t)
 Y (t)
 =  (t)dt  (t)dWt  
Z
 R
  (t; z)
 1 +  (t; z)
 ~ (dt; dz)
 where
  (t) =  ar(t; T ) rn(t) + rr(t)  
r(t; T1)b
 I(t)
  
Z
 R
  r(t; z; T1)c
 I(t; z) (dz) +  2(t) +
 Z
 jzj<R
  2(t; z)
 1 +  (t; z)
  (dz)
  (t) =  r(t; T1) + b
 I(t)
  (t; z) =  r(t; z; T1) + c
 I(t; z) +  r(t; z; T1)c
 I(t; z)
3.2 In ation Linked Swaps 67
 Applying Corollary 2.13 to X(t) = pn(t; T1)Y (t)
 dX(t)
 X(t )
 = [rn(t) +  (t)  
n(t; T1) (t)] dt 
Z
 R
  n(t; z; T1)
  (t; z)
 1 +  (t; z)
  (dt; dz)
 + [ n(t; T1)  (t)] dWt
 +
 Z
 R
  
 n(t; z; T1) 
 (t; z)
 1 +  (t; z)
   n(t; z; T1)
  (t; z)
 1 +  (t; z)
  
~ (dt; dz)
 = [rn(t) +  (t)  
n(t; T1) (t)] dt 
Z
 R
  n(t; z; T1)
  (t; z)
 1 +  (t; z)
  (dt; dz)
 + [ n(t; T1)  (t)] dWt +
 Z
 R
  n(t; z; T1)  (t; z)
 1 +  (t; z)
 ~ (dt; dz)
 Since
 dX(t)
 X(t )
 =
 dZT1;n=T1;rt
 ZT1;n=T1;rt 
, ZT1;n=T1;rt is a martingale under Q
 T1;r and a change of measure does
 not change either the volatility or the jump component; the dynamics of ZT1;n=T1;rt under Q
 T1;r are
 given by:
 dZT1;n=T1;rt
 ZT1;n=T1;rt 
=
  
 n(t; T1)  
r(t; T1) b
 I(t)
  
dWT1;r(t)
 +
 Z
 R
  n(t; z; T1) 
 
 r(t; z; T1) + cI(t; z) +  r(t; z; T1)cI(t; z)
  
1 +  r(t; z; T1) + cI(t; z) +  r(t; z; T1)cI(t; z)
 ~ T1;r(dt; dz)
 Since both
 pr(t; T2)
 pr(t; T1)
 and ZT1;n=T1;rt are Q
 T1;r-martingales, Theorem 2.23 and Equation (3.30)give
 that
 ET1;nt [pr(T1; T2] =
 pr(t; T2)eC(t;T1;T2)
 pr(t; T1)
 where
 C(t; T1; T2) =
 Z T1
 t
  
 
 n;1s   
r;1
 s  b
 I
 s
   
 r;2s   
r;1
 s
  
+
 Z
 R
  1;2s  
T1;r(ds; dz)
  
ds
 with the notation  i;js =  
i(s; Tj),  i;js =  
i(s; Tj), bIs = b
 I(s), cIt = c
 I(t; z) and
  1;2s =
  r;2t   
r;1
 t
 1 +  r;1t
  n;1t  
h
  r;1t + c
 I
 t +  
r;1
 t c
 I
 t
 i
 1 +  r;1t + c
 I
 t +  
r;1
 t c
 I
 t
 :
 Inserting this into Equation (3.29) gives that
  [t;XT2 ] =  2
 pn(t; T1)pr(t; T2)eC(t;T1;T2)
 pr(t; T1)
 Changing back to the general case with 1 = i and 2 = i+ 1 we have that
  [t;XTi+1 ] =  i+1
 pn(t; Ti)pr(t; Ti+1)eC(t;Ti;Ti+1)
 pr(t; Ti)
3.3 In ation Linked Caplets/Floorlets 68
 Hence the pricing Equation (3.25) is found to be
 YMm (t;K) =
 M 1X
 i=1
  i+1
 pn(t; Ti)pr(t; Ti+1)eC(t;Ti;Ti+1)
 pr(t; Ti)
  (K + 1)
 M 1X
 i=1
  i+1pn(t; Ti+1) (3.31)
 and the forward swap rate (3.26) is found to be
 RMm (t) =
 M 1X
 i=1
  i+1
 pn(t; Ti)pr(t; Ti+1)eC(t;Ti;Ti+1)
 pr(t; Ti)
  
M 1X
 i=1
  i+1pn(t; Ti+1)
 M 1X
 i=1
  i+1pn(t; Ti+1)
 For calibration purposes, it is more convenient to rewrite these formulas in function of the IL bond
 using the relation pIP (t; T ) = pr(t; T ). The new formulas are
 YMm (t;K) =
 M 1X
 i=1
  i+1
 pn(t; Ti)pIP (t; Ti+1)eC(t;Ti;Ti+1)
 pIP (t; Ti)
  (K + 1)
 M 1X
 i=1
  i+1pn(t; Ti+1)
 RMm (t) =
 M 1X
 i=1
  i+1
 pn(t; Ti)pIP (t; Ti+1)eC(t;Ti;Ti+1)
 pIP (t; Ti)
  
M 1X
 i=1
  i+1pn(t; Ti+1)
 M 1X
 i=1
  i+1pn(t; Ti+1)
 3.3 In ation Linked Caplets/Floorlets
 An in ation indexed (II) caplet (resp.  oorlet) written at time t over the period [Ti 1; Ti] (i.e. with
 maturity Ti) is a call (resp. put) on the in ation rate Xi =
 I(Ti)
 I(Ti 1)
 implied by an CPI index. The
 payo at time Ti of such an option with strike k is
 N i [!(Xi  1 k)]
 + (3.32)
 where N is the contract nominal,  i is the contract year fraction for the interval [Ti 1; Ti], and ! = 1
 for a caplet and ! =  1 for a  oorlet.
 The no-arbitrage price at time t of the i-th caplet is
 CFleti(t; k) = N iEQ
 (
 Bn(t)
 Bn(Ti)
 [! (Xi  1 k)]
 +
  
 
 
 
Ft
 )
 = N ipn(t; Ti)Ei
  
[! (Xi  K)]
 +
  
 
 
Ft
  
; (3.33)
 where K = k+ 1 and Ei is the short form of E
 Qn
 Ti
 , which is the conditional expectation with respect
 to the nominal risk neutral forward measure at time Ti.
 The next sections investigate di erent paths for computing this price.
3.3 In ation Linked Caplets/Floorlets 69
 3.3.1 Lognormally Distributed CPI
 Throughout this section the next assumption is supposed true.
 Assumption 7. The CPI is only driven by a Brownian motion without any jumps.
 The previous assumption leads to the Jarrow-Yildrim framework where the CPI is lognormal under
 Qn. The pricing formula given by Equation (3.33) becomes a simple Black and Scholes (BS) formula.
 Just as with standard BS pricing the next theorem will be useful.
 Theorem 3.9. Let X be a random variable that is lognormally distributed, and denote by M and
 V the mean and standard deviation of Y = ln(X). Then
 E
 n
 [!(X  K)]+
 o
 = !eM+
 1
 2V
 2
  
 
!
 M  lnK + V 2
 V
  
 !K 
 
!
 M  lnK
 V
  
;
 for each K > 0; ! 2 f 1; 1g, where E denotes expectation with respect to X?s distribution and  
denotes the cumulative standard normal distribution function.
 Under Assumption 7 (i.e. cI(t; z) = 0), and by Ito^?s formula
 d ln I(t) =
  
rn(s) rr(s) 
1
 2
 (bI(t))2
  
dt+ bI(t)dWt
 and the CPI is of the form
 I(T ) = I(t) exp
 (Z T
 t
  
rn(s) rr(s) 
1
 2
 (bI(s))2
  
ds+
 Z T
 t
 bI(s)dWs
 )
 ;
 where t < T . Therefore, ln
 I(T )
 I(t)
  
 
 
 
Ft
 and ln
 I(Ti)
 I(Ti 1)
  
 
 
 
Ft
 are lognormal under QTi;n. From Theorem
 3.9, if Xi is a lognormal random variable with mean E(X) = m and standard deviation of the
 logarithm distribution Std[ln(X)] = v, then
 E
 n
 [!(Xi  K)]
 +
 o
 = !m 
 
!
 ln mK +
 1
 2v
 2
 v
  
 !K 
 
!
 ln mK  
1
 2v
 2
 v
  
; (3.34)
 with K = 1 + k. The conditional expectation of I(Ti)I(Ti 1) is obtained from Equation (3.31)
 ETi;nt [Xi] =
 pn(t; Ti 1)
 pn(t; Ti)
 pr(t; Ti)
 pr(t; Ti 1)
 eC(t;Ti 1;Ti)
 The variable v is given by the following corollary.
 Corollary 3.10. The variance of the logarithm of the ratio I(Ti)I(Ti 1) under the (nominal) risk neutral
 measure is given by
 V arTi;nt [lnXi] = V
 2(t; Ti 1; Ti)
3.3 In ation Linked Caplets/Floorlets 70
 where
 V 2(t; Ti 1; Ti) =
  2n
 2a3n
 h
 1 e an(Ti Ti 1)
 i2 h
 1 e 2an(Ti 1 t)
 i
 + (bI)2(Ti  Ti 1)
 +
  2r
 2a3r
 h
 1 e ar(Ti Ti 1)
 i2 h
 1 e 2ar(Ti 1 t)
 i
  2 n;r
  n r
 anar(an + ar)
 h
 1 e an(Ti Ti 1)
 i h
 1 e ar(Ti Ti 1)
 i h
 1 e (an+ar)(Ti 1 t)
 i
 +
  2n
 a2n
  
Ti  Ti 1 +
 2
 an
 e an(Ti Ti 1)  
1
 2an
 e 2an(Ti Ti 1)  
3
 2an
  
+
  2r
 a2r
  
Ti  Ti 1 +
 2
 ar
 e ar(Ti Ti 1)  
1
 2ar
 e 2ar(Ti Ti 1)  
3
 2ar
  
 2 n;r
  n r
 anar
  
Ti  Ti 1  
1 e an(Ti Ti 1)
 an
  
1 e ar(Ti Ti 1)
 ar
  
1 e (an+ar)(Ti Ti 1)
 an + ar
  
+ 2 n;I
  n I
 an
  
Ti  Ti 1  
1 e an(Ti Ti 1)
 an
  
 2 r;I
  r I
 ar
  
Ti  Ti 1  
1 e ar(Ti Ti 1)
 ar
  
Proof. See [81, 23]
 Hence, by Equation (3.34)
 ETi+1;nt [Xi+1] = !m 
 
!
 ln mK +
 1
 2v
 2
 v
  
 !K 
 
!
 ln mK  
1
 2v
 2
 v
  
= !
 pn(t; Ti)
 pn(t; Ti+1)
 pr(t; Ti+1)
 pr(t; Ti)
 eC(t;Ti;Ti+1) 
2
 4!
 ln
  
pn(t;Ti)
 Kpn(t;Ti+1)
 pr(t;Ti+1)
 pr(t;Ti)
 eC(t;Ti;Ti+1)
  
+ 12v
 2
 v
 3
 5
  !K 
2
 4!
 ln
  
pn(t;Ti)
 Kpn(t;Ti+1)
 pr(t;Ti+1)
 pr(t;Ti)
 eC(t;Ti;Ti+1)
  
 12v
 2
 v
 3
 5
 = !
 pn(t; Ti)
 pn(t; Ti+1)
 pr(t; Ti+1)
 pr(t; Ti)
 eC(t;Ti;Ti+1) 
2
 4!
 ln pn(t;Ti)pr(t;Ti+1)Kpn(t;Ti+1)pr(t;Ti) + C(t; Ti; Ti+1) +
 1
 2v
 2
 v
 3
 5
  !K 
2
 4!
 ln pn(t;Ti)pr(t;Ti+1)Kpn(t;Ti+1)pr(t;Ti) + C(t; Ti; Ti+1) 
1
 2v
 2
 v
 3
 5 :
 Hence
 CFlet(Ti+1) =  i+1pn(t; Ti+1)
  
!
 pn(t; Ti)
 pn(t; Ti+1)
 pr(t; Ti+1)
 pr(t; Ti)
 eC(t;Ti;Ti+1)
   
2
 4!
 ln pn(t;Ti)pr(t;Ti+1)Kpn(t;Ti+1)pr(t;Ti) + C(t; Ti; Ti+1) +
 1
 2v
 2
 v
 3
 5
  !K 
2
 4!
 ln pn(t;Ti)pr(t;Ti+1)Kpn(t;Ti+1)pr(t;Ti) + C(t; Ti; Ti+1) 
1
 2v
 2
 v
 3
 5
 9
 =
 ;
 :
3.3 In ation Linked Caplets/Floorlets 71
 Using the equality pIP (t; T ) = I(t)Pr(t; T ), the caplet/ oorlet can be rewritten has
 CFlet(Ti+1) =  i+1pn(t; Ti+1)
  
!
 pn(t; Ti)
 pn(t; Ti+1)
 pIP (t; Ti+1)
 pIP (t; Ti)
 eC(t;Ti;Ti+1)
   
2
 4!
 ln pn(t;Ti)pIP (t;Ti+1)Kpn(t;Ti+1)pIP (t;Ti) + C(t; Ti; Ti+1) +
 1
 2v
 2
 v
 3
 5
  !K 
2
 4!
 ln pn(t;Ti)pIP (t;Ti+1)Kpn(t;Ti+1)pIP (t;Ti) + C(t; Ti; Ti+1) 
1
 2v
 2
 v
 3
 5
 9
 =
 ;
 :
 3.3.2 Pricing with the Bilateral Laplace Transform
 To price caplets and  oorlets under the assumption of exponential L evy distribution, the method-
 ology proposed by Eberlein and Kluge in [44] will be followed. First of all, recall that the forward
 rate dynamics in Assumption (3) can be rewritten as
 dfi(t; T ) =  
i(t; T )dt+  i(t; T )dLt i = r; n (0  t  T ); (3.35)
 with some common integrability conditions. A cap (resp.  oor) is a series of call (resp. put) options
 on subsequent variable rates. These single options are called caplets (resp.  oorlets). Each caplet
 (resp.  oorlet) is equivalent to a put (resp. call) option on the in ation rate. Thus, deriving suitable
 formulas for calls and puts on the in ation rate immediately gives formulas for caps and  oors.
 As described previously, the discounted bond price process pi( ; T ) for i = n; r are martingales with
 respect to the measure Q and the corresponding  ltration for each T . However, this is not the case
 for the in ation process and consequently the in ation rate. This di culty can be avoided through
 a change of probability measure. Moreover, because the \unwanted" term is the real interest rate
 that can be evaluated from market data, the calibration process will still be possible. In this new
 probability measure, which will be denoted by Qj;r, the pricing of a caplet can be achieved by taking
 the conditional expectation of the discounted payo . The time-t value of a caplet/ oorlet with strike
 k over the period [Tj 1; Tj ] is given by
 CFletj(t; k) = N jpn(t; Tj)Ej
  
[! (Xj  K)]
 +
  
 
 
Ft
  
(t  Tj)
 where K = k+ 1. For simpli cation, henceforth w = 1, N = 1 and  j = 1. The caplet fair price can
 be rewritten as
 CFletj(t;K) = pn(t; Tj)Ej
  
(Xj  K)
 +
  
 
 
Ft
  
(t  Tj):
3.3 In ation Linked Caplets/Floorlets 72
 Having a closer look at the in ation?s dynamics, through Corollary 2.10 and Equation (3.2)
 dI(t)
 I(t )
 = [rn(t) rr(t)] dt+ b
 I(t)dWt +
 Z
 R
 cI(t; z)~ (dt; dz);
 d ln It =
  
rn(t) rr(t) 
1
 2
 (bI(t))2
  
dt+
 Z
 jzj<R
  
ln[1 + cI(t; z)] cI(t; z)
  
 (dt; dz)
 +bI(t)dWt +
 Z
 R
 ln[1 + cI(t; z)] (dt; dz)
 ln
 IT
 It
 =
 Z s=T
 s=t
 ( 
rn(s) rr(s) 
1
 2
 (bI(s))2
  
ds+
 Z
 jzj<R
  
ln[1 + cI(s; z)] cI(s; z)
  
 (ds; dz)
 )
 +
 Z s=T
 s=t
  
bI(s)dWs +
 Z
 R
 ln[1 + cI(s; z)] (ds; dz)
  
:
 Under Assumption 5, for the in ation process to be a Q-martingale, it has to be multiplied by1
 exp
  Z t
 0
 rr(s)ds
  
which is deterministic. The caplet price is now
 CFletj(t;K) = pn(t; Tj)fjEj;r
  
 
Xrj  Kf
  1
 j
  +
  
 
 
Ft
  
(t  Tj);
 where
 Xrj = Xjf
  1
 j ;
 fj = exp
 "
  
Z Tj
 Tj 1
 rr(s)ds
 #
 and the expectation is under Qj;r and not the initial nominal risk forward probability measure.
 A straightforward approach is to derive the joint (conditional) distribution of the random variables
 pn(t; T ) and Xj . Although this can easily be done analytically [48], the numerical evaluation of
 the resulting expression is extremely time consuming. Instead, the change-of-numeraire technique is
 used here to circumvent the calculation of the joint probability law, that is as previously mentioned,
 calculations are not conducted in the spot martingale measure Q, but in the \modi ed" forward
 martingale measure for the settlement day Ti 1 Qj;r (see Geman, El Karoui, and Rochet (1995)
 for details). More precisely, the measure Qj;r, equivalent to Q, is de ned by its Radon-Nikodym
 derivative
 dQj;r
 dQ
 =
 1
 Bn(t)Xj exp
 "Z Tj
 Tj 1
 rr(s)ds
 # :
 The previous expression can be rewritten as
 dQt
 dQ
 = exp
  
 
Z t
 0
 AIP (s; t)ds+
 Z t
 0
  IP (s; t)dLIPs
  
1Eventually scaled by a multiplicative constant.
3.3 In ation Linked Caplets/Floorlets 73
 and, when restricted to the  - eld Fs,
 dQt
 dQ
  
 
 
 
Fs
 = exp
  
 
Z s
 0
 AIP (u; t)du+
 Z s
 0
  IP (u; t)dLIPu
  
:
 Equation (3.22) leads to
 CFletj(t;K) = pn(t; Tj)Ej
  
[D exp(Y ) K]+
  
 
 
Ft
  
(t  Tj):
 where
 D =
 pn(t; Ti)
 pn(t; Ti 1)
 exp
 (Z Ti 1
 t
  
AIP (s; Ti 1) A
 IP (s; Ti)
  
ds 
Z Tj
 Tj 1
 rr(s)ds
 )
 is deterministic and
 Y =
 Z Ti 1
 0
  
 IP (s; Ti)  
IP (s; Ti 1)
  
dLs
 is Ft-measurable. Notice that the expectation is now in the forward risk neutral probability measure.
 To calculate the option price, the distribution of Y under the measure Qt is required; and it can be
 estimated. If this distribution is represented by the Lebesgue-density ? in R, then
 CFletj(t;K) = pn(t; Tj)
 Z
 R
 (Dey  K)+?(y)dy:
 To get a numerical estimate of the caplet price, either the Laplace transform method or the Fourier
 transform method can be used. The Laplace technique was developed in [94] and used to derive
 exact pricing formulas for pricing caps,  oors and swaptions in [44]. The similar and rather simpler
 technique was previously proposed by Carr and Madan [26] using Fourier transforms. First, the
 option price is expressed as a convolution. The Laplace transform of this convolution equals the
 product of the Laplace transforms of the convolution factors. These factors are easy to calculate in
 this case. Then, to get the price of the option, an inverse Laplace transformation will be performed.
 Theorem 3.11. Let MYt denote the moment generating function of the random variable Y with
 respect to the measure Qt. If R is chosen R <  1 such that MYt ( R) <1, then
 CFletj(t;K) =
 1
 2 
Kpn(t; Tj)e
 R 
Z 1
  1
 eiu MYt ( R iu)
 (R+ iu)(R+ 1 + iu)
 du
 with
  = ln
 pn(t; Ti 1)
 pn(t; Ti)
  
Z Ti 1
 t
  
AIP (s; Ti 1) A
 IP (s; Ti)
  
ds+ lnK:
 Proof. See [44] Theorem 12.
3.4 Conclusion 74
 Theorem 3.12. Considering R > 0 such that MYt ( R) < 1, the price of a  oorlet with strike K
 and exercise period [Ti 1; Ti]
 CFletj(t;K)j!= 1 =
 1
 2 
Kpn(t; Tj)e
 R 
Z 1
  1
 eiu MYt ( R iu)
 (R+ iu)(R+ 1 + iu)
 du
 with
  = ln
 pn(t; Ti 1)
 pn(t; Ti)
  
Z Ti 1
 t
  
AIP (s; Ti 1) A
 IP (s; Ti)
  
ds+ lnK:
 Proof. See [44] Corollary 14.
 The formulas to price the caplet and  oorlet are similar except for the values for R. The accuracy
 of the numerically estimated security price relies on the right choice of R, which has already been
 discussed in Section 2.5.
 3.4 Conclusion
 This chapter has presented a non-trivial generalisation of the work by Hinnerich on pricing in ation
 linked securities. It started with the extension of the HJM framework to L evy processes, with
 the underlying proof of the foreign exchange analogy. Afterwards, some in ation linked derivative
 pricing formulas were derived in the new framework. Some calibration to market data is presented
 in Chapter 6, where both South African and American market data are used. These results reinforce
 the fact that L evy distributions are more appropriate than the conventional normal (or log normal)
 distribution, providing an improved accuracy and a more straightforward intuition when building
 and tuning models. However, the main issue encountered was the lack of data necessary for the
 calibration. This prevented the calibration process to be completed for some of the IL securities;
 but results of this calibration will be provided in upcoming work.
Chapter 4
 Stochastic Monetary Economy
 Models
 Virtually all asset pricing models are special cases of the fundamental equation [53]
 Pt = Et[mt+1(Pt+1 +Dt+1)]; (4.1)
 where Pt is the value of an asset at time t, Dt+1 is the amount of any dividends, interest or other
 cash ows received at time t + 1 and mt+1 is the stochastic discount factor (SDF) between time t
 and time t + 1. Equation (4.1) implies that the price process is a martingale under an appropriate
 measure.
 If mt+1 is a strictly positive random variable, equation (4.1) becomes equivalent to the no-arbitrage
 principle, which states that all portfolios of assets with non-negative payo s and positive probability
 of positive payo s, must have positive prices. While the no-arbitrage principle places restrictions
 on mt+1, other work explores the implications of equilibrium models for the SDF based on the
 investor? s utility optimization. A typical consumer/investor?s utility optimization involves the
 Bellman equation:
 J(Wt; st)  maxEt [U(Ct;  ) + J(Wt+1; st+1)] ;
 where U(Ct;  ) is the utility of consumption expenditures at time t, and J( ;  ) is the indirect utility
 of wealth [53].
 In the case that an asset pays dividends on a continuous basis, the pricing formula is given by
 75
76
 [66, 68]:
 Mt =
 1
  t
 Et
 "
  TST +
 Z T
 t
  sDsds
 #
 0  t  T;
 where (Mt)t 0 is a martingale, ( t)t 0 is the pricing kernel process, (St)t 0 is the value of the
 dividend-paying asset, and (Dt)t 0 is the dividend process.
 The stochastic monetary economy models built by Hughston and Macrina assume a positive nominal
 interest rate that was advocated for by Flesaker and Hughston (FH) [54]. Flesaker and Hughston
 were among the  rst to propose an entirely new approach to interest-rate modelling resulting in
 concrete models that are not part of the short-rate world. Instead of modelling instantaneous
 forward rates they model pricing kernels (also known as state-price densities or pricing operators).
 Assuming Nt is the conventional num eraire, the corresponding pricing kernel is given by  t =
  t
 Nt
 in
 the real world probability measure, where ( t)t 0 denotes the Radon-Nikodym density martingale
 transforming the real world measure into the risk neutral measure. The latter equation implies that
 under the real probability measure the asset price process multiplied with the pricing kernel process
 is a martingale. The process ( t)t 0 is a decreasing and positive supermartingale (i.e.  t   t+h
 with h > 0) thus ensures interest rate positivity.
 Proof. Consider a standard  ltered probability space ( ;F ;F;P) where F = (Ft)t 0 and P is
 the real world measure with equivalent risk neutral measure Q.
 With the standard num eraire approach the arbitrage-free price of a European contingent claim,
 (Yt)t 0, paying YT at its maturity T is given by:
 Yt = BtEQ
 "
 YT
 BT
  
 
 
 
Ft
 #
 ; (4.2)
 where the num eraire is the money market account de ned by:
 Bt = exp
  Z t
 0
 rsds
  
;
 with rs denoting the short rate at time s.
 The absence of arbitrage in a  nancial market is equivalent to the existence of a pricing kernel
 ( t)t 0. In fact, Equation (4.2) can be expressed in terms of a pricing kernel under the real world
 measure as shown in the next paragraphs.
 Let ( t)t 0 denote the Radon-Nikodym density martingale transforming the real world measure into
 the risk neutral measure, i.e.
  t =
 dQ
 dP
  
 
 
 
Ft
 :
77
 Application of Bayes? formula shows that:
 Yt = BtEQ
 "
 YT
 BT
  
 
 
 
Ft
 #
 = Bt
 EP
 "
 YT
 BT
  T
  
 
 
 
Ft
 #
  t
 :=
 EP
  
 TYT jFt
  
 t
 ;
 where the pricing kernel is de ned to be of form
  t =
  t
 Bt
 =
 dQ
 dP
  
 
 
 
Ft
 exp
  
 
Z t
 0
 rsds
  
:
 The IL framework proposed by Hughston and Macrina is based on the assumption that in ation is
 a purely monetary phenomenon. Thus the in uence of  uctuations in wages, supply and demand,
 foreign exchange and employment, etc. on in ation is not treated directly, but is rather re ected in
 the change of the rates of consumption and money supply, and the liquidity bene t of money supply.
 In a discrete time1 model, let the nominal money supply, the aggregate consumption and the nominal
 liquidity bene t be denoted respectively by (fMigi 0), (fkigi 1) and (f igi 0). At time ti, the real
 bene t (in units of goods and services) provided by the money supply is
 li =
  iMi
 Ci
 for i  0:
 Given that
 J = E
 "
 NX
 n=0
 e rtnU(kn; ln)
 #
 ;
 W = E
 "
 NX
 n=0
  n(Cnkn +  nMn)
 #
 ;
 where U( ;  ) is a bivariate utility function. Two examples of utility functions are considered below:
 the log-separable utility function and the separable power utility function.
 Note that the formulas obtained are not directly functions of any IL derivative?s price; therefore this
 novel framework could be a solution to pricing IL products despite the fact that they are illiquid.
 Sections 4.1 (resp. 4.2) studies the performances of this framework when the macroeconomic factors
 are L evy (resp. exponential L evy) processes.
 1The formulas in continuous time have also been derived and are similar to those obtained in discrete time.
4.1 L evy process distribution 78
 4.1 L evy process distribution
 Given the high  exibility of L evy distributions [7], it is reasonable to assume that the nominal money
 supply, the aggregate consumption and the nominal liquidity bene t can be reproduced using L evy
 processes (Assumption 8). Under the previous assumption, this section deduces the dynamics of the
 CPI, the pricing kernel and IL securities.
 The next assumption holds throughout this section.
 Assumption 8. Under the objective probability measure P, the dynamics of (Mt)t 0, (kt)t 0 and
 ( t)t 0 are given by:
 dMt =  M (t)dt+  M (t)dW
 P
 t +
 Z
 R
  M (t; z) (dt; dz);
 dkt =  k(t)dt+  k(t)dW
 P
 t +
 Z
 R
  k(t; z) (dt; dz);
 d t =   (t)dt+   (t)dW
 P
 t +
 Z
 R
   (t; z) (dt; dz);
 with
  (dt; dz) =
 8
 <
 :
 (   )(dt; dz); jzj < R
  (dt; dz); jzj  R
 where  i(t),  i(t),  i(t), ai(t), bi(t) and ci(t) are adapted processes with
 Z t
 0
 j i(s)jds <1;
 Z t
 0
 j i(s)j
 2ds <1;
 for all  nite t;  i(t; z) :   R+  R! R+ is a real valued function satisfying
 Z t
 0
 Z
 R
 j i(s; z)j
 2 (ds; dz) <1;
 for  nite t. These conditions guarantee integrability of the coe cients and are satis ed if the coe -
 cients are bounded for t from a bounded set and  ([0; t] R) <1 for  nite t.
 The real world Brownian motion W P will be denoted by W when there is no ambiguity.
 Section 4.1.1 (resp. 4.1.2) further assumes that the agent utility function is a log-separable (resp.
 separable power) utility function; then computes the IL pricing formulas and explicit formulas for
 the CPI and pricing kernel.
 4.1.1 Log-separable utility function
 Given two non-negative constants A and B, a log-separable utility function is of the form
 U(x; y) = A ln (x) +B ln (y):
4.1 L evy process distribution 79
 In the current pricing framework [68], the pricing kernel, the CPI and the no-arbitrage value of a
 contingent claim Ht, are respectively
 Cn =
 A
 B
  nMn
 kn
 ;
  n =
 Be rtn
   nMn
 ;
 H0 =  0M0e
  rtjE
  
Hj
  jMj
  
where  is an introduced Lagrange multiplier.
 The following propositions compute the dynamics of the CPI and pricing kernel, and an explicit
 formula for the value of the contingent.
 Proposition 4.1. The dynamics of the CPI and pricing kernel are given by
 dCt
 A=B
 =  C(t)dt+  C(t)dWt +
 Z
 R
  C(t; z) (dt; dz);
 d t
 B= 
=   (t)dt+   (t)dWt +
 Z
 R
   (t; z) (dt; dz);
 where
  C(t) =
 1
 kt
  
  (t)Mt +  M (t) t +   (t) M (t) +
 Z
 R
   (t; z) M (t; z) (dz)
  
+ Y (t) tMt  [  (t)Mt +  M (t) t]
  k(t)
 k2t
 +
 Z
 R
 [ M (t; z) t +   (t; z)Mt +   (t; z) M (t; z)]  Y (t; z) (dz);
  C(t) = [  (t)Mt +  M (t) t]
 1
 kt
   k(t)
  tMt
 k2t
 ;
  C(t; z) =  Y (t; z) t Mt + [ M (t; z) t +   (t; z)Mt +   (t; z) M (t; z)]
 1
 kt 
+ [ M (t; z) t +   (t; z)Mt +   (t; z) M (t; z)]) Y (t; z);
   (t) =
  
 
r
  tMt
  
1
  2tM
 2
 t
  
 2(t) (  (t)Mt +  M (t) t)
 2 1
  tMt
  
+
 Z
 jzj<R
  
1
  t Mt +  2(t; z)
  
1
  t Mt 
+
 1
  2t M
 2
 t 
 2(t; z)
  
 (dz)
 )
 e rt;
   (t) =  [  (t)Mt +  M (t) t]
 e rt
  2tM
 2
 t
 ;
   (t; z) =
  
1
  t Mt +  2(t; z)
  
1
  t Mt 
 
e rt;
4.1 L evy process distribution 80
  Y (t) =  
1
 k2t
  
 k(t)  
2
 k(t)
 1
 kt
  
+
 Z
 jzj<R
  
1
 kt +  k(t; z)
  
1
 kt 
+
 1
 k2t 
 k(t; z)
  
 (dz);
  Y (t; z) =
 1
 kt +  k(t; z)
  
1
 kt 
;
  2(t) =   (t)Mt +  M (t) t +   (t) M (t) +
 Z
 R
   (t; z) M (t; z) (dz);
  2(t; z) =  M (t; z) t +   (t; z)Mt +   (t; z) M (t; z):
 Proof. Applying Corollary 2.12 with Zt =  tMt
 dZt =
  
  (t)Mt +  M (t) t +   (t) M (t) +
 Z
 R
   (t; z) M (t; z) (dz)
  
dt+ [  (t)Mt +  M (t) t] dWt
 +
 Z
 R
 [ M (t; z) t +   (t; z)Mt +   (t; z) M (t; z)]  (dt; dz):
 Then applying Corollary 2.14
 dCt
 A=B
 =  C(t)dt+  C(t)dWt +
 Z
 R
  C(t; z) (dt; dz);
 where
  C(t) =
 1
 kt
  
  (t)Mt +  M (t) t +   (t) M (t) +
 Z
 R
   (t; z) M (t; z) (dz)
  
+ Y (t) tMt  [  (t)Mt +  M (t) t]
  k(t)
 k2t
 +
 Z
 R
 [ M (t; z) t +   (t; z)Mt +   (t; z) M (t; z)]  Y (t; z) (dz);
  C(t) = [  (t)Mt +  M (t) t]
 1
 kt
   k(t)
  tMt
 k2t
 ;
  C(t; z) =  Y (t; z) t Mt + [ M (t; z) t +   (t; z)Mt +   (t; z) M (t; z)]
 1
 kt 
+ [ M (t; z) t +   (t; z)Mt +   (t; z) M (t; z)]  Y (t; z);
  Y (t) =  
1
 k2t
  
 k(t)  
2
 k(t)
 1
 kt
  
+
 Z
 jzj<R
  
1
 kt +  k(t; z)
  
1
 kt 
+
 1
 k2t 
 k(t; z)
  
 (dz);
  Y (t; z) =
 1
 kt +  k(t; z)
  
1
 kt 
:
 The deterministic process de ned by Xt = e rt, i.e. lnXt =  rt has dynamics given by
 dXt =  re
  rtdt:
 Again applying Corollary 2.14
 d t
 B= 
=
  
 re rt
  tMt
 +  X(t)e
  rt
  
dt  2(t)
 e rt
  2tM
 2
 t
 dWt +
 Z
 R
  X(t; z)e
  rt (dt; dz)
 =
  
 
r
  tMt
 +  X(t)
  
e rtdt  2(t)
 e rt
  2tM
 2
 t
 dWt +
 Z
 R
  X(t; z)e
  rt (dt; dz);
4.1 L evy process distribution 81
 where
  X(t) =  
1
  2tM
 2
 t
  
 2(t)  
2
 2(t)
 1
  tMt
  
+
 Z
 jzj<R
  
1
  t Mt +  2(t; z)
  
1
  t Mt 
+
 1
  2t M
 2
 t 
 2(t; z)
  
 (dz);
  X(t; z) =
 1
  t Mt +  2(t; z)
  
1
  t Mt 
;
  2(t) =   (t)Mt +  M (t) t +   (t) M (t) +
 Z
 R
   (t; z) M (t; z) (dz);
  2(t) =   (t)Mt +  M (t) t;
  2(t; z) =  M (t; z) t +   (t; z)Mt +   (t; z) M (t; z):
 Proposition 4.2. The fair price at t = 0 of a contingent claim Ht is
 H0 = e
  rt
  
1 +  0M0
 Z t
 0
  X(s)ds
  
E[Ht] +  0M0e rtE
  
Ht
 Z t
 0
  X(s)dWs
  
+ 0M0e
  rtE
  
Ht
 Z t
 0
 Z
 R
  X(s; z) (ds; dz)
  
;
 where
  X(t) =  
1
  2tM
 2
 t
  
 Y (t) (  (t)Mt +  M (t) t)
 2 1
  tMt
  
+
 Z
 jzj<R
  
1
  t Mt +  Y (t; z)
  
1
  t Mt 
+
 1
  2t M
 2
 t 
 Y (t; z)
  
 (dz);
  X(t) =  
1
  2tM
 2
 t
 [  (t)Mt +  M (t) t] ;
  X(t; z) =
 1
  t Mt +  Y (t; z)
  
1
  t Mt 
;
  Y (t) =   (t)Mt +  M (t) t +   (t) M (t) +
 Z
 R
   (t; z) M (t; z) (dz);
  Y (t; z) =  M (t; z) t +   (t; z)Mt +   (t; z) M (t; z):
 Proof. Using the dynamics of Zt (see previous proof) in Corollary 2.8 with Xt = 1 tMt
 dXt =  X(t)dt+  X(t)dWt +
 Z
 R
  X(t; z) (dt; dz);
4.1 L evy process distribution 82
 where
  X(t) =  
1
  2tM
 2
 t
  
 Y (t) (  (t)Mt +  M (t) t)
 2 1
  tMt
  
+
 Z
 jzj<R
  
1
  t Mt +  Y (t; z)
  
1
  t Mt 
+
 1
  2t M
 2
 t 
 Y (t; z)
  
 (dz);
  X(t) =  
1
  2tM
 2
 t
 [  (t)Mt +  M (t) t] ;
  X(t; z) =
 1
  t Mt +  Y (t; z)
  
1
  t Mt 
;
  Y (t) =   (t)Mt +  M (t) t +   (t) M (t) +
 Z
 R
   (t; z) M (t; z) (dz);
  Y (t; z) =  M (t; z) t +   (t; z)Mt +   (t; z) M (t; z):
 Hence
 Xt = X0 +
 Z t
 0
  X(s)ds+
 Z t
 0
  X(s)dWs +
 Z t
 0
 Z
 R
  X(s) (ds; dz)
 and
 E[XtHt] = E
   Z t
 0
  X(s)ds+
 Z t
 0
  X(s)dWs +
 Z t
 0
 Z
 R
  X(s) (ds; dz)
  
Ht
  
+ E[X0Ht]:
 Therefore
 H0 = e
  rtE[Ht] +  0M0e rtE
  
Ht
  Z t
 0
  X(s)ds+
 Z t
 0
  X(s)dWs +
 Z t
 0
 Z
 R
  X(s) (ds; dz)
   
:
 If, additionally, the pricing kernel and the contingent claim are assumed to be independent, then
 the expression of the fair price can easily be compared to the standard no-arbitrage pricing formula
 as shown in the next proposition.
 Proposition 4.3. If a further assumption is made that
 1
  tMt
 and Ht are independent (i.e. the
 pricing kernel is independent of the contingent claim), the fair price at t = 0 of the contingent claim
 becomes
 H0 = e
  rt
 "
 1 +  0M0
  Z t
 0
  X(s)ds+
 Z t
 0
 Z
 jzj R
  X(s; z) (ds; dz)
 !#
 E[Ht]:
4.1 L evy process distribution 83
 where
  X(t) =  
1
  2tM
 2
 t
  
 Y (t) (  (t)Mt +  M (t) t)
 2 1
  tMt
  
+
 Z
 jzj<R
  
1
  t Mt +  Y (t; z)
  
1
  t Mt 
+
 1
  2t M
 2
 t 
 Y (t; z)
  
 (dz);
  X(t) =  
1
  2tM
 2
 t
 [  (t)Mt +  M (t) t] ;
  X(t; z) =
 1
  t Mt +  Y (t; z)
  
1
  t Mt 
;
  Y (t) =   (t)Mt +  M (t) t +   (t) M (t) +
 Z
 R
   (t; z) M (t; z) (dz);
  Y (t; z) =  M (t; z) t +   (t; z)Mt +   (t; z) M (t; z):
 The term e rt
 "
 1 +  0M0
  Z t
 0
  X(s)ds+
 Z t
 0
 Z
 jzj R
  X(s; z) (ds; dz)
 !#
 acts as the conventional
 discount factor e rt. The additional term
 At = e
  rt 0M0
  Z t
 0
  X(s)ds+
 Z t
 0
 Z
 jzj R
  X(s; z) (ds; dz)
 !
 can be thought of as due to the in ation linkage of the underlying securities. Note, however,
 that contrary to the standard pricing formula, the expectation is not computed in the risk neutral
 probability measure, but in the statistical probability measure.
 Proof. If Xt and Ht are independent, then
 E[XtHt] = E[Xt]E[Ht], with
 E[Xt] = X0 +
 Z t
 0
  X(s)ds+ E
  Z t
 0
  X(s)dWs
  
+ E
  Z t
 0
 Z
 R
  X(s; z) (ds; dz)
  
= X0 +
 Z t
 0
  X(s)ds+ E
 "Z t
 0
 Z
 jzj R
  X(s; z) (ds; dz)
 #
 since
 E
  Z t
 0
  X(s)dWs
  
= E
 "Z t
 0
 Z
 jzj<R
  X(s; z)(   )(ds; dz)
 #
 = 0:
 Hence
 E[Xt] = X0 +
 Z t
 0
  X(s)ds+
 Z t
 0
 Z
 jzj R
  X(s; z) (ds; dz):
 Therefore
 H0 = e
  rt
 "
 1 +  0M0
  Z t
 0
  X(s)ds+
 Z t
 0
 Z
 jzj R
  X(s; z) (ds; dz)
 !#
 E[Ht]:
4.1 L evy process distribution 84
 4.1.2 Separable power utility function
 Given two non-negative constants A and B, a separable power utility function has the form
 U(x; y) =
 A
 p
 xp +
 B
 q
 yq;
 with p; q 2] 1; 1]nf0g.
 Assuming that the previous utility function is the agent?s utility function in the market, the CPI,
 the pricing kernel and the fair value of a contingent claim Ht are respectively [68]
 Cn =
  
A
 B
  1 q  nMn
 k(1 q)=(1 p)n
 ;
  n =
 B
 1
 1 q
 A
 q
 1 q
 k
 q
 1 q (1 p)
 n
   nMn
 ;
 H0 =
  0M0
 kq(1 p)=(1 q)0
 e  tjE
 "
 Hjk
 q(1 p)=(1 q)
 j
  jMj
 #
 ;
 where  is a Lagrange multiplier.
 Proposition 4.4. The dynamics of the CPI and the pricing kernel are given by
 dCt
 (A=B)1 q
 =
  
 1(t)
 kat
 +  Y (t) tMt  
aka 1t  k(t)
 k2at
 [  (t)Mt +  M (t) t]
 +
 Z
 R
  1(t; z) Y (t; z) (dz)
  
dt+
  
  (t)Mt +  M (t) t
 kat
  aka 1t  k(t)
  tMt
 k2at
  
dWt
 +
 Z
 R
  
 Y (t; z) t Mt +
  1(t; z)
 kat 
+  1(t; z) Y (t; z)
  
 (dt; dz);
 d t
 D
 =
 "
  2(t)
  tMt
 +  X(t)k
 b
 t  
bkb 1t  k(t)
  2tM
 2
 t
 [  (t)Mt +  M (t) t] +
 Z
 R
  2(t; z) X(t; z) (dz)
 #
 dt
 +
 "
 bkb 1t  k(t)
  tMt
  [  (t)Mt +  M (t) t]
 kbt
  2tM
 2
 t
 #
 dWt
 +
 Z
 R
  
 X(t; z)k
 b
 t +
  2(t; z)
  t Mt 
+  2(t; z) X(t; z)
  
 (dt; dz);
4.1 L evy process distribution 85
 where
 a =
 1 q
 1 p
 ; D =
 B
 1
 1 q
  A
 q
 1 q
 ; b =
 q(1 p)
 1 q
 ;
  1(t) =   (t)Mt +  M (t) t +   (t) M (t) +
 Z
 R
   (t; z) M (t; z) (dz);
  1(t; z) =  M (t; z) t +   (t; z)Mt +   (t; z) M (t; z);
  2(t; z) = [kt +  k(t; z)]
 a  kat ;
  2(t) = ak
 a 1
 t  k(t) +
 1
 2
 a(a 1) 2k(t)k
 a 2
 t +
 Z
 jzj<R
  
 2(t; z) ak
 a 1
 t  k(t; z)
  
 (dz);
  Y (t; z) =
 1
 kat +  2(t; z)
  
1
 kat 
;
  Y (t) =  
1
 k2at
  
 2(t) 
 
aka 1t  k(t)
  2 1
 kat
  
+
 Z
 jzj<R
  
 Y (t; z) +
 1
 k2at 
 2(t; z)
  
 (dz);
  X(t; z) =
 1
  t Mt +  1(t; z)
  
1
  t Mt 
;
  X(t) =  
1
  2tM
 2
 t
  
 1(t) (  (t)Mt +  M (t) t)
 2 1
  tMt
  
+
 Z
 jzj<R
  
 X(t; z) +
 1
  2t M
 2
 t 
 1(t; z)
  
 (dz):
 Proof. By the one-dimensional Ito^ formula (Theorem 2.7), considering
 Yt(a) = f(t; kt) = k
 a
 t
 with a 2 R.
 Because @f@t = 0,
 @f
 @kt
 = aka 1t and
 @2f
 @2kt
 = a(a 1)ka 2t ,
 dYt = ak
 a 1
 t [ k(t)dt+  k(t)dWt] +
 1
 2
 a(a 1) [ k(t)]
 2 ka 2t dt
 +
 Z
 jzj<R
  
[kt +  k(t; z)]
 a  kat  ak
 a 1
 t  k(t; z)
  
 (dt; dz)
 +
 Z
 R
  
[kt +  k(t; z)]
 a  kat 
 
 (dt; dz)
 =
  
aka 1t  k(t) +
 1
 2
 a(a 1) 2k(t)k
 a 2
 t
  
dt+ aka 1t  k(t)dWt
 +
 Z
 jzj<R
  
[kt +  k(t; z)]
 a  kat  ak
 a 1
 t  k(t; z)
  
 (dt; dz)
 +
 Z
 R
  
[kt +  k(t; z)]
 a  kat 
 
 (dt; dz):
 Applying Corollary 2.14 with Xt =
 Zt
 kat
 , a =
 1 q
 1 p
 and the dynamics of Zt given in the proof of
4.1 L evy process distribution 86
 Proposition 4.1, gives
 dCt
 (A=B)1 q
 =
  
 1(t)
 kat
 +  Y (t) tMt  
aka 1t  k(t)
 k2at
 [  (t)Mt +  M (t) t]
 +
 Z
 R
  1(t; z) Y (t; z) (dz)
  
dt+
  
  (t)Mt +  M (t) t
 kat
  aka 1t  k(t)
  tMt
 k2at
  
dWt
 +
 Z
 R
  
 Y (t; z) t Mt +
  1(t; z)
 kat 
+  1(t; z) Y (t; z)
  
 (dt; dz);
 where
  1(t) =   (t)Mt +  M (t) t +   (t) M (t) +
 Z
 R
   (t; z) M (t; z) (dz);
  1(t; z) =  M (t; z) t +   (t; z)Mt +   (t; z) M (t; z);
  2(t; z) = [kt +  k(t; z)]
 a  kat ;
  2(t) = ak
 a 1
 t  k(t) +
 1
 2
 a(a 1) 2k(t)k
 a 2
 t +
 Z
 jzj<R
  
 2(t; z) ak
 a 1
 t  k(t; z)
  
 (dz);
  Y (t; z) =
 1
 kat +  2(t; z)
  
1
 kat 
;
  Y (t) =  
1
 k2at
  
 2(t) 
 
aka 1t  k(t)
  2 1
 kat
  
+
 Z
 jzj<R
  
 Y (t; z) +
 1
 k2at 
 2(t; z)
  
 (dz):
 Applying again Corollary 2.14 with  t = D
 kat
 Zt
 , D =
 B
 1
 1 q
  A
 q
 1 q
 and a = q(1 p)1 q , yields
 d t
 D
 =
  
 1(t)
  tMt
 +  Y (t)k
 a
 t  
aka 1t  k(t)
  2tM
 2
 t
 [  (t)Mt +  M (t) t] +
 Z
 R
  1(t; z) Y (t; z) (dz)
  
dt
 +
  
aka 1t  k(t)
  tMt
  [  (t)Mt +  M (t) t]
 kat
  2tM
 2
 t
  
dWt
 +
 Z
 R
  
 Y (t; z)k
 a
 t +
  1(t; z)
  t Mt 
+  1(t; z) Y (t; z)
  
 (dt; dz);
 where
  1(t; z) = [kt +  k(t; z)]
 a  kat ;
  1(t) = ak
 a 1
 t  k(t) +
 1
 2
 a(a 1) 2k(t)k
 a 2
 t +
 Z
 jzj<R
  
 1(t; z) ak
 a 1
 t  k(t; z)
  
 (dz);
  2(t) =   (t)Mt +  M (t) t +   (t) M (t) +
 Z
 R
   (t; z) M (t; z) (dz);
  2(t; z) =  M (t; z) t +   (t; z)Mt +   (t; z) M (t; z);
  Y (t; z) =
 1
  t Mt +  2(t; z)
  
1
  t Mt 
;
  Y (t) =  
1
  2tM
 2
 t
  
 2(t) [  (t)Mt +  M (t) t]
 2 1
  tMt
  
+
 Z
 jzj<R
  
 Y (t; z) +
 1
  2t M
 2
 t 
 2(t; z)
  
 (dz):
4.1 L evy process distribution 87
 Proposition 4.5. The fair price at t = 0 of a contingent claim Ht is given by
 H0 = e
  rtE[Ht]
  
1 +
  0M0
 ka0
 Z t
 0
  X(s)ds
  
+
  0M0
 ka0
 e rtE
  
Ht
 Z t
 0
  X(s)dWs
  
+
  0M0
 ka0
 e rtE
  
Ht
 Z t
 0
 Z
 R
  X(s) (ds; dz)
  
;
 where
 a =
 q(1 p)
 1 q
 ;
  X(t) =
  1(t)
  tMt
 +  Y (t)k
 a
 t  
aka 1t  k(t)
  2tM
 2
 t
 [  (t)Mt +  M (t) t] +
 Z
 R
  1(t; z) Y (t; z) (dz);
  X(t) =
 aka 1t  k(t)
  tMt
  [  (t)Mt +  M (t) t]
 kat
  2tM
 2
 t
 ;
  X(t; z) =  Y (t; z)k
 a
 t +
  1(t; z)
  t Mt 
+  1(t; z) Y (t; z);
  Y (t; z) =
 1
  t Mt +  2(t; z)
  
1
  t Mt 
;
  Y (t) =  
1
  2tM
 2
 t
  
 2(t) [  (t)Mt +  M (t) t]
 2 1
  tMt
  
+
 Z
 jzj<R
  
 Y (t; z) +
 1
  2t M
 2
 t 
 2(t; z)
  
 (dz);
  2(t) =   (t)Mt +  M (t) t +    M +
 Z
 R
   (t; z) M (t; z) (dz);
  2(t; z) =  M (t; z) t +   (t; z)Mt +   (t; z) M (t; z);
  1(t; z) = [kt +  k(t; z)]
 a  kat ;
  1(t) = ak
 a 1
 t  k(t) +
 1
 2
 a(a 1) 2k(t)k
 a 2
 t +
 Z
 jzj<R
  
 1(t; z) ak
 a 1
 t  k(t; z)
  
 (dz):
 Proof. Similarly to  t in the previous proof, considering a =
 q(1 p)
 1 q and Xt =
 kat
  tMt
 , its dynamics
 are given by
 dXt =  X(t)dt+  X(t)dWt +
 Z
 R
  X(t; z) (dt; dz);
4.1 L evy process distribution 88
 where
  X(t) =
  1(t)
  tMt
 +  Y (t)k
 a
 t  
aka 1t  k(t)
  2tM
 2
 t
 [  (t)Mt +  M (t) t] +
 Z
 R
  1(t; z) Y (t; z) (dz);
  X(t) =
 aka 1t  k(t)
  tMt
  [  (t)Mt +  M (t) t]
 kat
  2tM
 2
 t
 ;
  X(t; z) =  Y (t; z)k
 a
 t +
  1(t; z)
  t Mt 
+  1(t; z) Y (t; z);
  Y (t) =  
1
  2tM
 2
 t
  
 2(t) [  (t)Mt +  M (t) t]
 2 1
  tMt
  
+
 Z
 jzj<R
  
 Y (t; z) +
 1
  2t M
 2
 t 
 2(t; z)
  
 (dz);
  Y (t; z) =
 1
  t Mt +  2(t; z)
  
1
  t Mt 
;
  2(t) =   (t)Mt +  M (t) t +    M +
 Z
 R
   (t; z) M (t; z) (dz);
  2(t; z) =  M (t; z) t +   (t; z)Mt +   (t; z) M (t; z);
  1(t) = ak
 a 1
 t  k(t) +
 1
 2
 a(a 1) 2k(t)k
 a 2
 t +
 Z
 jzj<R
  
 1(t; z) ak
 a 1
 t  k(t; z)
  
 (dz);
  1(t; z) = [kt +  k(t; z)]
 a  kat :
 A reasoning similar to the one used in the proof of Proposition 4.2 yields
 E[XtHt] = E
   Z t
 0
  X(s)ds+
 Z t
 0
  X(s)dWs +
 Z t
 0
 Z
 R
  X(s) (ds; dz)
  
Ht
  
+ E[X0Ht]:
 Therefore
 H0 = e
  rtE[Ht] +
  0M0
 ka0
 e rtE
  
Ht
  Z t
 0
  X(s)ds+
 Z t
 0
  X(s)dWs +
 Z t
 0
 Z
 R
  X(s) (ds; dz)
   
= e rtE[Ht] +
  0M0
 ka0
 e rtE
  
Ht
 Z t
 0
  X(s)ds
  
+
  0M0
 ka0
 e rtE
  
Ht
 Z t
 0
  X(s)dWs
  
+
  0M0
 ka0
 e rtE
  
Ht
 Z t
 0
 Z
 R
  X(s) (ds; dz)
  
= e rtE[Ht]
  
1 +
  0M0
 ka0
 Z t
 0
  X(s)ds
  
+
  0M0
 ka0
 e rtE
  
Ht
 Z t
 0
  X(s)dWs
  
+
  0M0
 ka0
 e rtE
  
Ht
 Z t
 0
 Z
 R
  X(s) (ds; dz)
  
:
 An additional independence assumption similar to that made in Proposition 4.3 yields a similar
 result.
 Proposition 4.6. If an additional assumption is made that
 kat
  tMt
 and Ht are independent (i.e. the
 pricing kernel is independent of the contingent claim), the fair price at t = 0 of the contingent claim
 becomes
 H0 = e
  rt
 "
 1 +
  0M0
 ka0
  Z t
 0
  X(s)ds+
 Z t
 0
 Z
 jzj R
  X(s; z) (ds; dz)
 !#
 E[Ht]:
4.1 L evy process distribution 89
 where
 a =
 q(1 p)
 1 q
 ;
  X(t) =
  1(t)
  tMt
 +  Y (t)k
 a
 t  
aka 1t  k(t)
  2tM
 2
 t
 [  (t)Mt +  M (t) t] +
 Z
 R
  1(t; z) Y (t; z) (dz);
  X(t) =
 aka 1t  k(t)
  tMt
  [  (t)Mt +  M (t) t]
 kat
  2tM
 2
 t
 ;
  X(t; z) =  Y (t; z)k
 a
 t +
  1(t; z)
  t Mt 
+  1(t; z) Y (t; z);
  Y (t) =  
1
  2tM
 2
 t
  
 2(t) [  (t)Mt +  M (t) t]
 2 1
  tMt
  
+
 Z
 jzj<R
  
 Y (t; z) +
 1
  2t M
 2
 t 
 2(t; z)
  
 (dz);
  Y (t; z) =
 1
  t Mt +  2(t; z)
  
1
  t Mt 
;
  2(t) =   (t)Mt +  M (t) t +    M +
 Z
 R
   (t; z) M (t; z) (dz);
  2(t; z) =  M (t; z) t +   (t; z)Mt +   (t; z) M (t; z);
  1(t) = ak
 a 1
 t  k(t) +
 1
 2
 a(a 1) 2k(t)k
 a 2
 t +
 Z
 jzj<R
  
 1(t; z) ak
 a 1
 t  k(t; z)
  
 (dz);
  1(t; z) = [kt +  k(t; z)]
 a  kat :
 The additional term due to the in ation linkage is now
 At = e
  rt 0M0
 ka0
  Z t
 0
  X(s)ds+
 Z t
 0
 Z
 jzj R
  X(s; z) (ds; dz)
 !
 :
 The fair price of a contingent claim Ht is of the form
 H0 = e
  rt (1 +At)E[Ht];
 where At is function of the agent?s utility function considered.
 Proof. Similar to that of Proposition 4.3.
 4.1.3 Arithmetic Brownian distribution
 Working in an environment without jumps and assuming that the nominal money supply, aggregate
 consumption and nominal liquidity bene t are normally distributed, i.e. Assumption 8 with  =
  =  M =   =  k =  = 0 gives the following propositions, which are just particular cases of the
 previous propositions.
 Proposition 4.7. Considering a log-separable utility function, the dynamics of Ct,  t and the fair
4.1 L evy process distribution 90
 price at t = 0 of a contingent claim Ht are
 dCt
 A=B
 =  C(t)dt+  C(t)dWt;
 d t
 B= 
=   (t)dt+   (t)dWt;
 H0 = e
  rt
  
1 +  0M0
 Z t
 0
  X(s)ds
  
E[Ht] +  0M0e rtE
  
Ht
 Z t
 0
  X(s)dWs
  
;
 where
  C(t) =
 1
 kt
 [  (t)Mt +  M (t) t +   (t) M (t)] +  Y (t) tMt  [  (t)Mt +  M (t) t]
  k(t)
 k2t
 ;
  C(t) = [  (t)Mt +  M (t) t]
 1
 kt
   k(t)
  tMt
 k2t
 ;
   (t) =
  
 
r
  tMt
  
1
  2tM
 2
 t
  
 2(t) (  (t)Mt +  M (t) t)
 2 1
  tMt
   
e rt;
   (t) =  [  (t)Mt +  M (t) t]
 e rt
  2tM
 2
 t
 ;
  X(t) =  
1
  2tM
 2
 t
  
  (t)Mt +  M (t) t +   (t) M (t) (  (t)Mt +  M (t) t)
 2 1
  tMt
  
;
  X(t) =  
1
  2tM
 2
 t
 [  (t)Mt +  M (t) t] ;
  Y (t) =  
1
 k2t
  
 k(t)  
2
 k(t)
 1
 kt
  
;
  2(t) =   (t)Mt +  M (t) t +   (t) M (t);
  2(t) =   (t)Mt +  M (t) t:
 Proposition 4.8. Considering a log-separable utility function and assuming that the pricing kernel
 is independent of the contingent claim, the fair price at t = 0 becomes
 H0 = e
  rt
  
1 +  0M0
 Z t
 0
  X(s)ds
  
E[Ht]:
 where
  X(t) =  
1
  2tM
 2
 t
  
  (t)Mt +  M (t) t +   (t) M (t) (  (t)Mt +  M (t) t)
 2 1
  tMt
  
:
4.1 L evy process distribution 91
 Proposition 4.9. Considering a separable power utility function, the dynamics of Ct,  t and the
 fair price at t = 0 of a contingent claim Ht are given by
 dCt
 (A=B)1 q
 =
  
 1(t)
 kat
 +  Y (t) tMt  
aka 1t  k(t)
 k2at
 [  (t)Mt +  M (t) t]
  
dt+
  
  (t)Mt +  M (t) t
 kat
  aka 1t  k(t)
  tMt
 k2at
  
dWt;
 d t
 D
 =
 "
  2(t)
  tMt
 +  X(t)k
 b
 t  
bkb 1t  k(t)
  2tM
 2
 t
 [  (t)Mt +  M (t) t]
 #
 dt
 +
 "
 bkb 1t  k(t)
  tMt
  [  (t)Mt +  M (t) t]
 kbt
  2tM
 2
 t
 #
 dWt;
 H0 = e
  rt
  
1 +
  0M0
 ka0
 Z t
 0
  Z(s)ds
  
E[Ht] +
  0M0
 ka0
 e rtE
  
Ht
 Z t
 0
  Z(s)dWs
  
;
 where
 a =
 1 q
 1 p
 ; D =
 B
 1
 1 q
  A
 q
 1 q
 ; b =
 q(1 p)
 1 q
 ;
  X(t) =  
1
  2tM
 2
 t
  
 1(t) [  (t)Mt +  M (t) t]
 2 1
  tMt
  
;
  Y (t) =  
1
 k2at
  
 2(t) 
 
aka 1t  k(t)
  2 1
 kat
  
;
  Z(t) =
 1
  tMt
  
bkb 1t  k(t) +
 1
 2
 b(b 1) 2k(t)k
 b 2
 t
  
 
bkb 1t  k(t)
  2tM
 2
 t
 [  (t)Mt +  M (t) t]
  
kbt
  2tM
 2
 t
  
  (t)Mt +  M (t) t +   (t) M (t) [  (t)Mt +  M (t) t]
 2 1
  tMt
  
;
  Z(t) =
 bkb 1t  k(t)
  tMt
  [  (t)Mt +  M (t) t]
 kbt
  2tM
 2
 t
 ;
  2(t) = ak
 a 1
 t  k(t) +
 1
 2
 a(a 1) 2k(t)k
 a 2
 t ;
  1(t) =   (t)Mt +  M (t) t +   (t) M (t):
 Proposition 4.10. Considering a separable separable power utility function and assuming that the
 pricing kernel is independent of the contingent claim, the fair price at t = 0 becomes
 H0 = e
  rt
  
1 +
  0M0
 ka0
 Z t
 0
  X(s)ds
  
E[Ht]:
 where
 a =
 1 q
 1 p
 ;
  X(t) =  
1
  2tM
 2
 t
  
 1(t) [  (t)Mt +  M (t) t]
 2 1
  tMt
  
:
4.2 Exponential L evy distribution 92
 4.2 Exponential L evy distribution
 This section is similar to Section 4.1, but instead of following L evy processes, the macroeconomic
 factors are assumed to be exponential L evy processes. Since the nominal money supply, the aggregate
 consumption and the nominal liquidity bene t are always positive, this assumption is appropriate.
 Moreover, formulas obtained under the assumption of exponential L evy distribution are generally
 highly tractable.
 The following assumption is made throughout this section.
 Assumption 9. Under the objective probability measure P, the dynamics of Mt, kt and  t for every
 t > 0 are given by:
 dMt
 Mt
 =  M (t)dt+  M (t)dW
 P
 t +
 Z
 R
  M (t; z) (dt; dz);
 dkt
 kt
 =  k(t)dt+  k(t)dW
 P
 t +
 Z
 R
  k(t; z) (dt; dz);
 d t
  t
 =   (t)dt+   (t)dW
 P
 t +
 Z
 R
   (t; z) (dt; dz);
 with the standard integrability conditions (See Assumption 8).
 4.2.1 Log-separable utility function
 Recall that if the agent utility function is a log-separable utility function, then the pricing kernel,
 the CPI and the fair price of a contingent claim are given by
 Ct =
 A
 B
  tMt
 kt
 ;
  t =
 Be rt
   tMt
 ;
 H0 =  0M0e
  rtE
  
Ht
  tMt
  
;
 where A and B are two non-negative constants that de ne the utility function.
4.2 Exponential L evy distribution 93
 Proposition 4.11. The dynamics of CPI and the pricing kernel are
 dCt
 Ct 
=
  
  (t) +  M (t)  k(t) +   (t) M (t) +
 Z
 R
   (t; z) M (t; z) (dz)
 + 2k(t) +
 Z
 jzj<R
  2k(t; z)
 1 +  k(t; z)
  (dz) [  (t) +  M (t)] k(t)
 )
 dt
 +
 Z
 R
 [  (t; z) +  M (t; z) +   (t; z) M (t; z)]
  k(t; z)
 1 +  k(t; z)
  (dt; dz)
 + [  (t) +  M (t)  k(t)] dWt
 +
 Z
 R
  
  (t; z) +  M (t; z) +   (t; z) M (t; z) +
  k(t; z)
 1 +  k(t; z)
 + [  (t; z) +  M (t; z) +   (t; z) M (t; z)]
  k(t; z)
 1 +  k(t; z)
  
 (dt; dz);
 d t
  t 
=
  
 r    (t)  M (t)   (t) M (t) 
Z
 R
   (t; z) M (t; z) (dz) + [  (t) +  M (t)]
 2
 +
 Z
 jzj<R
 [  (t; z) +  M (t; z) +   (t; z) M (t; z)]
 2
 1 +   (t; z) +  M (t; z) +   (t; z) M (t; z)
  (dz)
 #
 dt [  (t) +  M (t)] dWt
 +
 Z
 R
  
  (t; z) +  M (t; z) +   (t; z) M (t; z)
 1 +   (t; z) +  M (t; z) +   (t; z) M (t; z)
  
 (dt; dz):
 Proof. Applying Corollary 2.13 with Zt =  tMt
 dZt
 Zt 
=
  
  (t) +  M (t) +   (t) M (t) +
 Z
 R
   (t; z) M (t; z) (dz)
  
dt+ [  (t) +  M (t)] dWt
 +
 Z
 R
 [  (t; z) +  M (t; z) +   (t; z) M (t; z)]  (dt; dz):
 Then applying Corollary 2.15
 dCt
 Ct 
=
  
  (t) +  M (t) +   (t) M (t) +
 Z
 R
   (t; z) M (t; z) (dz)  k(t)
 + 2k(t) +
 Z
 jzj<R
  2k(t; z)
 1 +  k(t; z)
  (dz) [  (t) +  M (t)] k(t)
 )
 dt
 +
 Z
 R
 [  (t; z) +  M (t; z) +   (t; z) M (t; z)]
  k(t; z)
 1 +  k(t; z)
  (dt; dz)
 + [  (t) +  M (t)  k(t)] dWt
 +
 Z
 R
  
  (t; z) +  M (t; z) +   (t; z) M (t; z) +
  k(t; z)
 1 +  k(t; z)
 + [  (t; z) +  M (t; z) +   (t; z) M (t; z)]
  k(t; z)
 1 +  k(t; z)
  
 (dt; dz):
 The deterministic process Xt = e rt has dynamics
 dXt
 Xt 
=  rdt:
4.2 Exponential L evy distribution 94
 Applying again Corollary 2.15
 d t
  t 
=
  
 r    (t)  M (t)   (t) M (t) 
Z
 R
   (t; z) M (t; z) (dz) + [  (t) +  M (t)]
 2
 +
 Z
 jzj<R
 [  (t; z) +  M (t; z) +   (t; z) M (t; z)]
 2
 1 +   (t; z) +  M (t; z) +   (t; z) M (t; z)
  (dz)
 #
 dt [  (t) +  M (t)] dWt
 +
 Z
 R
  
  (t; z) +  M (t; z) +   (t; z) M (t; z)
 1 +   (t; z) +  M (t; z) +   (t; z) M (t; z)
  
 (dt; dz):
 Proposition 4.12. The fair price at t = 0 of a contingent claim Ht is given by
 H0 = e
  rtE
  
Ht exp
  Z t
 0
  X(s)ds+
 Z t
 0
  X(s)dWs +
 Z t
 0
 Z
 R
  X(s; z) (ds; dz)
   
;
 where
  X(t) =    (t)  M (t)   (t) M (t) 
Z
 R
   (t; z) M (t; z) (dz) + [  (t) +  M (t)]
 2
 +
 Z
 jzj<R
  2(t; z)
 1 +  (t; z)
  (dz);
  X(t) =    (t)  M (t);
  X(t; z) =  
 (t; z)
 1 +  (t; z)
 ;
  (t; z) =   (t; z) +  M (t; z) +   (t; z) M (t; z):
 Proof. Using the dynamics of Zt (see previous proof) in Corollary 2.9 with Xt =
 1
  tMt
 yields
 dXt
 Xt 
=
  
   (t)  M (t)   (t) M (t) 
Z
 R
   (t; z) M (t; z) (dz) + [  (t) +  M (t)]
 2
 +
 Z
 jzj<R
  2(t; z)
 1 +  (t; z)
  (dz)
 )
 dt [  (t) +  M (t)] dWt  
Z
 R
  (t; z)
 1 +  (t; z)
  (dt; dz)
 =  X(t)dt+  X(t)dWt +
 Z
 R
  X(t; z) (dt; dz)
 where
  X(t) =    (t)  M (t)   (t) M (t) 
Z
 R
   (t; z) M (t; z) (dz)
 + [  (t) +  M (t)]
 2 +
 Z
 jzj<R
  2(t; z)
 1 +  (t; z)
  (dz);
  X(t) =    (t)  M (t);
  X(t; z) =  
 (t; z)
 1 +  (t; z)
 ;
  (t; z) =   (t; z) +  M (t; z) +   (t; z) M (t; z):
4.2 Exponential L evy distribution 95
 Hence
 Xt = X0 exp
  Z t
 0
  X(s)ds+
 Z t
 0
  X(s)dWs +
 Z t
 0
 Z
 R
  X(s; z) (ds; dz)
  
and
 E[XtHt] = E
  
Ht
  0M0
 exp
  Z t
 0
  X(s)ds+
 Z t
 0
  X(s)dWs +
 Z t
 0
 Z
 R
  X(s; z) (ds; dz)
   
:
 Therefore
 H0 = e
  rtE
  
Ht exp
  Z t
 0
  X(s)ds+
 Z t
 0
  X(s)dWs +
 Z t
 0
 Z
 R
  X(s; z) (ds; dz)
   
:
 Proposition 4.13. Further, assuming that 1 tMt and Ht are independent (i.e. the pricing kernel is
 independent of the priced security), the fair price at t = 0 of the contingent claim becomes
 H0 = e
  rt exp
  Z t
 0
  X(u)du 
1
 2
 Z t
 0
  2X(u)du+
 Z t
 0
 Z
 R
  
ez  1 z jzj<R
  
  X (dz)du
  
E[Ht];
 where
  X(t) =    (t)  M (t)   (t) M (t) 
Z
 R
   (t; z) M (t; z) (dz) + [  (t) +  M (t)]
 2
 +
 Z
 jzj<R
  2(t; z)
 1 +  (t; z)
  (dz);
  X(t) =    (t)  M (t);
  X(t; z) =  
 (t; z)
 1 +  (t; z)
 ;
  (t; z) =   (t; z) +  M (t; z) +   (t; z) M (t; z):
 Recall that the standard no-arbitrage price of a derivative is e rtE[Ht], the term
 Dt = exp
  Z t
 0
  X(u)du 
1
 2
 Z t
 0
  2X(u)du+
 Z t
 0
 Z
 R
  
ez  1 z jzj<R
  
  X (dz)du
  
in the previous equation can be interpreted as a correction factor due to in ation linkage. Note that
 the standard expectation is computed under the risk neutral probability measure while here it is
 computed under the statistical (i.e. real world) probability measure.
 Proof. If Xt and Ht are independent, then
 E[XtHt] = E[Xt]E[Ht], with
4.2 Exponential L evy distribution 96
 E[Xt] = X0 exp
  Z t
 0
  X(s)ds
  
E
  
exp
  Z t
 0
  X(s)dWs +
 Z t
 0
 Z
 R
  X(s; z) (ds; dz)
   
= X0 exp
  Z t
 0
  X(s)ds
  
E
  
exp
  Z t
 0
  X(s)dWs
   
E
  
exp
  Z t
 0
 Z
 R
  X(s; z) (ds; dz)
   
since dWt and  (dt; dz) are independent.
 E
  
exp
  Z t
 0
 Z
 R
  X(u; z) (du; dz)
   
= E
 "
 exp
  Z t
 0
 Z
 jzj<R
  X(u; z)(   )(du; dz)
 +
 Z t
 0
 Z
 jzj R
  X(u; z) (du; dz)
 !#
 = E
 "
 exp
  Z t
 0
 Z
 jzj<R
  X(u; z)(   )(du; dz)
 !#
 E
 "
 exp
  Z t
 0
 Z
 jzj R
  X(u; z) (du; dz)
 !#
 since fjzj < Rg and fjzj  Rg are disjoint.
 E
  
exp
  Z t
 0
 Z
 R
  X(u; z) (du; dz)
   
= exp
 (Z t
 0
 Z
 jzj<R
 [ez  1 z]  X (dz)du
 )
 exp
 "Z t
 0
 Z
 jzj R
 (ez  1)  X (dz)du
 #
 = exp
  Z t
 0
 Z
 R
  
ez  1 z jzj<R
  
  X (dz)du
  
by Corollary 2.21 with   X =   ( X)
  1.
 Recall that dWt  N (0; dt), hence
 Z t
 0
  X(u)dWu  N
  
0;
 Z t
 0
  2X(u)du
  
. Let h ;  i denote the
 Euclidean distance, i.e. for x; y 2 Rd such as x = (x1; x2;    ; xd) and y = (y1; y2;    ; yd), hx; yi =
 dX
 i=1
 xiyi. Let  N (m;V ) denote the log-characteristic function of a normal distributed process with
 mean m and variance V , de ned by
  N (m;V )(u) = i hm;ui  
1
 2
 hu; V ui
 with m;u 2 Rd. In particular
  N (m;V )(u) = imu 
1
 2
 V u2 with m;u 2 R:
 Using Theorem 2.19
 E
  
exp
  Z t
 0
  X(u)dWu
   
= exp
  
 
1
 2
 Z t
 0
  2X(u)du
  
:
 Combining these expectations yields
 E[Xt] = X0 exp
  Z t
 0
  X(u)du 
1
 2
 Z t
 0
  2X(u)du+
 Z t
 0
 Z
 R
  
ez  1 z jzj<R
  
  X (dz)du
  
:
4.2 Exponential L evy distribution 97
 Therefore
 H0 = e
  rt exp
  Z t
 0
  X(u)du 
1
 2
 Z t
 0
  2X(u)du+
 Z t
 0
 Z
 R
  
ez  1 z jzj<R
  
  X (dz)du
  
E[Ht]:
 For simplicity, the parameter estimation will always assume  (t; z) to be deterministic. In this case,
 the previous expression becomes more tractable as shown in the next proposition.
 Proposition 4.14. If  (t; z) is deterministic and 1 tMt and Ht are independent, the fair price at
 t = 0 of the contingent claim is
 H0 = e
  rt exp
  Z t
 0
  X(u)du 
1
 2
 Z t
 0
  2X(u)du+
 Z t
 0
 Z
 R
 h
 e X(t;z)  1  X(t; z) jzj<R
 i
  (dz)du
  
E[Ht];
 where the coe cients are given in Proposition 4.13.
 4.2.2 Power utility function
 Recall that if the agent utility function is a separable power utility function, the CPI, the pricing
 kernel and the fair price of an IL security are respectively
 Ct =
  
A
 B
  1 q  tMt
 k(1 q)=(1 p)t
 ;
  t =
 B
 1
 1 q
 A
 q
 1 q
 k
 q
 1 q (1 p)
 t
   tMt
 ;
 H0 =
  0M0
 kq(1 p)=(1 q)0
 e rtE
 "
 Htk
 q(1 p)=(1 q)
 t
  tMt
 #
 ;
 where A and B are two non-negative constants; p; q 2] 1; 1]nf0g de ning the utility function.
 The next proposition computes the dynamics of the CPI and the pricing kernel assuming that the
 macroeconomic factors are exponential L evy processes.
 Proposition 4.15. The dynamics of Ct and  t are given by
 dCt
 Ct 
=  C(t)dt+  C(t)dWt +
 Z
 R
  C(t; z) (dt; dz);
 d t
  t 
=   (t)dt+   (t)dWt +
 Z
 R
   (t; z) (dt; dz);
4.2 Exponential L evy distribution 98
 where
  C(t) =   (t) +  M (t) +   (t) M (t) +
 Z
 R
   (t; z) M (t; z) (dz) + a
 2 k(t)
 2
  a k(t) 
1
 2
 a(a 1) 2k(t) 
Z
 jzj<R
 f[1 +  k(t; z)]
 a  1 a k(t; z)g  (dz)
 +
 Z
 jzj<R
  22(t; z)
 1 +  2(t; z)
  (dz) a [  (t) +  M (t)] k(t) +
 Z
 R
  1(t; z)
  2(t; z)
 1 +  2(t; z)
  (dz);
  C(t) =   (t) +  M (t) a k(t);
  C(t; z) =  2(t; z) +
  1(t; z)
 1 +  1(t; z)
 +  2(t; z)
  1(t; z)
 1 +  1(t; z)
 ;
   (t) =  1(t)  2(t) + [  (t) +  M (t)]
 2 +
 Z
 jzj<R
  22(t; z)
 1 +  2(t; z)
  (dz)
  a k(t) [  (t) +  M (t)] +
 Z
 R
 f[1 +  k(t; z)]
 a  1g
  2(t; z)
 1 +  2(t; z)
  (dz);
   (t) = a k(t)   (t)  M (t);
   (t; z) = [1 +  k(t; z)]
 a  1 +
  2(t; z)
 1 +  2(t; z)
 + f[1 +  k(t; z)]
 a  1g
  2(t; z)
 1 +  2(t; z)
 ;
  1(t) = a k(t) +
 1
 2
 a(a 1) 2k(t) +
 Z
 jzj<R
 f[1 +  k(t; z)]
 a  1 a k(t; z)g  (dz);
  2(t) =   (t) +  M (t) +   (t) M (t) +
 Z
 R
   (t; z) M (t; z) (dz);
  1(t; z) = [1 +  k(t; z)]
 a  1;
  2(t; z) =   (t; z) +  M (t; z) +   (t; z) M (t; z):
 Proof. Applying the one-dimensional Ito^ formula (Theorem 2.7) with Yt(a) = f(t; kt) = kat for
 a 2 R. We have
 @f
 @t
 = 0,
 @f
 @kt
 = aka 1t and
 @2f
 @2kt
 = a(a 1)ka 2t ,
 dYt = ak
 a 1
 t [ k(t)kt dt+  k(t)kt dWt] +
 1
 2
 a(a 1) [ k(t)kt ]
 2 ka 2t dt
 +
 Z
 jzj<R
  
[kt +  k(t; z)kt ]
 a  kat  ak
 a 1
 t  k(t; z)kt 
 
 (dt; dz)
 +
 Z
 R
  
[kt +  k(t; z)kt ]
 a  kat 
 
 (dt; dz)
 = a [ k(t)dt+  k(t)dWt] +
 1
 2
 a(a 1) 2k(t)dt+
 Z
 jzj<R
 f[1 +  k(t; z)]
 a  1 a k(t; z)g (dt; dz)
 +
 Z
 R
 f[1 +  k(t; z)]
 a  1g  (dt; dz)
 = a
  
 k(t) +
 1
 2
 (a 1) 2k(t)
  
dt+ a k(t)dWt +
 Z
 jzj<R
 f[1 +  k(t; z)]
 a  1 a k(t; z)g  (dz)dt
 +
 Z
 R
 f[1 +  k(t; z)]
 a  1g  (dt; dz):
4.2 Exponential L evy distribution 99
 Hence
 dYt
 Yt 
=
 (
 a k(t) +
 1
 2
 a(a 1) 2k(t) +
 Z
 jzj<R
 f[1 +  k(t; z)]
 a  1 a k(t; z)g  (dz)
 )
 dt
 +a k(t)dWt +
 Z
 R
 f[1 +  k(t; z)]
 a  1g  (dt; dz):
 Applying Corollary 2.15 with Xt =
 Zt
 kat
 and a =
 1 q
 1 p
 (Zt from the proof of Proposition 4.11)
 dCt
 Ct 
=
  
  (t) +  M (t) +   (t) M (t) +
 Z
 R
   (t; z) M (t; z) (dz) + a
 2 k(t)
 2
  a k(t) 
1
 2
 a(a 1) 2k(t) 
Z
 jzj<R
 f[1 +  k(t; z)]
 a  1 a k(t; z)g  (dz)
 +
 Z
 jzj<R
  22(t; z)
 1 +  2(t; z)
  (dz) a [  (t) +  M (t)] k(t)
 )
 dt
 +
 Z
 R
  1(t; z)
  2(t; z)
 1 +  2(t; z)
  (dt; dz) + [  (t) +  M (t) a k(t)] dWt
 +
 Z
 R
  
 1(t; z) +
  2(t; z)
 1 +  2(t; z)
 +  1(t; z)
  2(t; z)
 1 +  2(t; z)
  
 (dt; dz)
 where
  1(t; z) =   (t; z) +  M (t; z) +   (t; z) M (t; z);
  2(t; z) = [1 +  k(t; z)]
 a  1:
 Again applying Corollary 2.15 with  t =
 B
 1
 1 q
  A
 q
 1 q
 kat
 Zt
 and a = q(1 p)1 q yields
 d t
  t 
=   (t)dt+   (t)dWt +
 Z
 R
   (t; z) (dt; dz);
 where
   (t) =  1(t)  2(t) + [  (t) +  M (t)]
 2 +
 Z
 jzj<R
  22(t; z)
 1 +  2(t; z)
  (dz)
  a k(t) [  (t) +  M (t)] +
 Z
 R
 f[1 +  k(t; z)]
 a  1g
  2(t; z)
 1 +  2(t; z)
  (dz);
   (t) = a k(t)   (t)  M (t);
   (t; z) = [1 +  k(t; z)]
 a  1 +
  2(t; z)
 1 +  2(t; z)
 + f[1 +  k(t; z)]
 a  1g
  2(t; z)
 1 +  2(t; z)
 ;
  1(t) = a k(t) +
 1
 2
 a(a 1) 2k(t) +
 Z
 jzj<R
 f[1 +  k(t; z)]
 a  1 a k(t; z)g  (dz);
  2(t) =   (t) +  M (t) +   (t) M (t) +
 Z
 R
   (t; z) M (t; z) (dz);
  2(t; z) =   (t; z) +  M (t; z) +   (t; z) M (t; z):
4.2 Exponential L evy distribution 100
 The next proposition computes the fair price of an Il derivative under the current assumptions.
 Proposition 4.16. The fair price at t = 0 of a contingent claim Ht is
 H0 = e
  rtE
  
Ht exp
  Z t
 0
  X(s)ds+
 Z t
 0
  X(s)dWs +
 Z t
 0
 Z
 R
  X(s; z) (ds; dz)
   
;
 where
  X(t) =  1(t)  2(t) + [  (t) +  M (t)]
 2 +
 Z
 jzj<R
  22(t; z)
 1 +  2(t; z)
  (dz)
  a k(t) [  (t) +  M (t)] +
 Z
 R
 f[1 +  k(t; z)]
 a  1g
  2(t; z)
 1 +  2(t; z)
  (dz);
  X(t) = a k(t)   (t)  M (t);
  X(t; z) = [1 +  k(t; z)]
 a  1 +
  2(t; z)
 1 +  2(t; z)
 + f[1 +  k(t; z)]
 a  1g
  2(t; z)
 1 +  2(t; z)
 ;
  1(t) = a k(t) +
 1
 2
 a(a 1) 2k(t) +
 Z
 jzj<R
 f[1 +  k(t; z)]
 a  1 a k(t; z)g  (dz);
  2(t) =   (t) +  M (t) +   (t) M (t) +
 Z
 R
   (t; z) M (t; z) (dz);
  2(t; z) =   (t; z) +  M (t; z) +   (t; z) M (t; z):
 Proof. Similarly to  t in the previous proof, the process de ned byXt =
 kat
  tMt
 with a =
 q(1 p)
 1 q
 has dynamics
 dXt
 Xt 
=  X(t)dt+  X(t)dWt +
 Z
 R
  X(t; z) (dt; dz);
 where
  X(t) =  1(t)  2(t) + [  (t) +  M (t)]
 2 +
 Z
 jzj<R
  22(t; z)
 1 +  2(t; z)
  (dz)
  a k(t) [  (t) +  M (t)] +
 Z
 R
 f[1 +  k(t; z)]
 a  1g
  2(t; z)
 1 +  2(t; z)
  (dz);
  X(t) = a k(t)   (t)  M (t);
  X(t; z) = [1 +  k(t; z)]
 a  1 +
  2(t; z)
 1 +  2(t; z)
 + f[1 +  k(t; z)]
 a  1g
  2(t; z)
 1 +  2(t; z)
 ;
  1(t) = a k(t) +
 1
 2
 a(a 1) 2k(t) +
 Z
 jzj<R
 f[1 +  k(t; z)]
 a  1 a k(t; z)g  (dz);
  2(t) =   (t) +  M (t) +   (t) M (t) +
 Z
 R
   (t; z) M (t; z) (dz);
  2(t; z) =   (t; z) +  M (t; z) +   (t; z) M (t; z):
 Thus
 H0 = e
  rtE
  
Ht exp
  Z t
 0
  X(s)ds+
 Z t
 0
  X(s)dWs +
 Z t
 0
 Z
 R
  X(s; z) (ds; dz)
   
:
4.2 Exponential L evy distribution 101
 When the pricing kernel is independent of the derivative security, a mapping can be done between
 IL securities and nominal securities as shown in the next proposition.
 Proposition 4.17. Further, assuming that k
 a
 t
  tMt
 and Ht are independent (i.e. the pricing kernel is
 independent of the contingent claim), the fair price at t = 0 of a contingent claim Ht is
 H0 = e
  rt exp
  Z t
 0
  X(u)du 
1
 2
 Z t
 0
  2X(u)du+
 Z t
 0
 Z
 R
  
ez  1 z jzj<R
  
  X (dz)du
  
E[Ht];
 where
  X(t) =  1(t)  2(t) + [  (t) +  M (t)]
 2 +
 Z
 jzj<R
  22(t; z)
 1 +  2(t; z)
  (dz)
  a k(t) [  (t) +  M (t)] +
 Z
 R
 f[1 +  k(t; z)]
 a  1g
  2(t; z)
 1 +  2(t; z)
  (dz);
  X(t) = a k(t)   (t)  M (t);
  X(t; z) = [1 +  k(t; z)]
 a  1 +
  2(t; z)
 1 +  2(t; z)
 + f[1 +  k(t; z)]
 a  1g
  2(t; z)
 1 +  2(t; z)
 ;
  1(t) = a k(t) +
 1
 2
 a(a 1) 2k(t) +
 Z
 jzj<R
 f[1 +  k(t; z)]
 a  1 a k(t; z)g  (dz);
  2(t) =   (t) +  M (t) +   (t) M (t) +
 Z
 R
   (t; z) M (t; z) (dz);
  2(t; z) =   (t; z) +  M (t; z) +   (t; z) M (t; z):
 The correction factor due to in ation linkage is
 Dt = exp
  Z t
 0
  X(u)du 
1
 2
 Z t
 0
  2X(u)du+
 Z t
 0
 Z
 R
  
ez  1 z jzj<R
  
  X (dz)du
  
:
 Note that the correction factor has the same form as with the log-separable utility function and can
 readily be written as a function of Nt =
 1
  t
 which is the standard num eraire in Finance.
 Proof. Similar to that of Proposition 4.13.
 Proposition 4.18. If  (t; z) is deterministic and k
 a
 t
  tMt
 and Ht are independent, the fair price at
 t = 0 of a contingent claim Ht is
 H0 = e
  rt exp
  Z t
 0
  X(u)du 
1
 2
 Z t
 0
  2X(u)du+
 Z t
 0
 Z
 R
 h
 e X(t;z)  1  X(t; z) jzj<R
 i
  (dz)du
  
E[Ht];
 where the coe cients are given in Proposition 4.17.
 4.2.3 Geometric Brownian distribution
 Working in an environment without jumps and assuming that the nominal money supply, aggregate
 consumption and nominal liquidity bene t follow a geometric distribution, i.e. Assumption 9 with
  =  =  M =   =  k =  = 0 gives the following results.
4.2 Exponential L evy distribution 102
 Proposition 4.19. Considering a log-separable utility function, the dynamics of Ct,  t and the fair
 price at t = 0 of a contingent claim Ht are
 dCt
 Ct 
=
  
  (t) +  M (t)  k(t) +   (t) M (t) +  
2
 k(t) [  (t) +  M (t)] k(t)
  
dt
 + [  (t) +  M (t)  k(t)] dWt;
 d t
  t 
=
 n
  r    (t)  M (t)   (t) M (t) + [  (t) +  M (t)]
 2
 o
 dt
  [  (t) +  M (t)] dWt;
 H0 = e
  rtE
  
Ht exp
  Z t
 0
  X(s)ds+
 Z t
 0
  X(s)dWs
   
;
 where
  X(t) =    (t)  M (t)   (t) M (t) + [  (t) +  M (t)]
 2 ;
  X(t) =    (t)  M (t):
 Proposition 4.20. Considering a log-separable utility function and assuming that the pricing kernel
 is independent of the contingent claims, the fair price at t = 0 becomes
 H0 = e
  rt exp
  Z t
 0
  X(u)du 
1
 2
 Z t
 0
  2X(u)du
  
E[Ht];
 where
  X(t) =    (t)  M (t)   (t) M (t) + [  (t) +  M (t)]
 2 ;
  X(t) =    (t)  M (t):
 Proposition 4.21. Considering a separable power utility function, the dynamics of Ct,  t and the
 fair price at t = 0 of a contingent claim Ht are
 dCt
 Ct 
=  C(t)dt+  C(t)dWt;
 d t
  t 
=   (t)dt+   (t)dWt;
 H0 = e
  rtE
  
Ht exp
  Z t
 0
  X(s)ds+
 Z t
 0
  X(s)dWs
   
;
 where
  C(t) =   (t) +  M (t) +   (t) M (t) a k(t) 
1
 2
 a(a 1) 2k(t) a [  (t) +  M (t)] k(t);
  C(t) =   (t) +  M (t) a k(t);
4.3 Conclusion 103
  X(t) =  1(t)  2(t) + [  (t) +  M (t)]
 2  a k(t) [  (t) +  M (t)] ;
  X(t) = a k(t)   (t)  M (t);
   (t) =  1(t)  2(t) + [  (t) +  M (t)]
 2  a k(t) [  (t) +  M (t)] ;
   (t) = a k(t)   (t)  M (t);
  1(t) = a k(t) +
 1
 2
 a(a 1) 2k(t);
  2(t) =   (t) +  M (t) +   (t) M (t):
 Proposition 4.22. Considering a separable power utility function and assuming that the pricing
 kernel is independent of the contingent claims the fair price at t = 0 becomes
 H0 = e
  rt exp
  Z t
 0
  X(u)du 
1
 2
 Z t
 0
  2X(u)du
  
E[Ht]
 where
  X(t) =  1(t)  2(t) + [  (t) +  M (t)]
 2  a k(t) [  (t) +  M (t)] ;
  X(t) = a k(t)   (t)  M (t);
  1(t) = a k(t) +
 1
 2
 a(a 1) 2k(t);
  2(t) =   (t) +  M (t) +   (t) M (t):
 4.3 Conclusion
 In this chapter, pricing formulas for IL securities were successfully derived. As initially expected,
 these formulas are mainly function of the selected macroeconomic factors and less function of the
 observed market prices. One of the main advantage of this approach is that there is a unique pricing
 formula for \every" IL derivatives. This fair price is not a function of the actual security being
 priced and in theory might be applied even to customised and exotic IL securities.
 The calibration process has been started in Chapter 6. However, it was not pushed as far as planned
 because of the unavailability of the necessary market data. The preliminary study conducted justi es
 our choice of L evy distributions as the driving factor and source of randomness of all processes.
Chapter 5
 Reverse Engineering
 In this chapter, the real and nominal pricing kernels are modelled without the use of utility func-
 tions. In 1994, Backus and Zin [8] developed the methodology of \Reverse engineering the yield
 curve" with application to the nominal yield curve. In 2003, Craig and Haubrich [36] conducted
 an implementation with the real term structure. Both studies were made in discrete time with un-
 derlying AR and ARMA processes; whereas the framework built here is in continuous time and its
 underlying processes follow L evy and exponential L evy distributions. The pricing kernels? dynamics
 are deduced from the market observed yield curves and the in ation. The obtained estimates of the
 nominal and real pricing kernels (See Section 5.5) re ect their relatively complex dynamics and more
 importantly the existing interaction between nominal economy and real economy. This provides a
 new perspective on how the market yield curves reveals nominal and real in uences.
 5.1 In ation Breakeven Rate
 Despite the fact that the approach presented in this chapter can be applied to any contingent claim,
 the current study only focuses on bonds (nominal and in ation linked). This sections brie y reviews
 some standard concepts related to bonds in general and others speci c to in ation linked (IL) bonds.
 Among the latter is the less known in ation breakeven rate or in ation compensation, which is the
 di erence between the yield of nominal bonds and the real yield obtained from IL bonds with
 same maturity. It has been widely used as a proxy of in ation expectation because of the Fischer
 hypothesis that is recalled later.
 104
5.1 In ation Breakeven Rate 105
 From now on, the subscript/exponent n (resp. r and IP ) stands for nominal (resp. real and in ation
 protected).
 For a \smooth" yield curve to be deductible from the market bonds, the following initial assumption
 is made.
 Assumption 10. There exists a market for nominal and IL bonds for all maturities T > 0 or at
 least for T  > T > 0 with T   xed. Furthermore, for every  xed t 2 [0; T ], pn(t; T ) and pIP (t; T )
 are di erentiable with respect to the maturity T .
 The instantaneous forward rates, contracted at time t are de ned by
 fi(t; T ) =  
@ ln pi(t; T )
 @T
 for i = r; n;
 and the instantaneous interest rates
 ri(t) = fi(t; t) for i = r; n:
 The continuously compounded yields yi for i = n; r are de ned by
 pi(t; T ) = exp [ yi(t; T )  (T  t)] :
 Therefore
 yi(t; T ) =  
ln pi(t; T )
 T  t
 :
 The continuously compounded nominal yield can also be computed from the real bonds by [57]
 yn(t; T ) =  
1
 T  t
 ln
 pr(t; T )I(t)
 I(T )
 =  
1
 T  t
 ln
 pIP (t; T )
 I(T )
 :
 The previous equation can be decomposed as follow
 yn(t; T ) =  
1
 T  t
  
ln pr(t; T ) + ln
 I(t)
 I(T )
  
= yr(t; T ) +
 1
 T  t
 ln
 I(T )
 I(t)
 :
 In the last equation, all the components except the I(T ) are known at time t; thus the last term is
 random at time t and its expected value will be used instead. The expression
 ie(t; T ) =
 1
 T  t
 Et
  
ln
 I(T )
 I(t)
  
5.1 In ation Breakeven Rate 106
 is the standard expectation of the average in ation between t and T . Hence, the Fisher hypothesis
 is retrieved
 yn(t; T ) = yr(t; T ) + i
 e(t; T ):
 Another relationship between the nominal and real yields can be computed using the nominal and
 real pricing kernels. Let  i = ( i(t))t 0 with i = n; r denote the pricing kernels; the shorthand
 notation  it for  
i(t) will also be used.
 Considering t < T , the real pricing kernel is de ned by [68, 37]
 pr(t; T ) =
 Et [ rT pr(T; T )]
  rt
 ;
 where Et denotes the expectation conditional on the information available at time t.
 Considering  > 0 and s = t+  such that t < s < T , then
 pr(t; T ) =
 Et [ rspr(s; T )]
  rt
 : (5.1)
 However, real bond prices are not directly observable on the market. The only way to ensure
 purchasing power is by taking a position in in ation protected securities (here IL bonds). At time
 t, the real and in ation protected bonds with maturity T are related by
 pr(t; T ) =
 pIP (t; T )
 I(t)
 ;
 where I(t) is the in ation at time t. Substituting pr(t; T ) in equation (5.1), gives
 pIP (t; T ) =
 Et
  
 rspIP (s; T )
 It
 Is
  
 rt
 =
 Et
  
 rspIP (s; T )
 1
 Gs
  
 rt
 ; (5.2)
 where Gs =
 Is
 Is  
is the gross in ation return over [s  ; s].
 Since the nominal pricing kernel  n is also given by
 pIP (t; T ) =
 Et [ ns pIP (s; T )]
  nt
 ; (5.3)
 because pIP (t; T ) is a nominal contingent claim. Identi cation between the last two equations breaks
 the nominal pricing kernel into the real pricing kernel and another component1 due to in ation. The
 1The term in ation is not directly used because of Equation (5.9) that will be used instead in our pricing framework.
5.1 In ation Breakeven Rate 107
 obtained no-arbitrage relationship between the pricing kernels is
  ns =
  rs
 Is
 (5.4)
 This relationship is model independent (i.e. satis ed without any assumption on the dynamics of
 the pricing kernels and in ation) and has been stated in [37]. Combined with the in ation dynamics,
 Equation (5.1) enables the partition of the nominal pricing kernel into real and nominal components
 which can both be estimated.
 A relationship similar to Equation (5.1) exists between the nominal bonds and the nominal pricing
 kernel
 pn(t; T ) =
 Et [ ns pn(s; T )]
  nt
 : (5.5)
 Using Equation (5.4), the previous expression can be rewritten has
 pn(t; T ) = Et
  
 rs
  rt
 1
 Gs
 pn(s; T )
  
:
 Taking s = T yields
 pn(t; s) = Et
  
 rs
  rt
 1
 Gs
  
= Et
  
 rs
  rt
  
Et
  
1
 Gs
  
+ Cov
  
 rs
  rt
 ;
 1
 Gs
  
= pr(t; s)Et
  
1
 Gs
  
+ Cov
  
 rs
  rt
 ;
 1
 Gs
  
by Equation (5.1). Taking the logarithm of the last equality, then taking the negation of its division
 by the time to maturity gives
 yn(t; T ) = yr(t; T ) + i
 e(t; T ) + pI(t; T ); (5.6)
 where ie(t; T ) and pI(t; T ) are respectively the expected in ation and the in ation risk premium
 over [t; T ]. The in ation risk premium is generally decomposed in two components: the Jensen?s
 e ect and the covariance e ect [37] which are respectively de ned by
 J(t; T ) =  
lnEt
  
G 1s
  
 Et
  
lnG 1s
  
T  t
 and
 c(t; T ) =  
1
 T  t
 ln
 0
 B
 B
 @1 +
 Cov
  
 rs
  rt
 ; G 1s
  
Et
  
G 1s
  
1
 C
 C
 A :
5.2 Theoretical framework 108
 If the in ation premium is zero, then the Fisher Hypothesis is recovered. However, recall from the
 Subsection 1.1.1 that the existence of a \non-zero" in ation risk premium in nominal bonds was
 one of the reasons for the issuance of IL bonds [38]. This suggest that the Fisher Hypothesis should
 in \general" mis-estimate the expected in ation because it ignores the covariance between the real
 pricing kernel and in ation. Nevertheless, the improvement is not without disadvantage; the pricing
 kernel approach requires models for (i.e. assumptions on) the pricing kernels and the CPI, while the
 Fisher hypothesis gives a simple (no models for yields) means to estimated in ation expectation.
 The next section presents an assumption on the \information  ow", which simpli es the computation
 of formulas and the calibration of the pricing kernel approach. Afterwards Sections 5.4 and 5.3 each
 makes assumptions on the dynamics of the pricing kernels and in ation for data  tting.
 5.2 Theoretical framework
 The existence of a pricing kernel is synonymous to the well known no-arbitrage principle. In fact,
 given a pricing kernel the price of any bond or derivative security can be computed. Here the reverse
 operation is accomplished: from bond prices, the nominal and real pricing kernels are inferred.
 Following Backus et al. [8], to simplify the current framework and the  tting process, the next
 assumption will be made on the  ltration.
 Assumption 11. In Equations (5.1) and (5.3), the pricing kernels  i for i = n; r are contained in
 Et, i.e.  rt is considered to be part of the information known at time t that is represented by Ft with
 Et[ ] = EFt [ ].
 Note that derivative pricing is generally conducted at t = 0, where  i0 = 1 for i = n; r. In which case
 the previous assumption is satis ed, therefore this property has been extended to forward pricing,
 i.e. pricing a security forward in time. Another interpretation of the previous assumption is that
 instead of modelling the pricing kernels, the change in the pricing kernel (in these equations between
 times t and s) is modelled. If t = 0, because  i0 = 1 for i = n; r, this comes back to modelling the
 pricing kernels. Equation (5.1) becomes
 pr(t; T ) = Et [ rspr(s; T )]
 or similarly to [36]
 Et [ rsR(t; s; T )] = 1; (5.7)
5.2 Theoretical framework 109
 where R(t; s; T ) =
 pr(s; T )
 pr(t; T )
 is the gross real return over the period [t; s]. Inserting the IL bonds
 gives
 pIP (t; T ) = Et
  
 rspIP (s; T )
 I(t)
 I(s)
  
:
 Equation (5.3) is now
 pIP (t; T ) = Et [ ns pIP (s; T )] : (5.8)
 Identi cation of these two expressions of the IL bonds yields the new no-arbitrage relationship
 between the pricing kernels
  ns =
  rs
 Gs
 ; (5.9)
 which is still model independent.
 The reverse engineering approach [8] then proceeds as follows. It  rst speci es processes for  rt and
 Gt (or It); and then uses those to price the term structure, i.e. derive the yield on zero-coupon bonds
 of di erent maturities. This contrasts with the consumption based view (see examples in Chapter
 4), in which the asset pricing equation takes the form
 pi(t; T ) = Et
  
B
 U(Cs)
 U(Ct)
 pi(s; T )
  
for a time separable utility function U( ). In this case, the stochastic process for consumption and
 the form of the utility function determine the pricing kernel.
 Secondly, the time series and cross section properties implied by the theory are then matched with
 the data to derive the parameters of the underlying process and to deduce the pricing kernel and
 in ation. Once deduced, the two halves of the pricing kernel can function as a metric for assessing
 asset pricing theories and as an engine for pricing securities.
 The general asset pricing condition (5.7) becomes a theory of bond pricing once the pricing kernel
  r and the gross in ation rate G = (Gt)t 0 are characterized.
 The gross in ation return can be approximated as a function of the logarithm in ation rate in the
 following way:
 Gs =
 Is
 Is  
=
 Is  Is  
Is  
+ 1  ln
  
Is  Is  
Is  
 1
  
+ 1
 = ln
  
Is
 Is  
 
+ 1:
5.3 Exponential L evy Gross return 110
 The approximation holds because
 Is  Is  
Is  
is assumed \small enough". Note that the last line
 expresses the gross return in terms of the absolute return on in ation that will be denoted by rIs .
 So instead of studying the gross returns process Gs, the di erenced log-price process rIs is studied.
 Recall that taking the log returns of a data set is a standard procedure from time-series analysis,
 which transforms a non stationary sequence to one that is plausibly modelled as stationary [24].
 Sections 5.3 and 5.4 follow Craig et al. [36] by making assumptions on the distribution (of the
 logarithm) of the in ation gross return. If the gross return has an exponential L evy growth (Section
 5.3), then in ation has an exponential of exponential of L evy growth. This is rather unrealistic
 especially given the \good" results other works [81, 82, 14, 71] obtained when pricing IL securities
 under the hypothesis that in ation had an exponential growth. Furthermore, this fast growth
 assumption might explain the fact that Craig et al. had good results on short maturities, but
 worsening results the longer the maturity considered. The more \realistic" hypothesis of in ation
 exponential growth is used in Section 5.4.
 5.3 Exponential L evy Gross return
 The in ation gross return is a positive process, thus a reasonable choice of distribution is an ex-
 ponential L evy distribution. The following assumption extends the work of Craig et al. [36] by
 assuming the logarithm of in ation gross return follows a L evy process instead of an AR(ARMA)
 process. Note that this makes the in ation an exponential of exponential of L evy process.
 Assumption 12. Under the objective probability measure P the in ation gross return G and the
 real pricing kernel  r follow exponential L evy processes. Their dynamics are given by
 d rt
  rt 
=  r(t)dt+  r(t)dW Pt +
 Z
 R
  r(t; z) (dt; dz)
 dGt
 Gt 
=  G(t)dt+  G(t)dW Pt +
 Z
 R
  G(t; z) (dt; dz):
 with
  (dt; dz) =
 8
 <
 :
 (   )(dt; dz); jzj < R
  (dt; dz); jzj  R
 where the coe cients  i(t),  i(t) and  i(t; z) are adapted processes with
 Z t
 0
 j i(s)jds <1 and
 Z t
 0
 j i(s)j2ds <1
5.3 Exponential L evy Gross return 111
 for all  nite t;  i(t; z) :   R+  R! R is a real valued function satisfying
 Z t
 0
 Z
 R
 j i(s; z)j2 (ds; dz) <1;
 for  nite t. These conditions guarantee integrability of the coe cients and are satis ed if the coe -
 cients are bounded for t from a bounded set and  ([0; t] R) <1 for  nite t.
 The following propositions compute the dynamics and analytic formulas for the pricing kernels and
 forward rates.
 Proposition 5.1. Under Assumption 12, the dynamics of the nominal pricing kernel are given by
 d nt
  nt 
=  n(t)dt+  n(t)dW Pt +
 Z
 R
  n(t; z) (dt; dz);
 where
  n(t) =  r(t)  G(t) + [ G(t)]2 +
 Z
 jzj<R
 [ G(t; z)]2
 1 +  G(t; z)
  (dz)  r(t) G(t)
 +
 Z
 R
  r(t; z)
  G(t; z)
 1 +  G(t; z)
  (dz);
  n(t) =  r(t)  G(t);
  n(t; z) =  r(t; z) +
  G(t; z)
 1 +  G(t; z)
 +  r(t; z)
  G(t; z)
 1 +  G(t; z)
 :
 Proof. Direct application of Corollary 2.15.
 Under Assumption 11, the nominal and real bonds are given in terms of their corresponding pricing
 kernels by
 pi(t; T ) = Et
  
 ispi(s; T )
  
for i = n; r:
 For a maturity T = s, i.e. pn(s; T ) = 1
 pi(t; T ) = Et
  
 iT
  
for i = n; r:
 Hence, the time t price of a bond (nominal or real) is the expected value of its corresponding pricing
 kernel at the bond?s maturity. Without Assumption 11, this expectation would have been multiplied
 by the pricing kernel at time t to get the bond price. This simpli ed relationship is used in the next
 proposition to compute the nominal and real bonds prices.
5.3 Exponential L evy Gross return 112
 Proposition 5.2. At time t, the nominal and real bonds with maturity T are worth
 pi(t; T ) = exp
 (Z T
 0
  i(u)du 
1
 2
 Z T
 t
 [ i(u)]2du+
 Z T
 t
 Z
 R
  
ez  1 z jzj<R
  
  i(dz)du
 +
 Z t
 0
  i(s)dW Ps +
 Z t
 0
 Z
 R
  i(s; z) (ds; dz)
  
for i = n; r;
 where   i =   ( i) 1.
 The dynamics of pi(t; T ) are
 dpi(t; T )
 pi(t ; T )
 = ai(t)dt+ bi(t)dW Pt +
 Z
 R
 ci(t; z) (dt; dz) for i = n; r;
 where
 ai(t) = [ i(t)]2  
Z
 R
  
ez  1 z jzj<R
  
  i(dz) +
 Z
 jzj<R
  
exp[ i(t; z)] 1  i(t; z)
  
 (dz);
 bi(t) =  i(t); ci(t; z) = exp[ i(t; z)] 1;
 with constraints
 Z t
 0
  n(u)du+
 Z t
 0
  n(s)dW Ps +
 Z t
 0
 Z
 R
  n(s; z) (ds; dz) = 0 8t 2 R:
 If the coe cient  i(t; z) is deterministic, then
 ai(t) = [ i(t)]2  
Z
 R
 h
 e 
i(t;z)  1  i(t; z) jzj<R
 i
  (dz) +
 Z
 jzj<R
  
exp[ i(t; z)] 1  i(t; z)
  
 (dz);
 Proof. For i = n; r
 Et[ iT ] =  
i
 t exp
  Z T
 t
  i(s)ds
 !
 Et
 "
 exp
  Z T
 t
  i(s)dWs +
 Z T
 t
 Z
 R
  i(s; z) (ds; dz)
 !#
 =  it exp
  Z T
 t
  i(s)ds
 !
 Et
 "
 exp
  Z T
 t
  i(s)dWs
 !#
 Et
 "
 exp
  Z T
 t
 Z
 R
  i(s; z) (ds; dz)
 !#
 since dWt and  (dt; dz) are independent.
 Et
 "
 exp
  Z T
 t
 Z
 R
  i(u; z) (du; dz)
 !#
 = Et
 "
 exp
  Z T
 t
 Z
 jzj<R
  i(u; z)(   )(du; dz)
 +
 Z T
 t
 Z
 jzj R
  i(u; z) (du; dz)
 !#
 = Et
 "
 exp
  Z T
 t
 Z
 jzj<R
  i(u; z)(   )(du; dz)
 !#
  Et
 "
 exp
  Z T
 t
 Z
 jzj R
  i(u; z) (du; dz)
 !#
5.3 Exponential L evy Gross return 113
 since fjzj < Rg and fjzj  Rg are disjoint. Using Corollary 2.21 for the  rst equality
 E
 "
 exp
  Z T
 t
 Z
 R
  i(u; z) (du; dz)
 !#
 = exp
 (Z T
 t
 Z
 jzj<R
 [ez  1 z]  n(dz)du
 )
  exp
 "Z T
 t
 Z
 jzj R
 (ez  1)  n(dz)du
 #
 = exp
 (Z T
 t
 Z
 R
  
ez  1 z jzj<R
  
  n(dz)du
 )
 with   i =   ( i) 1.
 Recall that dWt  N (0; dt), hence
 Z T
 t
  i(u)dWu  N
  
0;
 Z T
 t
 [ i(u)]2du
 !
 . Let h ;  i be the inner
 product, i.e. if x; y 2 Rd such as x = (x1; x2;    ; xd) and y = (y1; y2;    ; yd), then hx; yi =
 dX
 i=1
 xiyi.
 Let  N (m;V ) denote the log-characteristic function of a normal distributed process with mean m and
 variance V , de ned by
  N (m;V )(u) = i hm;ui  
1
 2
 hu; V ui with m;u 2 Rd:
 In particular
  N (m;V )(u) = imu 
1
 2
 V u2 with m;u 2 R:
 Using Theorem 2.19
 Et
 "
 exp
  Z T
 t
  i(u)dWu
 !#
 = exp
 (
  
1
 2
 Z T
 t
 [ i(u)]2du
 )
 :
 Combining these expectations yields
 Et[ iT ] =  
i
 t exp
 (Z T
 t
  i(u)du 
1
 2
 Z T
 t
 [ i(u)]2du+
 Z T
 t
 Z
 R
  
ez  1 z jzj<R
  
  i(dz)du
 )
 :
 Therefore
 pi(t; T ) = exp
 (Z T
 t
  i(u)du 
1
 2
 Z T
 t
 [ i(u)]2du+
 Z T
 t
 Z
 R
  
ez  1 z jzj<R
  
  i(dz)du
 )
  exp
  Z t
 0
  i(s)ds+
 Z t
 0
  i(s)dW Ps +
 Z t
 0
 Z
 R
  i(s; z) (ds; dz)
  
= exp
 (Z T
 0
  i(u)du 
1
 2
 Z T
 t
 [ i(u)]2du+
 Z T
 t
 Z
 R
  
ez  1 z jzj<R
  
  i(dz)du
 +
 Z t
 0
  i(s)dW Ps +
 Z t
 0
 Z
 R
  i(s; z) (ds; dz)
  
:
5.3 Exponential L evy Gross return 114
 Thus
 ln pi(t; T ) =
 Z T
 0
  i(u)du 
1
 2
 Z T
 t
 [ i(u)]2du+
 Z T
 t
 Z
 R
  
ez  1 z jzj<R
  
  i(dz)du
 +
 Z t
 0
  i(s)dW Ps +
 Z t
 0
 Z
 R
  i(s; z) (ds; dz):
 Denoting ln pi(t; T ) by X(t; T )
 dX(t; T ) =
 1
 2
 [ i(t)]2dt 
Z
 R
  
ez  1 z jzj<R
  
  i(dz)dt+  
i(t)dW Pt +
 Z
 R
  i(t; z) (dt; dz);
 with constraints
 X(t; t) =
 Z t
 0
  i(u)du+
 Z t
 0
  i(s)dW Ps +
 Z t
 0
 Z
 R
  i(s; z) (ds; dz):
 But, by de nition of nominal and real bonds, ln pi(t; t) = 0; therefore, the constraint becomes
 Z t
 0
  i(u)du+
 Z t
 0
  i(s)dW Ps +
 Z t
 0
 Z
 R
  i(s; z) (ds; dz) = 0:
 Using pi(t; T ) = f [X(t; T )] = exp[X(t; T )] in the one-dimensional Ito^ formula (Theorem 2.7), we
 have
 @f
 @t
 = 0 and
 @f
 @x
 =
 @2f
 @2x
 = pn(t; T ). Thus
 dpi(t; T ) = pi(t; T )
  
1
 2
 [ i(t)]2dt 
Z
 R
  
ez  1 z jzj<R
  
  i(dz)dt+  
i(t)dW Pt +
 1
 2
 [ i(t)]2dt
  
+
 Z
 jzj<R
  
pi(t ; T ) exp[ 
i(t; z)] pi(t ; T ) pi(t ; T ) 
i(t; z)
  
 (dt; dz)
 +
 Z
 R
  
pi(t ; T ) exp[ 
i(t; z)] pi(t ; T )
  
 (dt; dz):
 The dynamics of pi are given by
 dpi(t; T )
 pi(t ; T )
 =
  
1
 2
 [ i(t)]2dt 
Z
 R
  
ez  1 z jzj<R
  
  i(dz)dt+  
i(t)dW Pt +
 1
 2
 [ i(t)]2dt
  
+
 Z
 jzj<R
  
exp[ i(t; z)] 1  i(t; z)
  
 (dt; dz) +
 Z
 R
  
exp[ i(t; z)] 1
  
 (dt; dz)
 =
  
[ i(t)]2  
Z
 R
  
ez  1 z jzj<R
  
  i(dz) +  
i(t)dW Pt
 +
 Z
 jzj<R
  
exp[ i(t; z)] 1  i(t; z)
  
 (dz)
 )
 dt+
 Z
 R
  
exp[ i(t; z)] 1
  
 (dt; dz);
 with
 Z t
 0
  i(u)du+
 Z t
 0
  i(s)dW Ps +
 Z t
 0
 Z
 R
  i(s; z) (ds; dz) = 0:
 If  i(t; z) is assumed deterministic, then Theorem 2.22 is used.
5.4 Exponential L evy In ation 115
 Proposition 5.3. The nominal and real forward rate are
 fi(t; T ) =   
i(T ) +
 1
 2
 [ i(T )]2  
Z
 R
  
ez  1 z jzj<R
  
  i(dz) for i = n; r:
 If  i is assumed deterministic, then
 fi(t; T ) =   
i(T ) +
 1
 2
 [ i(T )]2  
Z
 R
 h
 e 
i(t;z)  1  i(t; z) jzj<R
 i
  (dz):
 Proof. For i = r; n, the instantaneous forward rates contracted at time t are given by
 fi(t; T ) =  
@ ln pi(t; T )
 @T
 =   i(T ) +
 1
 2
 [ i(T )]2  
Z
 R
  
ez  1 z jzj<R
  
  i(dz):
 If  i is deterministic, then Theorem 2.22 is again used.
 5.4 Exponential L evy In ation
 Many studies [81, 82, 14, 71] have used the assumption that in ation is lognormal, which implies an
 exponential distribution for IL securities2 instead of the previously assumed exponential of exponen-
 tial distribution. This suggests that although the CPI is fast growing, it does grow at an exponential
 of exponential pace. The former is more realistic especially given that \most" can still a ord daily
 expenses. The already successful lognormal distribution is here extended to an exponential L evy
 process for the the in ation. By similitude, the pricing kernel is also assumed to be an exponential
 L evy process. The dynamics of in ation?s gross return will be implied by those of in ation as is
 shown in the next assumption.
 Assumption 13. Under the objective probability measure P the in ation It and the real pricing
 kernel follow exponential L evy processes. Their dynamics are given by
 d rt
  rt 
=  r(t)dt+  r(t)dW Pt +
 Z
 R
  r(t; z) (dt; dz);
 dIt
 It 
=  I(t)dt+  I(t)dW Pt +
 Z
 R
  I(t; z) (dt; dz):
 with the standard integrability conditions imposed on the coe cients.
 The log formulation is useful because yields and interest rates are easier to work with than bond
 prices, and it allows us to exploit some property of exponential L evy distributions. Furthermore, in
 2See Chapter 3 for example.
5.4 Exponential L evy In ation 116
 this form,  r and G are not independent, but they do not depend directly on one another. Splitting
 in ation into nominal and real parts involves more than merely using more complicated distributions;
 it requires explicit consideration of the interactions between real and nominal rates.
 The nominal pricing kernel is given by
  nt =
  rt
 rIt + 1
 :
 This, in conjunction with equation (5.7), prices assets. It describes how the pricing kernel evolves
 over time, or equivalently, how the discount rate depends on both real and nominal shocks.
 Proposition 5.4. The dynamics of the nominal pricing kernel are given by
 d nt
  nt 
=  n(t)dt+  n(t)dW Pt +
 Z
 R
  n(t; z) (dt; dz);
 where
  n(t) =  r(t) Yt
  
 I(t) [ I(t)]2Yt
  
  r(t)Yt 
I(t)
 +
 Z
 jzj<R
  
1
 1 +  I(t; z)Yt 
 1 + Yt  
I(t; z)
  
 (dz)
 +
 Z
 R
  r(t; z)
  
1
 1 + Yt  I(t; z)
  1
  
 (dz);
  n(t) =  r(t) Yt 
I(t);
  n(t; z) =
  r(t; z) + 1
 1 + Yt  I(t; z)
  1;
 Yt =
 1
 rIt + 1
 =
 1
 Gt
 :
 Proof.
 Assuming that  is small enough, from the in ation dynamics (Assumption 13), the absolute in ation
 return
 rIt = ln
  
It
 It  
 
has dynamics
 drIt =  
I(t)dt+  I(t)dW Pt +
 Z
 R
  I(t; z) (dt; dz):
 Applying the one-dimensional Ito^ formula (Theorem 2.7) with Yt = f(t; rIt ) =
 1
 rIt + 1
 whose deriva-
5.4 Exponential L evy In ation 117
 tives are
 @f
 @t
 = 0,
 @f
 @rIt
 =  
1
 (rIt + 1)2
 and
 @2f
 @2rIt
 =
 2
 (rIt + 1)3
 ; we have
 dYt =  
1
 (rIt + 1)2
  
 I(t)dt+  I(t)dWt
  
+
 1
 2
 [ I(t)]2
 2
 (rIt + 1)3
 dt
 +
 Z
 jzj<R
  
1
 rIt +  I(t; z) + 1
  
1
 rIt + 1
 +
  I(t; z)
 (rIt + 1)2
  
 (dt; dz)
 +
 Z
 R
  
1
 rIt +  I(t; z) + 1
  
1
 rIt + 1
  
 (dt; dz)
 =  
1
 (rIt + 1)2
  
 I(t) [ I(t)]2
 1
 rIt + 1
  
dt 
1
 (rIt + 1)2
  (t)dWt
 +
 Z
 jzj<R
  
1
 rIt +  I(t; z) + 1
  
1
 rIt + 1
 +
  I(t; z)
 (rIt + 1)2
  
 (dt; dz)
 +
 Z
 R
  
1
 rIt +  I(t; z) + 1
  
1
 rIt + 1
  
 (dt; dz)
 =  Y 2t
  
 I(t) [ I(t)]2Yt
  
dt Y 2t  
I(t)dWt
 +
 Z
 jzj<R
  
Yt 
1 +  I(t; z)Yt 
 Yt + Y
 2
 t  
I(t; z)
  
 (dt; dz) +
 Z
 R
  
Yt 
1 + Yt  I(t; z)
  Yt 
 
 (dt; dz):
 Now applying the multidimensional Ito^ formula (Theorem 2.11) with f(t;  rt ; Yt) =  
r
 t Yt
 d nt =
  
 rt  
r(t)Yt  Y
 2
 t
  
 I(t) [ I(t)]2Yt
  
 rt   
r
 t  
r(t)Y 2t  
I(t)
 +
 Z
 jzj<R
  
Yt 
1 +  I(t; z)Yt 
 Yt + Y
 2
 t  
I(t; z)
  
 (dz) rt
 +
 Z
 R
  rt  
r(t; z)
  
Yt 
1 + Yt  I(t; z)
  Yt 
 
 (dz)
  
dt
 +
  
 rt  
r(t)Yt  Y
 2
 t  
I(t) rt
  
dWt
 +
 Z
 R
  
Yt 
1 + Yt  I(t; z)
  rt  Yt  
r
 t +  
r
 t  
r(t; z)Yt 
+ rt  
r(t; z)
 Yt 
1 + Yt  I(t; z)
   rt  
r(t; z)Yt 
 
 (dt; dz)
 d nt =
  
 nt  
r(t) Yt
  
 I(t) [ I(t)]2Yt
  
 nt   
n
 t  
r(t)Yt 
I(t)
 +
 Z
 jzj<R
  
 nt 
1 +  I(t; z)Yt 
  nt +  
n
 t Yt  
I(t; z)
  
 (dz)
 +
 Z
 R
  r(t; z)
  
 nt 
1 + Yt  I(t; z)
   nt 
 
 (dz)
  
dt
 +
  
 nt  
r(t) Yt 
I(t) nt
  
dWt
 +
 Z
 R
  
 nt 
1 + Yt  I(t; z)
   nt +  
r(t; z)
  nt 
1 + Yt  I(t; z)
  
 (dt; dz):
 From here on, a reasoning similar to that of Section 5.3 gives similar formulas for the IL bonds,
 nominal and real forward instantaneous forward rates.
5.5 Parameter estimation 118
 5.5 Parameter estimation
 At time t, the zero coupon bond price pi(t; T ) for i = n; r is known for a maturity T . To get the
 entire yield curve, forward rates over the period [T; T  ] are needed, where T  is the maturity up to
 which the yield curve is required. This comes back to estimating the unique driving process in the
 nominal and real yield curves just as in Subsection 6.1.8. The methodology of parameter estimation
 will not be repeated here.
 Work in progress aims at  rst obtaining the historical real yield curve data for the South African
 market. Afterward, the calibration and interpretation of results will be conducted.
Chapter 6
 Empirical Study and Calibration
 Chapters 3, 4 and 5 each presented a di erent framework to price in ation linked (IL) derivatives
 taking into account the in ation market?s illiquidity. This chapter now performs parameter estima-
 tion from market data for each of the previous models. The chapter is divided into two main sections.
 The  rst section conducts an empirical study of historical data to highlight the shortcomings of the
 normality assumption and the appropriateness of L evy distributions. The second section deals with
 the actual pricing of derivative securities (swaps, caps and  oors).
 6.1 Empirical study
 Let (Xi)i2N denote an observed macroeconomic factor and  t be the  xed time step between the
 observations. Depending on the process being considered,  t will be a day, a week or even a month.
 The observation Xi represents the value taken by the factor X at time i t and pX(xi) is the
 probability that X has the value xi.
 6.1.1 Data
 The  nancial data used for parameter estimation is from both the Johannesburg Stocks Exchange
 (JSE) securities exchange and the New York Stock Exchange (NYSE). The former is an emerging
 (African) market while the latter is a developed (American) market. Thus the overall performances
 of the models in both types of markets can be tested. A detailed description of the data is provided
 in the Subsections 6.1.9 and 6.1.10.
 119
6.1 Empirical study 120
 The statistical study is carried out on the log returns of the macroeconomical factors for two main
 reasons. Firstly,  nancially, the log return corresponds to the continuously compounded return of
 the factor. It is dimensionless and assumed smooth \enough". Secondly, numerically, this is done to
 constrain return values to be positive. Others arguments for this approach are the extensive evidence
 of returns stationarity in the literature [34]; plus the direct transferability of return independence
 and identical distribution to its logarithm. Considering a discrete process (Xi), with i = 1; 2;    ,
 the corresponding log return process ri, with i = 1; 2;    , is de ned by:
 ri =
 1
  t
 ln
  
Xi+1
 Xi
  
:
 The time interval  t is generally constant and equal to one time step and is thus ignored.
 6.1.2 Statistics
 This section brie y reviews some common descriptive statistics that will be used. A more de-
 tailed coverage can be found in Hamilton (1994). For illustration, the South African (SA) monthly
 Consumer Price Index (CPI) data between January 1965 and March 2008 (Figure 6.4(b)) is used.
 Monthly SA CPI between January 1960 and December 1964 were mostly constant over periods of
 at least six months and thus ignored because they do not provide any \useful" information for the
 study.
 De nition 6.1 (Mean). The (sample) mean or  rst moment of X is de ned by
 E[X] =
 X
 i
 xipX(xi);
 where (xi)i2I is the set of attainable values by X with I  N.
 From here on,  represents the mean of the distribution X.
 Example 4. The sample mean of South African (SA) Consumer Price Index (CPI) monthly log
 return between January 1965 and March 2008 is 0:74% which is rather low given that it is monthly
 and South Africa?s target for monthly CPIX1 is about 0:8666% 1:7321%, i.e. 3% 6% annually.
 De nition 6.2 (Variance). The (sample) variance or central second moment of X is de ned by
 V ar(X) = E[(X   )2] =
 X
 i
 (xi   )
 2pX(xi);
 1See Subsection 6.1.9 for a de nition.
6.1 Empirical study 121
 where (xi)i2I is the set of attainable values by X with I  N.
 The variance is characterized by
 V ar(X) = E[X2] (E[X])2
 where E[X2] =
 X
 i
 x2i pX(xi) is the second non-central moment of X.
 The variance and its square root, the standard deviation or volatility, are measures of the uncertainty
 of the return of a speci c factor. In the market, a period of relatively low (resp. high) risk is generally
 followed by periods of relatively low (resp. high) risk. This phenomenon is referred to as volatility
 clustering [59].
 From here on,  represents the standard deviation of the distribution X.
 Example 5. The standard deviation of SA CPI log returns over the period from January 1965 to
 March 2008 is 0:70% which is only 4 basis points (bp) less than its mean. This can be explained by
 the rather low average and the high volatility of the earliest CPI (Figure 6.4(b)). Unfortunately the
 small amount of data (all recorded SA CPI) prevent the exclusion of the earliest data sample.
 De nition 6.3 (Skewness). The (sample) skewness is a measure of the asymmetry of a distribution
 with respect to (in short w.r.t.) its mean. It is de ned by
 S(X) =
 E[(X   )3]
  3
 :
 Note that for a normal distribution the skewness is zero. A distribution with positive skewness (right
 skewed) has a fatter right tail, i.e. it is more likely to have its values above the mean value than
 below. Likewise, a distribution with negative skewness (left skewed) has a fatter left tail, i.e. it is
 more likely to have its values below the mean value than above.
 Example 6. The skewness of the SA CPI log returns is 0:9163. Therefore, the CPI is asymmetric
 (i.e. non-normal) and has mass concentrated on the right.
 De nition 6.4 (Kurtosis). The (sample) Kurtosis is a measure of the peakedness (i.e. tail behavior)
 of a distribution. It is de ned by
 K(X) =
 E[(X   )4]
  4
 :
 A normal distribution has a kurtosis of 3. A distribution with kurtosis greater than 3 is said to
 have \fat tails" or to be leptokurtic, i.e. it is more peaked than a Gaussian around the mean. A
 distribution with kurtosis less than 3 is said to be platykurtic.
6.1 Empirical study 122
 Some programs return the excess kurtosis which is the kurtosis minus 3 instead of the kurtosis.
 Example 7. The kurtosis of the SA CPI log returns is 4:5877 > 3. Hence the log return innovations?
 density function is more peaked than the normal density function.
 A  nancial time series can be viewed as a sequence of random observations. This random sequence,
 or stochastic process, may exhibit some degree of correlation from one observation to the next. This
 correlation structure can be used to predict future values of the process based on past observations.
 Exploiting the correlation structure, if any, allows the decomposition of the time series into a de-
 terministic component (i.e. the forecast), and a random component (i.e. the error, or uncertainty,
 associated with the forecast). Autocorrelation and partial autocorrelation are important tools for
 studying stationary time series such as simple autoregressive (AR) models, moving-average (MA)
 models, autoregressive moving-average (ARMA) models and seasonal models [59].
 De nition 6.5 (Autocorrelation). The autocorrelation function (ACF) is a measure of cross-
 dependence of a distribution with itself given a time lag. It is useful to  nd repeating patterns
 in a distribution. The jth (sample) autocorrelation of the distribution X is de ned by
  j =
 Cov(Xi; Xi+j)
 p
 V ar(Xi)V ar(Xi+j)
 ;
 where
 Cov(X;Y ) = E[(X   x)(Y   y)]
 represents the (sample) covariance of the distributions X and Y ; with  x (resp.  y) the average of
 X (resp. Y ).
 Example 8. Figures 6.1 show that the South African CPI log returns and its square are slightly
 autocorrelated. The highest correlation (at lag 12) is due to annual seasonality of the CPI. This
 translates to the fact that during festive periods (Christmas, end of year, etc) prices tend to increase
 because of the higher demand; and (most of) agriculture follow an annual cycle and thus the prices
 of related goods is seasonal, this annual seasonality is not surprising. Moreover, it is present in
 most (if not all) CPIs worldwide. Notice the similar distribution of the spikes (of log returns) for
 the subsets f1; 2; 3g, f4; 5; 6g and f10; 11; 12g. The spikes at lag 1, 4 and 10 (resp. 2, 5 and 11) are
 almost (resp. perfectly) identical. The spike at lag 12 is surely greater than those at lags 3 and 6
 just because of the annual seasonality. Strangely enough, log returns are more correlated than their
 square this might be due to the small size of data.
6.1 Empirical study 123
 (a) Log returns (b) Squared returns
 Figure 6.1 Monthly SA CPI correlograms.
 De nition 6.6 (Partial autocorrelation). The partial autocorrelation function (PACF) is a mea-
 sure of the conditional cross-dependence of a distribution with itself given a time lag. The PACF
 removes the e ect of shorter lag autocorrelation from the correlation estimate at longer lags. The
 mth (sample) partial autocorrelation of the distribution X is de ned by
 ?m;m =
  m  
m 1X
 j=1
 ?m 1;j m 1
 1 
m 1X
 j=1
 ?m 1;j j
 ;
 where  j is the autocorrelation with a time lag of j.
 (a) Log returns (b) Squared returns
 Figure 6.2 Monthly SA CPI partial correlograms.
6.1 Empirical study 124
 Example 9. In Figure 6.2, as expected the monthly SA CPI partial correlograms? spikes are below
 that of the correlograms. The log returns autocorrelation spikes are only maintained at lags 1,
 2, 3, 5 and 6; meaning that the remaining higher-order autocorrelations are due to these initial
 autocorrelations. Hence, when forecasting monthly CPI, it is not \necessary" to use data beyond
 a year from the prediction date. This is a common and successful practise in the South African
 market. However, this should only give \good" results for a year or less forecast. For a longer period
 forecast and a more accurate forecasting framework, L evy distributions should be used. Recall that
 the normal distribution is a particular case of a L evy distribution, a generalisation of the standard
 forecast will thus be obtained by using L evy processes.
 The autocorrelations at lags 1 and 5 are \quite" small and can be ignored. Between the raw squared
 returns ACF and PACF, the only signi cant autocorrelation maintained is the one at lag 12. This
 corresponds to a year periodicity, i.e. the CPI seasonality. The non-existence of a high autocorre-
 lation in the squared innovations might be due to data sparsity (granularity and size). This issue is
 \solved" later by increasing the data size (See Subsection 6.1.6).
 6.1.3 Hypothesis Tests
 A hypothesis test is a procedure used to check if a certain criterion is satis ed by a given sample
 distribution. This study conducts two families of hypothesis tests:
 (i) Normality tests: Jarque-Bera test, Kolmogorov-Smirnov test, the Pearson?s Chi-squared test,
 Normal Probability Plots and Quantile-Quantile Plots;
 (ii) Heteroscedasticity tests: Ljung-Box-Pierce Q-test and Engle?s ARCH test.
 Normality tests investigate if a sample data comes from a normal distribution. While heteroscedas-
 ticity (i.e. ARCH=GARCH e ects) tests investigate if the sample data?s variance is non-constant
 (i.e. time varying). Heteroscedasticity tests are commonly used to quantify the correlation in a
 sample data. Here is a brief description of the selected hypothesis tests.
 Jarque-Bera test
 The Jarque-Bera (JB) test examines whether a speci c distribution is normal or not. The JB-value
 is calculated as:
 JB(X) =
 n l
 6
  
S2(X) +
 [K(X) 3]2
 4
  
;
6.1 Empirical study 125
 where n is the number of observations, S( ) the skewness function, K( ) the kurtosis function and l is
 the number of estimated parameters. The intuition behind this test is that the larger the JB-value
 is, the lower the probability is that the given series is drawn from a normal distribution. For large
 sample size, the test statistic of the Jarque-Bera test is  2-distributed with 2 degrees of freedom
 under the null hypothesis that the series is normally distributed.
 Example 10. The Jarque-Bera test of the monthly SA CPI log returns yields h = 1, p = 10 3. In
 fact the p-value is less than 10 3, which is the smallest value returned by the Matlab function jbtest.
 The p-value is below the default signi cance level of 5%, and the test rejects the null hypothesis that
 the distribution is normal.
 Kolmogorov-Smirnov test
 The Kolmogorov-Smirnov (KS) test is a goodness of  t test used to determine whether two underlying
 one-dimensional probability distributions di er, or whether an underlying probability distribution
 di ers from a hypothesized distribution, in either case based on  nite samples.
 The one-sample KS test compares the empirical distribution function with the cumulative distri-
 bution function speci ed by the null hypothesis. The main applications are testing goodness of  t
 with the normal and uniform distributions. For normality testing, minor improvements made by
 Lilliefors lead to the Lilliefors test. This test is sensitive to di erences in both location and shape
 of the empirical cumulative distribution functions of the two distributions.
 The Anderson-Darling (AD) [99] is another modi cation of the KS test. It gives more weight to the
 tails than does the KS test. Contrary to the KS test, the AD test?s critical values are functions of
 the speci c distribution being tested. This implies a more sensitive test, but critical values have to
 be computed for each distribution.
 Example 11. The Kolmogorov-Smirnov test of monthly SA CPI log returns yields h = 1, p =
 4:5267 10 112  0, k = 0:4970 and c = 0:0595 where k (resp. c) is the test statistic, i.e. the maximum
 di erence between the cumulative distribution functions (resp. the cuto value for determining if
 k is signi cant). Since h = 1, the test rejects the null hypothesis that the values come from a
 normal distribution at the 5% signi cance level. A look at the Kolmogorov-Smirnov test in Figure
 6.3 con rms the unsuitability of the normal distribution.
 Henceforth, p-value as small as the previous will be assimilated to 0.
 The Lilliefors test returns h = 1 and p = 0. This test also rejects the normality of the monthly SA
6.1 Empirical study 126
 Figure 6.3 SA CPI raw returns Kolmogorov-Smirnov test.
 CPI log returns. The AD test also reject the normality of the monthly SA CPI log returns.
 The Chi-squared Test
 The  2 test has as null hypothesis that the provided data comes from a speci ed distribution with
 unknown parameters. If the considered distribution is a normal distribution, then the null hypothesis
 is that the data sample is from a normal distribution with unknown mean and standard variance,
 which are estimated from the sample data. The test counts the number of sample points falling into
 certain intervals (referred to as bins) and compares these counts with the expected number in these
 intervals under the null hypothesis. The  2 statistic is given by
  2 =
 X
 i
 (Oi  Ei)2
 Ei
 ;
 where Oi and Ei are respectively the observed and expected counts.
 Normal Probability Plot
 A normal probability plot (See Figure 6.11(a)) is a useful graph for assessing that a data sample comes
 from a normal distribution. Many statistical procedures make the assumption that the underlying
6.1 Empirical study 127
 distribution of the data is normal, so this plot can provide some assurance that the assumption of
 normality is not being violated, or provide an early warning of a problem.
 Quantile-Quantile Plot
 A quantile-quantile plot (See Figure 6.11(b)) is useful for determining whether two samples come
 from the same distribution (the distribution can be normal or not).
 Even though the parameters and sample sizes are di erent, the straight line relationship shows that
 the two samples come from a similar kind of distribution. The set consisting of numerous pluses is
 the quantiles of each sample. By default the number of pluses is the number of data values in the
 smaller sample. The solid line joins the 25th and 75th percentiles of the samples. The dashed line
 extends the solid line to the extent of the sample.
 Ljung-Box-Pierce Q-test
 The Ljung-Box-Pierce (LBP) Q-test is performed to test jointly whether several autocorrelations of
 data series are signi cant or not. The LBP Q-value is calculated by:
 Qk = n(n+ 2)
 kX
 i=1
  2i
 n i
 ;
 where n is the sample size, k is the number of lags and  i the ith autocorrelation. If Qk is large then
 the probability that the process has uncorrelated data decreases. The null hypothesis for the test is
 that there exists no correlation and under that hypothesis, Qk is  2-distributed with k degrees of
 freedom.
 Example 12. The Ljung-Box-Pierce Q-test estimate of the autocorrelation present in the raw and
 squared SA CPI returns when tested for up to 10, 15, and 20 lags at 0:05 level of signi cance gives
 Raw return Squared Raw return
 Lag H p Stat Crit H p Stat Crit
 10 1 0 193:7546 18:3070 0 0:7479 6:7599 18:3070
 15 1 0 321:7936 24:9958 1 0 65:5773 24:9958
 20 1 0 405:1546 31:4104 1 0 69:8175 31:4104
 Table 6.1 Ljung-Box-Pierce Q-test for SA CPI raw and squared returns.
6.1 Empirical study 128
 The column \Stat" (resp. \Crit") is the vector of Q-statistics for each lag (resp. the vector of critical
 values of the  2 distribution for comparison with the corresponding element of \Stat").
 The correlation in the squared of raw returns translates the existing volatility clustering that will be
 captured by the GARCH(1; 1)  lter presented in Section 6.1.5.
 Engle?s ARCH Tests
 It is fairly easy to test whether the residuals from a regression have conditional heteroskedasticity
 or not. The test is based on Ordinary Least Squares (OLS) regression, where the OLS residuals
 u^t from the regression are saved. The process u^2t is thereafter regressed on a constant and its own
 m-lagged values. This is done for all samples t = 1; 2;    ; n. This regression has a corresponding
 R2-value. The distribution nR2 is then asymptotically  2-distributed with m degrees of freedom
 under the null hypothesis that u^t is i.i.d. N (0;  2) [50].
 This ARCH-test can also be performed as a test for GARCH-e ects. The ARCH-test for lag (p+ q)
 is locally equivalent to a test for GARCH e ects with lags (p; q). (MathWorks 2007)
 The null hypothesis, H0, is that no ARCH e ects exist.
 Example 13. The Engle?s ARCH test (Table 6.2) con rms the existence of autocorrelation and
 GARCH e ects only for time lags of 15 and 20.
 Raw return Squared Raw return
 Lag H p Stat Crit H p Stat Crit
 10 0 0:8166 5:9839 18:3070 0 1 0:5681 18:3070
 15 1 0 56:4319 24:9958 1 0 48:8665 24:9958
 20 1 0 59:2076 31:4104 1 0:0004 48:4818 31:4104
 Table 6.2 Engle?s ARCH test results for SA CPI raw and squared returns.
 6.1.4 Goodness of Fit
 This section  ts a wide variety of distributions to the empirical distribution of a data sample. To
 evaluate the performance of each distribution, multiple goodness of  t assessment measures are also
 used. The latter can be divided in two majors groups; visual assessment and quantitative assessment.
6.1 Empirical study 129
 The visual assessment indicators are the normal probability plots and quantile-quantile plots that
 have already been introduced.
 Similarly, most of the quantitative assessment indicators have already been presented. These are
 Kolmogorov-Smirnov test, the Pearson?s Chi-squared test, which are not only restricted to the test
 of normality. The log-likelihood estimate is a common measure generally associated to the maximum
 likelihood estimator which is described in Section 6.2.1.
 The Akaike information criterion (AIC) that comes with the ghyp package of R is also used. These
 goodness-of- t test will be used after the parameter estimation for the L evy distributions in Sub-
 section 6.1.7.
 6.1.5 Data Filtering
 The hypothesis of independent price returns is extremely important in  nancial modelling. So is
 the time varying volatility which can be reproduced by L evy models. To reinforce the observed
 volatility clustering, a GARCH  lter  rst captures the persistence in the volatility. Moreover,
 McNeil et al [80] argued that a GARCH(1; 1) model with Student t innovations is enough to remove
 the dependence in return series. This approach is used here to render the return series \more"
 independent and identically distributed (i.i.d.). In this study, GARCH(1; 1)  lters with normal and
 student-t innovations are considered.
 GARCH model
 The Generalized Autoregressive Conditional Heteroskedasticity (GARCH) is a generalization of the
 ordinary ARCH-model. The model structure was introduced by Bollerslev [22]. The generalization
 with respect to ARCH model is similar to the extension of an AR(p) to an ARMA(p; q).
 The intuitive introduction to GARCH models presented here is similar to that done by John Hull
 in [69]. GARCH models are generally used to reproduce and forecast volatility and correlation. As
 mentioned previously, the standard deviation  t or its square is a convenient measure of risk. The
 continuously compounded interest rate yt of the asset price represented by Xt is de ned by
 yi = ln
  
Xi
 Xi 1
  
:
 In this section  i denotes the standard deviation of the rate yi at time i t. An estimate of  i using
6.1 Empirical study 130
 the m most recent observations is
  2i =
 1
 m 1
 mX
 j=1
 (yi j   y)
 2 ; (6.1)
 where  y denotes the average of yi for i 2 [i m; i 1]:
  y =
 1
 m
 mX
 j=1
 yi j :
 From Equation (6.1), the following approximations are made
 (i) the rate yi is de ned as the percentage change between time (i 1) t and i t:
 yi =
 Xi  Xi 1
 Xi 1
 ; (6.2)
 (ii) the average  y is considered to be zero;
 (iii) the denominator m 1 is replaced by m.
 These changes simplify the variance formula to
  2i =
 1
 m
 mX
 j=1
 y2i j ;
 where yi is given by Equation (6.2). Moreover, they do not a ect the variance estimates much.
 However, the previous equation gives the same weight to all the used observations; because of
 volatility clustering it is more reasonable to give higher weights to recent observations. This yields
  2i =
 mX
 j=1
  jy
 2
 i j ;
 with
 mX
 j=1
  j = 1. The weights are all positive and  j <  k for j > k translates the fact that less
 weight is given to older observations.
 Under the further assumption that there is a long-run average rate yL which should be given a
 weight,
  2i =  yL +
 mX
 j=1
  jy
 2
 i j or
  2i = ! +
 mX
 j=1
  jy
 2
 i j ;
 with ! =  yL and  +
 mX
 j=1
  j = 1.
6.1 Empirical study 131
 The latter model is known as an ARCH(m) model. A GARCH(m;n) model extends the ARCH(m)
 model, by assuming that  i is not only a function of the long-run average rate and the last m
 observed rates, but also of the last n variances. The model is de ned by
  2i =  yL +
 mX
 j=1
  jy
 2
 i j +
 nX
 j=1
  j 
2
 i j or
  2i = ! +
 mX
 j=1
  jy
 2
 i j +
 nX
 j=1
  j 
2
 i j ;
 with ! =  yL and  +
 mX
 j=1
  j +
 nX
 j=1
  j = 1.
 GARCH (1; 1)
 In the case where m = n = 1, the model reduces to
  2i =  yL +  y
 2
 i 1 +   
2
 i 1 or
  2i = ! +  y
 2
 i 1 +   
2
 i 1;
 with ! =  yL and  +  +  = 1.
 When estimating the parameters, !;  and  are  rst evaluated and  deduced as 1     . A
 stable GARCH(1; 1) model requires  +  < 1 for  to be positive. Note that this model is mean
 reverting since it assumes that the variance rate is always pulled back to the long-run average.
 Parameter estimation
 The GARCH(1; 1)  lter is not directly applied to the log returns, but to the intermediate distribution
 r^i =
 ri   r
 p
 V ar(r)
 : (6.3)
 This distribution has an average of zero which agrees with the approximation (ii) made when building
 the GARCH model. After the GARCH(1; 1) parameters for r^i are estimated, the  ltered interest
 rate is obtained from the model generated r^i through Equation (6.3).
 Example 14. The presence of heteroscedasticity (GARCH e ects), shown in the previous analysis,
 indicates that GARCH modelling is appropriate. The Matlab function garch t is used with Student t
 innovations to estimate the GARCH(1; 1) parameters. After 19 iterations, the monthly SA CPI from
 January 1965 to 2008 gives the following parameters: C =  0:095175, K = 0:010126, GARCH(1) =
6.1 Empirical study 132
 0:94397 and ARCH(1) = 0:045577. Hence, the constant conditional mean/GARCH(1; 1) conditional
 variance model that best  ts the observed data is
 r^t =  0:095175 + "t
  ^2t = 0:010126 + 0:94397   ^
 2
 t 1 + 0:045577  "
 2
 t 1;
 where  ^t is the standard deviation of r^t and "t represents the student t innovations.
 (a) CPI (b) Log returns
 Figure 6.4 Filtered vs raw SA CPI data series.
 Figures 6.4 give plots of the raw vs  ltered simulations for respectively the SA CPI and its log
 return. The  ltered data has kept the general behaviour of the initial data without the unwanted
 trend at the beginning of the log returns.
 (a) Log returns (b) Squared returns
 Figure 6.5 Filtered monthly SA CPI correlograms.
6.1 Empirical study 133
 (a) Log returns (b) Squared returns
 Figure 6.6 Filtered monthly SA CPI partial correlograms.
 Figures 6.5 and 6.6 show that the ACF and PACF of both the  ltered log return series of SA CPI and
 its square have little serial correlation, thus the GARCH  lter is \good". Recall that by de nition
 the increments of L evy distributions are independent, i.e. not correlated. Therefore, this  ltered
 series is better suited for L evy distributions? parameter estimation than the initial series.
 6.1.6 Increasing the data size
 A \good" empirical study, generally requires a large data sample for many reasons. Firstly, the
 parameter estimation for most of the models used in this empirical study needs such a dataset. For
 example, it is advised on the Willmot forum to have at least 700 to 800 data points in the empirical
 series for a GARCH(1; 1) model  tting. Secondly, the bigger the sample data, the smoother the
 QQ-plots and density plots obtained, which will be used to assess the distributions? goodness of  t.
 Thirdly, because there are a lot of data points, there is no need to run multiple simulations as is
 commonly done with Monte Carlo simulations. However, for most of our South African data, the
 available dataset has less than 700 elements. To remedy to this, two alternatives are considered:
 (i) Linear interpolation is used to get a daily dataset from the monthly dataset. This is a common
 practice when dealing with CPI or CPIX, therefore the obtained results are still relevant.
 (ii) The GARCH(1; 1)  lter is used to increase the size of the data. This is done in two steps:
  rst parameters of the GARCH model are estimated; then when generating the  ltered data,
 a bigger dataset is generated.
6.1 Empirical study 134
 With the two approaches, there is no added information in the  ltered dataset. The  rst method
 will only use the most recent information (CPI over the last 4 years for example), thus ignoring old
 data whose behaviour is less likely to be related to today?s data behaviour. This is particularly true
 for SA who is having high CPI hikes nowadays against almost constant CPI in 1965. In general,
 the bigger the time span of the data, the more di erent the initial and  nal sub-data?s behaviour
 are. One inconvenience of this approach, is that the change of granularity of data through linear
 interpolation might generate more (positive) correlation.
 The second approach focuses on maintaining general volatility behaviour of the dataset. But given
 the small size of the data, the GARCH model used to increase the data?s size is not that \well"
  tted to the initial data. Therefore, the forecast (i.e. added data points) might tamper with the
 results. However, because the primary goal of IL securities is to protect against in ation risk, it
 seems reasonable to give priority to reproducing the in ation?s volatility.
 Daily South African CPI
 The descriptive statistics of the daily SA CPI between January 2005 (in fact the 31st December
 2004 for interpolation purpose) and March 2008 are provided in Table 6.8. In total the data has
 1187 data points. The SA CPI went from being more peaked than the normal density function for
 the monthly dataset to less peaked than the normal density function. The daily dataset is also more
 symmetric than its monthly counterpart. These changes suggest that assuming normal distribution
 should give better results with the daily CPI as compared to monthly CPI.
 As expected the only major change in the autocorrelation?s spikes compared to the monthly SA CPI
 is the appearance of spikes at lags 1  30. These are due to the linear interpolation performed in
 between the monthly (i.e. 30 days on average) CPI to get the daily CPI. The spikes for the monthly
 lags should not have change much compared to Figures 6.1 and 6.2.
 For daily CPI, the GARCH  lter also successfully reduces data autocorrelation (See Figure 6.8).
 The  at levels observed for the raw log returns in Figure 6.9(b) are due to the linear interpolation
 (i.e. almost constant return over a month). The  ltered simulations have a \quasi" zero volatility;
 this is more obvious when looking at the simulated CPI (Figure 6.9(a)). The simulated daily SA
 CPI are \perfectly" superposed and linear, thus the CPI is \fully" deterministic which is not wanted
 in the model. Taking a bigger sample size (back up to 2001, i.e. 2648 data points) did not solve the
 problem of linearity and non-zero volatility; however increasing the granularity of the data (weekly
6.1 Empirical study 135
 (a) Log returns ACF (b) Squared returns ACF
 (c) Log returns PACF (d) Squared returns PACF
 Figure 6.7 Daily SA CPI (partial) correlograms.
 instead of daily) might reduce the e ect of the linear interpolation. Taking weekly data should
 preserve some of the volatility and increase the data size following common market practise.
 The previous GARCH  lter is using normal innovations instead of student-t innovations. As can
 be seen in Figure 6.10, when using student-t innovations, the log returns are \almost" constant, i.e.
 zero volatility. The fact that the daily CPI is deterministic is more true with student-t innovations
 than with normal innovations. In the former case, it su ces to estimate the \constant" log return to
 be able to forecast CPI. However, the observed log return of the CPI on the market is not constant.
6.1 Empirical study 136
 (a) Log returns ACF (b) Squared returns ACF
 (c) Log returns PACF (d) Squared returns PACF
 Figure 6.8 Daily  ltered SA CPI (partial) correlograms.
 (a) Raw vs Filtered CPI (b) Raw vs Filtered CPI log return
 Figure 6.9 Daily raw vs  ltered SA CPI (normal innovations).
6.1 Empirical study 137
 (a) Raw vs Filtered CPI (b) Raw vs Filtered CPI log return
 Figure 6.10 Daily raw vs  ltered SA CPI (student-t innovqtions).
 6.1.7 L evy Distributions? Parameter Estimation
 The calibration assumes that the L evy characteristic triplet is not time dependent, i.e.  (t),  (t) and
  (t) are constants over time. Under this assumption, for 1-dimensional processes, there is a single
 risk neutral measure and thus a unique fair price for IL derivatives just as in the Black-Scholes pricing
 theory [47]. The parameter estimation of all L evy distributions was performed using the maximum
 likelihood parameter estimation under R with the package ghyp. Parameters are estimated both for
 the daily and the monthly South African consumer price index.
 Monthly SA CPI
 Figure 6.11 shows the empirical against the normal density functions of monthly SA CPI log returns
 with the corresponding QQ plot. The sample normal data is generated from a normal distribution
 with same mean and standard deviation as the empirical distribution. The plots indicates that
 the normality assumption is \highly" questionable for monthly SA CPI log returns. The normal
 distribution does not reproduce the peakedness of the market around its mean (Figure 6.11(a)),
 neither does it match the upper and lower tails behaviour (Figure 6.11(b)). The  t with the lower
 tail is worse than that with the upper tail and the peaks of the normality plots are not vertically
 aligned. If normal innovations were used in the GARCH  lter instead, then a better  t to the
 empirical data is obtained (see Appendix A.1.1). In summary, the empirical density function is
 taller than the corresponding normal density function, its peak is more to the left; however their
 support is \quite" similar.
6.1 Empirical study 138
 (a) Probability plot (b) QQ plot
 Figure 6.11 Monthly SA CPI probability and QQ plots: Empirical vs normal.
 (a) Probability plot (b) QQ plot
 Figure 6.12 Monthly SA CPI probability plots: Empirical vs L evy.
6.1 Empirical study 139
 The normal density and QQ plots?  t with L evy distributions (Figures 6.12 and A.3) is in general
 better than under the normality assumption. Despite the fact that the parameter estimation did
 not converge for the GH and Student-t distributions, all the L evy distributions did better than
 the normal distribution is reproducing the empirical peakedness. The VG distribution has the
 best performances in matching the empirical peak. However, it is not the best in reproducing the
 empirical tail behaviour. With respect to the latter, the best  t is under the assumption of NIG
 distribution. To accurately model the monthly SA CPI, the appropriate distribution in this case is
 either the VG or the NIG distribution according to what is believed to be more important.
 The L evy distributions parameters for the SA CPI log returns are given in Table 6.3. Notice that
 all the distributions agree on the \high" asymmetry (i.e. non-normality) since beta is non-zero.
 They also agree on the value of the location parameter  . Recall that the Student-t distribution
 calibration did not converge, the corresponding results can thus be ignored.
 Model    ( 105)   LLH
 NIG 32:01031  24:13984 0:22725 22724:94  0:5 773:7516
 H 13:86566  8:11529 3:31747 0:13085 1 803:5996
 GH 15:04463  8:36209 18:18795 0:13220 1:21520 806:6979
 VG 15:04654  8:35547 0 0:13224 1:21616 806:7002
 Sk.Std. 1347687  1347687 46:93758 0:70584  22:10530 771:4530
 Table 6.3 Monthly SA CPI L evy distributions? estimated parameters.
 The AIC model selection returns the VG model as having the best  t to the empirical data among
 all distributions (normal included). Unfortunately the KS test and Chi-squared goodness of  t test
 available under Matlab (resp. kstest and chi2gof ) do not allow a two-samples goodness of  t test.
 The corresponding functions under R (resp. ks.test and chisq.test) do perform a two samples plot,
 but they keep returning a warning when performing the test. The Wafo [108] library for Matlab
 was used instead for the Chi-squared goodness of  t test which is better suited than the KS test in
 this settings. The results of the test are presented in Table 6.9. All the distributions have the same
 p-value; the test was not decisive. Recall that the larger the p-value, the better the  t. However,
 the smallest test value is obtained with the hyperbolic distribution, not the VG. This is not that
 relevant since both distributions have the same p-value.
6.1 Empirical study 140
 Daily SA CPI
 Because of the linearity of  ltered daily SA CPI and their non-zero volatility, some will judge the
 L evy distributions? parameter estimation useless. In fact, if the  ltered SA CPI is linear, then a
 \good" characteristic of its evolution is its slope. However, the parameter estimation might provide
 a better insight into the evolution process.
 Since the probability density function of the  ltered daily SA CPI between 2005 and 2008 is not
 \well behaved" (see the red and solid line in Figure 6.13(a)), the daily SA CPI is extended from
 2001 to 2008. Contrary to the  ltered daily SA CPI, the raw daily CPI is volatile, therefore it might
 also be useful to estimate the L evy distributions? parameter estimation for the raw data.
 (a) Empirical vs Normal (b) Empirical vs L evy
 Figure 6.13 Filtered daily SA CPI 2005 2008 probability plots.
 Even when the sample data is not well behaved as with the daily CPI, the  t with L evy distributions
 proves to be better than that achieved under the normality assumption. When performing model
 calibration, monthly CPI will be preferred to daily CPI because of their previous behaviour. The
 daily CPI will be obtained by interpolation as is the convention in the market.
6.1 Empirical study 141
 (a) Probability plot (b) QQ plot
 Figure 6.14 Filtered daily SA CPI probability and QQ plots: Empirical vs normal.
 (a) Probability plot (b) QQ plot
 Figure 6.15 Filtered daily SA CPI probability plots: Empirical vs L evy.
6.1 Empirical study 142
 6.1.8 Forward rates
 The parameter estimation of the term structure is substantially more di cult than in the case of
 macroeconomic factors. The di culty is due to the fact that a number of di erent assets (in theory
 an in nite number) are driven by only \one" process. Therefore, to extract the parameters of the
 (unique) driving process from the assets (in this case log returns of \zero coupon" bond prices or
 discount factors) is not straightforward.
 For the real world study, the approach for L evy forward rate model proposed by Eberlein and
 Wolfgang [43] is used to deduce the unique L evy driving process from market zero coupon bonds.
 The methodology is  rst presented before providing the results obtained.
 The initial assumptions and notations used in this subsection were introduced in Section 3.1. Con-
 sidering the logarithm of the ratio between the bond price and its forward price on the day before,
 i.e.
 LRi(t; T ) = ln
 pi(t+ 1; t+ T )
 pi(t; t+ 1; t+ T )
 ;
 for i = n; r, where the subscript n (resp. r) stands for nominal (resp. real). The forward price of
 pi(t+ 1; t+ T ) at time t is
 pi(t; t+ 1; t+ T ) =
 pi(t; t+ T )
 pi(t; t+ 1)
 :
 The variable LRi(t 1; t) denotes the daily log return. Using Equation (3.21)
 ln pi(t; T ) = ln pi(0; T ) ln pi(0; t) 
Z t
 0
 Ai(s; t; T )ds+
 Z t
 0
  i(s; t; T )dLs:
 Therefore
 LRi(t; T ) = ln pi(t+ 1; t+ T ) ln pi(t; t+ T ) + ln pi(t; t+ 1)
 = ln pi(0; t+ T ) ln pi(0; t+ 1) 
Z t+1
 0
 Ai(s; t+ 1; t+ T )ds+
 Z t+1
 0
  i(s; t+ 1; t+ T )dLs
  ln pi(0; t+ T ) + ln pi(0; t) +
 Z t
 0
 Ai(s; t; t+ T )ds 
Z t
 0
  i(s; t; t+ T )dLs
 + ln pi(0; t+ 1) ln pi(0; t) 
Z t
 0
 Ai(s; t; t+ 1)ds+
 Z t
 0
  i(s; t; t+ 1)dLs
 =  
Z t+1
 0
 Ai(s; t+ 1; t+ T )ds+
 Z t
 0
 Ai(s; t; t+ T )ds 
Z t
 0
 Ai(s; t; t+ 1)ds
 +
 Z t+1
 0
  i(s; t+ 1; t+ T )dLs  
Z t
 0
  i(s; t; t+ T )dLs +
 Z t
 0
  i(s; t; t+ 1)dLs
6.1 Empirical study 143
 LRi(t; T ) =  
Z t+1
 0
 Ai(s; t+ T )ds+
 Z t+1
 0
 Ai(s; t+ 1)ds+
 Z t
 0
 Ai(s; t+ T )ds
  
Z t
 0
 Ai(s; t)ds 
Z t
 0
 Ai(s; t+ 1)ds+
 Z t
 0
 Ai(s; t)ds
 +
 Z t+1
 0
  i(s; t+ T )dLs  
Z t+1
 0
  i(s; t+ 1)dLs  
Z t
 0
  i(s; t+ T )dLs
 +
 Z t
 0
  i(s; t)dLs +
 Z t
 0
  i(s; t+ 1)dLs  
Z t
 0
  i(s; t)dLs
 =  
Z t+1
 0
 Ai(s; t+ T )ds+
 Z t+1
 0
 Ai(s; t+ 1)ds+
 Z t
 0
 Ai(s; t+ T )ds 
Z t
 0
 Ai(s; t+ 1)ds
 +
 Z t+1
 0
  i(s; t+ T )dLs  
Z t+1
 0
  i(s; t+ 1)dLs  
Z t
 0
  i(s; t+ T )dLs +
 Z t
 0
  i(s; t+ 1)dLs
 =  
Z t+1
 t
 Ai(s; t+ T )ds+
 Z t+1
 t
 Ai(s; t+ 1)ds+
 Z t+1
 t
  i(s; t+ T )dLs  
Z t+1
 t
  i(s; t+ 1)dLs
 =  
Z t+1
 t
 Ai(s; t+ 1; t+ T )ds+
 Z t+1
 t
  i(s; t+ 1; t+ T )dLs:
 The next \stationarity" assumptions allows to get rid of the cumbersome integrals.
 Assumption 14. (i) The volatility structure is stationary, i.e.  i(s; T ) depends only on (T  s)
 for s < T .
 (ii) Similarly, the drift term satisfy some stationarity condition, namely
 A(s; T ) = A(0; T  s) for s < T:
 Note that the second assumptions follows from the  rst assumption and Equation (3.23). It yields
  
Z t+1
 t
 Ai(s; t+ 1; t+ T )ds =  
Z 1
 0
 Ai(s; 1; T )ds := f(T );
 where \:=" means \denoted by" and is used to de ne the function f which is independent of t.
 For the second integral, let?s consider the Ho-Lee volatility structure, i.e.  i(s; T ) =  i  (T  s)
 for constants  i, which will be set equal to one henceforth without loss of generality. The second
 integral becomes
 Z t+1
 t
  i(s; t+ 1; t+ T )dLs =
 Z t+1
 t
  i(s; t+ T )dLs  
Z t+1
 t
  i(s; t+ 1)dLs
 = (t+ T  s)
 Z t+1
 t
 dLs  (t+ 1 s)
 Z t+1
 t
 dLs
 = (T  1)
 Z t+1
 t
 dLs = (T  1)(Lt+1  Lt)
 Hence
 LRi(t; T ) = f(T ) + (T  1)Yt+1 (6.4)
6.1 Empirical study 144
 where Yt+1 = Lt+1  Lt  L1 is Ft+1 measurable and does not depend on T .
 Let D (resp. T) denote the set of days (resp. set of bonds? maturity in years) for which data is
 available. Considering d 2 D and n 2 T, the daily log returns are determined by
 LR(d; d+ n) = lnB(d+ 1; d+ n) + ln
 B(d; d+ 1)
 B(d; d+ n)
 :
 Since B(d; d+ 1) and B(d+ 1; d+ n) are not provided (in the initial discount factors), the negative
 of the logarithm of the bond prices is interpolated with a cubic spline to get those. That is because
 bond prices decrease exponentially with the time to maturity, thus linear interpolation will generate
 errors. On the other hand, the negative of the logarithm of the bond prices is linear in the time
 to maturity for constant interest rates, therefore linear interpolation will introduce less error. The
 transformation, then the interpolation, are conducted for each considered day (i.e. d 2 D) separately.
 For the South African nominal yield curve, daily market coupon bearing bonds between the 31st July
 2000 and the 30th May 2008 (i.e. 1953 trading days) were initially used for computation. Hermite
 polynomials were applied on the interest rates to get zero coupon interest rates with maturities
 ranging from one to thirty years in steps of one year. This dataset was generously provided by
 Nicolette Roussos from Standard Bank, South Africa.
 Zero coupon interest rates at maturities 1, 3, 6 and 9 month(s) were also provided, but not used for
 the calibration process. However, the discount factor of one for the zero year maturity was included
 for the calibration. The short term (i.e. less than a year) interest rates were surely computed
 from the money market (i.e. JIBAR and other) which is quite di erent from the bonds market.
 Furthermore, the concern here, in ation, is more in the long term than in the short term. Figure
 6.16(a) (resp. 6.16(b)) gives the South African nominal yield curve (resp. interpolated negative log
 bond price) on the 30th May 2008.
 Taking the expectation of Equation (6.4) gives
 E[LRi(t; T )] = f(T );
 since E[L1] = 0. Therefore
 LRi(t; T ) E[LRi(t; T )] = (T  1)Yt+1:
 Recall that Yt+1 for t 2 D are independent and equal to L1 in distribution, thus the last equation
 means that the centred log returns are a ne linear in T .
6.1 Empirical study 145
 (a) Yield curve (b) interpolated \-ln"
 Figure 6.16 Nominal yield and interpolated \-ln" on the 30th May 2008.
 For a  xed n 2 T, the sample value yd+1 corresponding to Yd+1 should be computed as
 yd+1 =
 LR(d; d+ n) xn
 n 1
 ;
 where
 xn =
 1
 jDj
 X
 D
 LR(d; d+ n):
 However, since the centred log returns are not exactly linear in n (See Figure 6.17), the sample
 values yd+1 will depend on n. This is not the case of L1 which does not depend on the bonds?
 maturity. Thus, a di erent approach is used: a linear regression through the origin and with the
 points [n  1; LRi(d; d + n)  xn] for n 2 T is performed. The value of yd+1 is the gradient of the
 straight line. Recall that in the expression n 1, n is in years while 1 is in days. Figure 6.18 gives
 the estimated values of L1 between the 31st July 2000 and the 29th May 2008.
 The linear regression is conducted under Matlab with the function poly t. For the 29th May 2008,
 the linear regression of the centred empirical daily log return return the following model:
 y =  0:002458x+ 0:003171:
 The linear regression?s slope estimation is performed for each of trading days (except the last), then
 the gradient are used for the parameter estimation of L1. The procedure is similar to that of the
 macroeconomic factors where the gradients replace the log returns.
6.1 Empirical study 146
 Figure 6.17 SA centred empirical daily return and regression line 29th May 2008.
 Model    ( 105)  ( 105)  LLH
 NIG 621:6958 24:04922 64:19035  2:46877  0:5 10974:58
 H 1499:659 35:21145 9:70121  3:19967 1 10934:69
 GH 25:52232 19:69284 102:9604  2:32664  1:46109 10994:66
 VG 1388:358 36:26896 0  3:36557 0:90868 10934:82
 Sk.Std. 19:03043 19:03043 103:2756  2:32732  1:46824 10994:69
 Table 6.4 Estimated parameters for empirical L1 for SA nominal forward rate.
 The log likelihood estimate and the AIC return the student-t as the distribution (normal included)
 having the best  t with the sample data.
6.1 Empirical study 147
 Figure 6.18 SA estimated L1 between the 31st July 2000 and the 29th May 2008.
 (a) Probability plot (b) QQ Plot
 Figure 6.19 Estimated L1: Empirical vs normal.
6.1 Empirical study 148
 (a) Density (b) Log Density
 Figure 6.20 Estimated L1 probability plots: Empirical vs L evy.
6.1 Empirical study 149
 6.1.9 South African data
 Monthly and daily South African consumer price index have already been studied in details and
 thus will not be mentioned again. The other South African data investigated are the CPIX and the
 money supply aggregates. The SA CPIX is the SA CPI without interest rates on mortgage bonds; it
 is generally used interchangeably with the SA CPI. The money supply aggregates have had di erent
 roles in monetary policy as their reliability as guides has changed. They are mainly indicators of
 the monetary structure and  ow of a given country. Here is a brief description of the main money
 supply aggregates in South Africa:
 1. M0: Deposit of banks, mutual banks with the South African Reserve Bank (SARB) and notes
 and coins outside the SARB and SA mint.
 2. M1A: Coins and banknotes in circulation outside the monetary sector, cheque and transmission
 deposit with banking institutions and post o ce savings bank.
 3. M1: M1A plus other demand deposit with banking institutions.
 4. M2: M1 plus other short term deposits, and all medium term deposits (including savings
 deposits) with the monetary banking institutions.
 5. M3: M2 plus all long term deposit with monetary banking institutions.
 The following subsections present each of these data series more in details. Most of the corresponding
 plots and parameter estimates are provided in Appendix A.
 Consumer Price Index (CPIX)
 The monthly South African Consumer Price Index Metropolitan and urban areas excluding interest
 rates on mortgage bonds (CPIX) time series data is obtained from the South African Reserve Bank
 [97]. The data is from January 1997 to February 2008 normalised at 100 in 2000, which is 134 data
 points. That is not enough data points for a \good" statistical study; the GARCH  lter will be used
 to increase the number of data points. The data series code is KBP7113J and its unit R millions.
 In Figure 6.21, the only positive autocorrelations in the CPIX are at three months intervals, most
 before the 12th month. Given that the spikes at lags 3 and 9 are \quite" small, there might be a semi
 annual cycle in the CPIX instead of a clear annual seasonality as for the SA CPI. This possibility is
 reinforced by Figure 6.22(a), where the spike at lag 6 is higher than that at lag 12. This suggest that
6.1 Empirical study 150
 (a) Log returns (b) Squared returns
 Figure 6.21 Monthly SA CPIX (1997-2008) correlograms.
 most of the previous 12 months? lag autocorrelation was due to the 6 months? lag autocorrelation.
 However, this remark is not true for the squared log returns where the PACF spike at lag 12 is
 higher than that at lag 6.
 (a) Log returns (b) Squared returns
 Figure 6.22 Monthly SA CPIX (1997-2008) partial correlograms.
 The LBP Q-test identi es GARCH e ects only in the raw returns and not in their square. However,
 the Engle?s ARCH test  nds GARCH e ects neither in the log returns nor their square. It is the
 reverse LBP Q-test?s results (i.e. no GARCH e ects in log returns and some GARCH e ects in their
 square) that is more appropriate for a GARCH model calibration. The Engle?s ARCH test output
 just con rms the fact that the sample?s volatility does not vary \much" with time. Nevertheless,
 L evy distributions? parameter estimation will be performed to compare their  t with that of the
6.1 Empirical study 151
 Raw return Squared Raw return
 Lag H p Stat Crit H p Stat. Crit.
 10 1 0 41:8128 18:3070 0 0:8504 5:5649 18:3070
 15 1 0 76:9280 24:9958 0 0:8661 9:2125 24:9958
 20 1 0 104:7414 31:4104 0 0:8752 13:0499 31:4104
 Table 6.5 Ljung-Box-Pierce Q-test for SA Monthly CPIX (1997-2008) raw and squared
 returns.
 Raw return Squared Raw return
 Lag H p Stat Crit H p Stat. Crit.
 10 0 0:7774 6:4354 18:3070 0 0:9992 1:4237 18:3070
 15 0 0:8397 9:6740 24:9958 0 1:0000 1:8261 24:9958
 20 0 0:8456 13:6963 31:4104 0 0:9991 5:8234 31:4104
 Table 6.6 Engle?s ARCH test results for SA Monthly CPIX (1997-2008) raw and squared
 returns.
 normal distribution and the in ation index in the South African settings will always be the CPI and
 not the CPIX.
 (a) CPIX (b) CPIX Log returns
 Figure 6.23 Filtered vs raw monthly SA CPIX (1997-2008) data series.
 Figures 6.24 and 6.25 show that despite the small size of the sample, the GARCH  lter has reduced
 its autocorrelation.
6.1 Empirical study 152
 (a) Log returns (b) Squared returns
 Figure 6.24 Filtered monthly SA CPIX (1997-2008) correlograms.
 (a) Log returns (b) Squared returns
 Figure 6.25 Filtered monthly SA CPIX (1997-2008) partial correlograms.
 For the L evy distributions? parameter estimation, the GARCH  lter is used to multiply the sample
 size by 10.
 Figures 6.26 and 6.27 give the normality plots and QQ-plots with the empirical distribution. Un-
 fortunately, none of the parameter estimation for L evy distributions did converge. However, the  t
 with the L evy distributions is still better than that with the normal distribution.
 Money Supply aggregate M1A
 The monthly South African Monetary aggregate M1(A) time series data is obtained from the South
 African Reserve Bank [97]. The data is from March 1979 to December 2007 that is 346 data points
 overall. That is not enough data points for a \good" statistical study; the GARCH  lter will be used
6.1 Empirical study 153
 (a) Probability plot (b) QQ plot
 Figure 6.26 Filtered monthly SA CPIX probability and QQ plots: Empirical vs normal.
 (a) Probability plot (b) QQ plot
 Figure 6.27 Filtered monthly SA CPIX probability plots: Empirical vs L evy.
6.1 Empirical study 154
 Model      LLH
 NIG 4303:523  4206:992 0:04419 0:20696  0:5 2683:354
 H 797:9752  704:9338 0:08325 0:16626 1 2683:190
 GH 1128:630  1033:692 0:07289 0:17756 1:32246 2683:241
 VG 1383:623  1254:324 0 0:13672 18:34853 2683:819
 Sk.Std. 3812:483  3812:483 0:09803 0:26698  70:0834 2682:768
 Table 6.7 Estimated parameters for monthly SA CPIX log returns.
 to triple the number of data points. The data series code is KBP1374M and its unit R millions.
 Money Supply aggregate M1, M2 and M3
 The monthly South African Monetary aggregates M1 (resp. M2, M3) time series data is obtained
 from the South African Reserve Bank [97]. The data is from March 1965 to December 2007 that is
 514 data points overall. The GARCH  lter will be used to double the number of data points. The
 data series code is KBP1373M (resp. KBP1372M , KBP1370M) and its unit R millions.
 Series  (%)  (%) Skew. Kurt. Min.(%) Max.(%)
 M1A 1:36 4:14  0:1212 2:8675  10:16 12:85
 M1 1:21 3:40  0:0826 3:4912  9:07 13:28
 M2 1:20 1:65 0:0645 3:2218  3:41 6:44
 M3 1:14 1:23 0:0815 3:4580  2:74 5:44
 CPI(Monthly) 0:74 0:70 0:9198 4:6009  0:74 4:21
 CPI(Daily) 0:017537 0:013374 0:2031 2:4956  0:0052327 0:051089
 CPIX 0:53 0:40 0:3912 3:1446  0:37 1:75
 Table 6.8 Descriptive statistics of S.A. Data series log returns.
 Table 6.9 Chi squared Pearson?s test.
 Normal GH H VG NIG Sk.t
 p Test p Test P Test P Test P Test P Test
 CPI 0 345.3730 0 211.4566 0 192.0096 0 231.0064 0 295.7781 0 320.7814
6.1 Empirical study 155
 6.1.10 United State of America data
 The American data studied is similar to the South African data seen in the previous subsection.
 However, although the US money aggregates are named identically to their South African counter-
 part, they are not exactly identical. The following details their principal components [102]:
 1. M0: The total of all physical currency, plus accounts at the central bank that can be exchanged
 for physical currency.
 2. M1: M0 minus those portions of M0 held as reserves or vault cash plus the amount in demand
 accounts (\checking" or \current" accounts).
 3. M2: M1 plus most savings accounts, money market accounts, and small denomination time
 deposits (certi cates of deposit of under $100; 000).
 4. M3: M2 plus all other CDs (large time deposits, institutional money market mutual fund
 balances), deposits of eurodollars and repurchase agreements.
 The CPIX is particular to South Africa, therefore there is only one potential in ation index for US.
 Table 6.10 (resp. 6.11) contains descriptive statistics (resp. hypothesis tests) of all our considered
 data samples. For a brief overview of these descriptive statistics and hypothesis tests see Subsection
 6.1.2.
 Series  (%)  (%) Skew. Kurt. Min.(%) Max.(%)
 CPI (186 yrs) 0:22 1:45 1:7607 70:1835  16:83 19:72
 CPI (70 yrs) 0:32 0:47 2:2667 24:2856  1:40 5:72
 M1 0:082719 2:08  0:4315 4:1987  9:22 10:81
 M1 (Adj.) 0:085279 0:62 1:6958 63:5971  6:84 10:14
 M2 0:11 0:59 0:1760 3:2690  2:56 3:02
 M2 (Adj.) 0:11 0:19 2:4551 55:2031  1:47 3:22
 M3 0:12 0:43 0:0537 3:1615  1:19 2:18
 M3 (Adj.) 0:12 0:19 1:3564 18:6892  0:88 2:31
 Table 6.10 Descriptive statistics of USA Data series log returns.
 In Table 6.10, the kurtosis is always larger than three, which would have been the kurtosis if the
 sample data series were taken from normal distributions. This behaviour is commonly observed
6.1 Empirical study 156
 in the market [8, 27, 109, 55, 9]. In particular the money supply aggregates?s kurtosis increases
 considerably when adjusting it for seasonality. However, the maximum-likelihood estimators (see
 Subsection 6.2.1) used for L evy distributions? parameter estimation assumes that the data points
 are independent identically distributed, i.e. no autocorrelation. Therefore the seasonally adjusted
 data series, which are less normal, are more appropriate than their non-adjusted counterpart for the
 parameter estimation.
 The non-adjusted money aggregate M1 is the only one with a negative skewness. Its general be-
 haviour might di er from that of the other money aggregates, therefore it will not be used later
 when modelling the money aggregate as a macroeconomic factor.
 Series JB K-S Lill. AD
 H p H p H p H p
 CPI (186 yrs) 1 10 3 1 0 1 10 3 1 10 3
 CPI (70 yrs) 1 10 3 1 0 1 10 3 1 10 3
 M1 1 10 3 1 0 1 10 3 1 10 3
 M1 (Adj.) 1 10 3 1 0 1 10 3 1 10 3
 M2 1 0:0054 1 0 1 10 3 1 10 3
 M2 (Adj.) 1 10 3 1 0 1 10 3 1 10 3
 M3 0 0:3464 1 0 0 0:5000 0 5%
 M3 (Adj.) 1 10 3 1 0 1 10 3 1 10 3
 Table 6.11 Hypothesis Tests of US Data series log returns.
6.1 Empirical study 157
 Money Supply aggregate M1 and M2 (Adjusted and unadjusted)
 The weekly American Monetary aggregates M1, M2 (both adjusted and unadjusted) time series data
 is obtained from the Federal Reserve System [103]. The data is from the 5th January 1981 to the
 21st April 2008 that is 1425 data points overall. The data was obtained on the 1st May 2008 and is
 measured in billions of US dollars.
 Some of the results obtained with theses data series are presented in Appendix B.
 Money Supply aggregate M3
 The weekly seasonally adjusted and unadjusted American Monetary aggregates M3 time series data
 were obtained from the Federal Reserve System [103]. The data is from the 5th January 1981 to the
 13tH March 2006 that is 1315 data points overall. The data is for the 1st May 2008 and measured
 in billions of US dollars.
 The Federal Reserve ceased publishing M3 statistics in March 2006, claiming that M3 did not appear
 to convey additional information about economic activity compared to M2, had not been used in
 determining economic policy, and that the costs to collect M3 data outweighed the bene ts. Some
 of the data used to calculate M3 are still collected and published on a regular basis.
 The results obtained with all the previous US macroeconomic factors con rms the better  t of L evy
 distributions compare to that of the conventional normal distribution. Only results obtained with
 the real and nominal US yield curves are presented in this subsection.
 Nominal and Real Yields
 The nominal and real daily historical yield curves were downloaded from the U.S. treasury website
 [107] from January 2003 to September 2008. The sample data has 1430 data points, with only
 the trading days considered. The  t with the L evy distributions are still better than that under
 normality assumption (Figures 6.29 and 6.29). In both cases the best  t according to the AIC is
 obtained with the Student-t distribution. This is also the case when using the log-likelihood estimate.
 Further calibration results are provided in Appendix B.3.
6.1 Empirical study 158
 (a) Empirical vs Normal (b) Empirical vs L evy
 Figure 6.28 Nominal yield curve: Empirical vs model.
 (a) Empirical vs Normal (b) Empirical vs L evy
 Figure 6.29 Real yield curve: Empirical vs model.
6.1 Empirical study 159
 Model    ( 105)   LLH
 NIG 2728:92 24:64427 53:96559  4:83289  10 6  0:5 9056:900
 H 3938:503 26:19406 34:55043  5:20755  10 6 1 9050:542
 GH 73:83758 70:64118 91:40627  1:40600  10 5  3:19922 9072:226
 VG 4437:195 3:61602 0  6:79980  10 7 35 9046:039
 Sk.Std. 67:62844 67:62844 91:2076  1:36021  10 5  3:18674 9072:230
 Table 6.12 Estimated parameters for US nominal forward rate.
 Model    ( 105)  ( 105)  LLH
 NIG 1890:162  51:50919 60:56153 1:65484  0:5 8725:085
 H 2909:199  54:34723 32:85109 1:68966 1 8720:764
 GH 52:25032  44:33243 93:34401 1:46063  2:33771 8729:895
 VG 3304:674  45:19182 0 1:42259 1:72004 8717:872
 Sk.Std. 39:62266  39:62266 94:60124 1:28412  2:38186 8729:890
 Table 6.13 Estimated parameters for US real forward rate.
 Table 6.14 Kolmogorov-Smirnov test.
 ( 103) Normal GH H VG NIG Sk.t
 p D p D P D P D P D P D
 US Nom. 2.423 68.5 43.45 51.7 693.6 26.6 537.6 30.1 840.9 23.1 86.63 46.9
 US Real 43.45 51.7 866.4 22.4 840.9 23.1 240.8 38.5 240.8 38.5 813.7 23.8
6.2 Option pricing 160
 6.2 Option pricing
 After the statistical study which was comparing the  t with the empirical data of normal distribution
 against that of L evy distributions, this section reviews some calibration tools for option pricing. It
 begins by the maximum-likelihood parameter estimation method that was used in the previous
 section without a speci c description. Afterward, the discretisation (i.e. numerical implementation)
 of the Fast Fourier Transform (FFT) is presented.
 6.2.1 Maximum-Likelihood Estimator
 Contrary to the normal distribution for which the parameters (sample?s average and variance) are
 easily computed, there is no formula to estimated a L evy distribution parameters. The maximum
 likelihood estimator (MLE) method described in this subsection can be used for L evy distributions?
 parameter estimation. It is a common method in statistic for curve  tting and parameter estimation.
 Considering independent and identically distributed (i.i.d.) observations x1; x2;    ; xn, the likeli-
 hood function of parameter  is
 lik( ) = f(x1; x2;    ; xnj );
 where f is the frequency function. If the distribution is discrete, the likelihood function gives the
 probability of observing the given data as function of the parameter  . With maximum likelihood
 estimator (MLE) a maximisation of the probability is performed. Since x1; x2;    ; xn are assumed
 i.i.d. and the natural logarithm is a monotonic function a maximisation is conducted on the log
 likelihood function
 l( ) =
 nX
 1
 ln[f(xij )]:
 The MLE also have good theoretical properties such as being asymptotically e cient according
 to Cramer-Rao Inequality. Of course, this parameter estimation can also be used for a normal
 distribution. In fact, this case yields unbiased estimations of  and  2.
 For the GH distribution, the log likelihood function is
 l( ) = ln a( ;  ;  ;  ) +
  
 
2
  
1
 4
  nX
 i=1
 ln[ 2 + (x  )2]
 +
 nX
 i=1
 h
 lnK  12 ( 
p
  2 + (xi   )2) +  (xi   )
 i
 ;
6.3 Conclusion 161
 with a de ned in Section 2.4.1. Simpler expressions are obtained for hyperbolic and NIG distributions
 by taking respectively  = 1 and  =  
1
 2
 .
 6.2.2 Discretisation of the FFT
 Recall from Section 2.5 that the formula to be numerically evaluated is
 cT (K) =
 exp(  K)
  
<
  Z +1
 0
 e ivK T (v)dv
  
:
 First an approximation of 1 is made. Let  represents the integration step and N 2 N be a \large"
 enough number, the previous equation can be approximated by
 cT (K)  
exp(  K)
  
<
 "Z N 
0
 e ivK T (v)dv
 #
 :
 The discretisation of the integral can be done using the Simpson?s weighting rule [26], the midpoint
 rule, the trapeze method or any other common integral discretisation scheme. The trapeze method
 will be used here. The call fair price is now
 cT (K)  
exp(  K)
  
<
 2
 4
 NX
 j=0
 e ivjK T (vj)  j
 3
 5 ;
 where
  j =
 8
 <
 :
 0:5 for j = 0; N
 1 otherwise:
 The Fast Fourrier Transform (FFT) returns the call price for N + 1 strike price with a regular
 interval. The FFT parameters (initial strike and strike step) are chosen such that the strike of the
 options to be priced are in the range of the estimated. An interpolation will also eventually be used
 to get the wanted option price(s).
 6.3 Conclusion
 An empirical study of the sample data from the South African and American markets was performed
 in this chapter. The results for the developing and developed markets all agree on the fact that
 market data is non-normal (and non-lognormal). This agrees with well documented stylised facts
 highlighting the non-normality of markets.
 It is shown here that a calibration using L evy distributions and speci cally Generalised Hyperbolic,
 Hyperbolic, Variance Gamma, Normal Inverse Gaussian and Student-t prove to give better results
6.3 Conclusion 162
 in \every" case. Moreover, the calibration cost with L evy distributions might eventually be \less"
 expensive than under normality assumption. A number of test of normality and goodness of  t test
 are also used to quantify how inappropriate the normal assumption is and how well each distribution
 performed. In most of the cases, the best  t is obtained with the Student-t distribution.
Appendix A
 Empirical Study SA
 This Appendix is divided in two sections which gives parameters estimated and other results ob-
 tained. Section A.1 (resp. A.2) presents results obtained from a GARCH  lter with normal (resp.
 student t) innovations.
 A.1 Normal innovations
 In this section a GARCH  lter with normal innovations was used. Recall that the  lter is meant to
 reduce the autocorrelation in the sample data.
 A.1.1 Monthly SA CPI
 When using normal innovations, the  t with the lower tail is better than that with the upper tail
 (Figure A.1(b)), i.e. the normality assumption will have more di culties predicting high in ation
 increases than low increases. But, an investor is more concerned about high in ation rates than low
 rates, these \forecasting" performances are contrary to what is needed. In summary, the empirical
 density function is taller than the corresponding normal density function, however their support and
 skin?s shape are \quite" similar.
 A.1.2 Consumer Price Index (CPIX)
 Figures A.6 and A.7 show that despite the the small size of the sample, the GARCH  lter has
 reduced its autocorrelation.
 163
A.1 Normal innovations 164
 (a) Probability plot (b) QQ plot
 Figure A.1 Monthly SA CPI probability and QQ plots: Empirical vs normal.
 (a) Probability plot (b) QQ plot
 Figure A.2 Monthly SA CPI probability plots: Empirical vs L evy.
A.1 Normal innovations 165
 (a) GH (b) H
 (c) NIG (d) VG
 (e) Skw. Std. (f) Normal
 Figure A.3 Monthly SA CPI QQ plots (normality assumption).
A.1 Normal innovations 166
 (a) GH (b) H
 (c) NIG (d) VG
 (e) Skw. Std. (f) Normal
 Figure A.4 Filtered daily SA CPI QQ plots.
A.1 Normal innovations 167
 (a) CPIX (b) CPIX Log returns
 Figure A.5 Filtered vs raw monthly SA CPIX (1997-2008) data series.
 (a) Log returns (b) Squared returns
 Figure A.6 Filtered monthly SA CPIX (1997-2008) correlograms.
 (a) Log returns (b) Squared returns
 Figure A.7 Filtered monthly SA CPIX (1997-2008) partial correlograms.
A.1 Normal innovations 168
 Model      
NIG 342:72583 18:73317 0:020182 0:0062122  0:5
 H 369:70173 18:44541 0:017584 0:0062295 1
 GH 465:85525 19:08309 1:90  10 5 0:0061911 6:38998
 VG 465:93679 19:09688 0 0:0061914 6:39349
 Sk.Std. 18:12631 18:12631 0:00423256 0:0062488 0:053276
 Table A.1 Monthly SA CPI L evy distributions? estimated parameters.
 For the L evy distributions? parameters estimation, the GARCH  lter is used to multiply the sample
 size by 10.
 A.1.3 Money Supply aggregate M1A
 (a) M1A (b) M1A Log returns
 Figure A.8 Filtered vs raw monthly SA M1A (1979-2007) data series.
 A.1.4 Money Supply aggregate M1
 A.1.5 Money Supply aggregate M2
A.1 Normal innovations 169
 (a) Log returns (b) Squared returns
 Figure A.9 Filtered monthly SA M1A (1979-2007) correlograms.
 (a) M1 (b) M1 Log returns
 Figure A.10 Filtered vs raw monthly SA M1 (1965-2007) data series.
 (a) Log returns (b) Squared returns
 Figure A.11 Filtered monthly SA M1 (1965-2007) correlograms.
A.1 Normal innovations 170
 (a) Log returns (b) Squared returns
 Figure A.12 Monthly SA Money Supply M2 (1965-2007) correlograms.
 (a) Log returns (b) Squared returns
 Figure A.13 Monthly SA Money Supply M2 (1965-2007) partial correlograms.
 Raw return Squared Raw return
 Lag H p Stat Crit H p Stat. Crit.
 10 1 0 87:7421 18:3070 0 0:6379 7:9071 18:3070
 15 1 0 161:2922 24:9958 0 0:1300 21:2190 24:9958
 20 1 0 188:6595 31:4104 0 0:1950 25:1705 31:4104
 Table A.2 Ljung-Box-Pierce Q-test for SA Monthly M2 (1965-2007) raw and squared
 returns.
A.1 Normal innovations 171
 Raw return Squared Raw return
 Lag H p Stat Crit H p Stat. Crit.
 10 0 0:6974 7:2942 18:3070 0 0:7489 6:7487 18:3070
 15 0 0:2219 18:8214 24:9958 0 0:6849 11:9225 24:9958
 20 0 0:3202 22:3815 31:4104 0 0:8147 14:3047 31:4104
 Table A.3 Engle?s ARCH test results for SA M2 (1965-2007) raw and squared returns.
 The LBP Q-test identi es GARCH e ects only in the raw returns and not in their square. However,
 the Engle?s ARCH test  nds GARCH e ects neither in the log returns nor their square. The Engle?s
 ARCH test output con rms the fact that the sample?s volatility does not vary \much" with time.
 Nevertheless, L evy distributions? parameters estimation will be performed to compare their  t with
 that of the normal distribution.
 (a) M2 (b) M2 Log returns
 Figure A.14 Filtered vs raw monthly SA M2 (1965-2007) data series.
 Money Supply aggregate M3
A.1 Normal innovations 172
 (a) Log returns (b) Squared returns
 Figure A.15 Filtered monthly SA M2 (1965-2007) correlograms.
 (a) Log returns (b) Squared returns
 Figure A.16 Filtered monthly SA M2 (1965-2007) partial correlograms.
 (a) Log returns (b) Squared returns
 Figure A.17 Monthly SA Money Supply M3 (1965-2007) correlograms.
A.1 Normal innovations 173
 (a) Log returns (b) Squared returns
 Figure A.18 Monthly SA Money Supply M3 (1965-2007) partial correlograms.
 Raw return Squared Raw return
 Lag H p Stat Crit H p Stat. Crit.
 10 1 0 98:2356 18:3070 0 0:3576 10:99907 18:3070
 15 1 0 155:7858 24:9958 0 0:0545 24:6732 24:9958
 20 1 0 178:4955 31:4104 1 0:0466 31:7032 31:4104
 Table A.4 Ljung-Box-Pierce Q-test for SA Monthly M3 (1965-2007) raw and squared
 returns.
 Raw return Squared Raw return
 Lag H p Stat Crit H p Stat. Crit.
 10 0 0:3916 10:5742 18:3070 0 0:4477 9:9184 18:3070
 15 0 0:1650 20:1843 24:9958 0 0:6191 12:7825 24:9958
 20 0 0:1324 27:1015 31:4104 0 0:8047 14:4929 31:4104
 Table A.5 Engle?s ARCH test results for SA M3 (1965-2007) raw and squared returns.
A.2 Student t innovations 174
 (a) M3 (b) M3 Log returns
 Figure A.19 Filtered vs raw monthly SA M3 (1965-2007) data series.
 (a) Log returns (b) Squared returns
 Figure A.20 Filtered monthly SA M3 (1965-2007) correlograms.
 A.2 Student t innovations
 A.2.1 SA CPIX
 A.3 Forward estimates
A.3 Forward estimates 175
 (a) Log returns (b) Squared returns
 Figure A.21 Filtered monthly SA M3 (1965-2007) partial correlograms.
 (a) Empirical vs Normal (b) Empirical vs L evy
 Figure A.22 Probability plots: Empirical vs model.
A.3 Forward estimates 176
 (a) GH (b) H
 (c) NIG (d) VG
 (e) Skw. Std. (f) Normal
 Figure A.23 Estimated L1 QQ plots.
Appendix B
 Empirical Study US
 B.1 Normal innovations
 B.1.1 Consumer Price Index
 The monthly United States of America Consumer Price Index is taken from the Bureau of Labor
 Statistics? (BLS) website [106]. The data is sampled from the 31rst December 1821 to the 30th
 November 2007. The entire historical data covers a period of 186 years corresponding to 1649
 observation points. The empirical study is  rst conducted on the entire data, then on the most
 recent half. The latter coincide with the period going from the 31rst January 1937 to the 30th
 November 2007, with 850 observations covering 70 years. The sample size is big enough for the
  ltering using the GARCH(1; 1) to give \good" results and 70 years is big enough to cover the
 investment of a particularly long lived client of a pension fund.
 Figures B.2 and B.1 show that there is no annual seasonality (i.e. high spike at lag 12). This is
 quite surprising, this might re ect the \success" of in ation targeting in US.
 Figure B.6 (resp. B.7) shows the empirical density and log density functions of monthly log returns
 of US CPI (1821-2007). Each graph also has the normal (resp. L evy) probability density and log
 density functions with parameters evaluated from the sample data and presented in Tables B.3
 and 6.10. The plots indicate that the monthly US CPI is non-normal and the L evy distributions
 are more suited for the calibration. The empirical density and log density functions are more
 peaked than the corresponding normal distribution, but L evy distributions reproduce \fairly" well
 the peakedness. The normal distribution?s density function also has fatter tails than its empirical
 177
B.1 Normal innovations 178
 (a) Log returns (b) Squared returns
 Figure B.1 Monthly USA CPI (1821-2007) correlograms.
 (a) Log returns (b) Squared returns
 Figure B.2 Monthly USA CPI (1821-2007) partial correlograms.
 (a) CPI (b) Log returns
 Figure B.3 Filtered vs raw monthly USA CPI (1821-2007) data series .
B.1 Normal innovations 179
 Raw return Squared Raw return
 Lag H p Stat Crit H p Stat. Crit.
 10 1 0 303:2123 18:3070 1 0 964:3 18:3
 15 1 0 324:1402 24:9958 1 0 1570:2 25:0
 20 1 0 400:5362 31:4104 1 0 1854:6 31:4
 Table B.1 Ljung-Box-Pierce Q-test for USA CPI (1821-2007) raw and squared returns.
 Raw return Squared Raw return
 Lag H p Stat Crit H p Stat. Crit.
 10 1 0 453:6761 18:3070 1 0 112:9745 18:3070
 15 1 0 609:4648 24:9958 1 0 280:9518 24:9958
 20 1 0 813:3873 31:4104 1 0 383:9034 31:4104
 Table B.2 Engle?s ARCH test results for USA CPI (1821-2007) raw and squared returns.
 (a) Log returns (b) Squared returns
 Figure B.4 Filtered monthly USA CPI (1821-2007) correlograms.
 counterpart; with a wider support. While L evy distribution have a support \almost" identical to
 that of the empirical density function. However, they do not perform so well in reproducing the tail
 behaviour of the empirical sample. This is more obvious when looking at the QQ plots (Figures B.8
 and B.9). In other words, the empirical density and log density functions are taller, skinnier, but
 with a smaller support than their normal counterpart. While the L evy distributions? density and
 log density functions have the same general structure as the corresponding empirical function, with
B.1 Normal innovations 180
 (a) Log returns (b) Squared returns
 Figure B.5 Filtered monthly USA CPI (1821-2007) partial correlograms.
 (a) Density (b) Log Density
 Figure B.6 Monthly USA CPI (1821-2007) probability plots: Empirical vs normal.
B.1 Normal innovations 181
 (a) Density (b) Log Density
 Figure B.7 Monthly USA CPI (1821-2007) probability plots: Empirical vs L evy.
 some mismatches on the tails.
 Model      LLH
 NIG 28:00664 1:153026 0:00339 0:00248  0:5 5805:90
 H 169:93788 3:39996 1:77  10 6 0:00238 1 5669:98
 GH 23:66755 1:06193 0:00362 0:00248  0:56210 5805:17
 VG 114:86472 2:85505 0 0:00239 0:54532 5726:15
 Sk.Std. 0:09226 0:09226 N/A 0:00249 0:49989 5788:29
 Table B.3 Estimated parameters for USA CPI (1821-2007).
B.1 Normal innovations 182
 (a) Normal (b) L evy
 Figure B.8 Monthly US CPI QQ plots (individually).
B.1 Normal innovations 183
 (a) GH (b) H
 (c) NIG (d) VG
 (e) Skw. Std. (f) Normal
 Figure B.9 Monthly US CPI QQ plots.
B.1 Normal innovations 184
 B.1.2 Consumer Price Index (End half)
 (a) Log returns (b) Squared returns
 Figure B.10 Monthly USA CPI (1937-2007) correlograms.
 (a) Log returns (b) Squared returns
 Figure B.11 Monthly USA CPI (1937-2007) partial correlograms.
 Raw return Squared Raw return
 Lag H p Stat Crit H p Stat. Crit.
 10 1 0 647:9843 18:3070 1 0 55:6300 18:3070
 15 1 0 816:7679 24:9958 1 0 72:3584 24:9958
 20 1 0 853:6211 31:4104 1 0 78:5209 31:4104
 Table B.4 Ljung-Box-Pierce Q-test for USA CPI (1937-2007) raw and squared returns.
B.1 Normal innovations 185
 Raw return Squared Raw return
 Lag H p Stat Crit H p Stat. Crit.
 10 1 0 40:6426 18:3070 0 1 0:4588 18:3070
 15 1 0 50:7111 24:9958 0 1 0:6084 24:9958
 20 1 0:0001 51:7990 31:4104 0 1 0:6089 31:4104
 Table B.5 Engle?s ARCH test results for USA CPI (1937-2007) raw and squared returns.
 (a) CPI (b) Log returns
 Figure B.12 Filtered vs raw monthly USA CPI (1937-2007) data series .
 (a) Log returns (b) Squared returns
 Figure B.13 Filtered monthly USA CPI (1937-2007) correlograms.
B.1 Normal innovations 186
 (a) Log returns (b) Squared returns
 Figure B.14 Filtered monthly USA CPI (1937-2007) partial correlograms.
 (a) Density (b) Log Density
 Figure B.15 Monthly USA CPI (1937-2007) probability plots: Empirical vs normal.
 Model      LLH
 NIG 42:17912  2:20823 0:00344 0:00278  0:5 3055:83
 H 190:76526  1:44290 1:24  10 5 0:00268 1 3020:98
 GH 55:97538  2:35718 0:00287 0:00279  0:32058 3056:39
 VG 137:34979 0:10904 0 0:00264 0:60524 3038:41
 Sk.Std. 0:51414  0:51414 N/A 0:00272 0:49952 3047:13
 Table B.6 Estimated parameters for USA CPI (1937-2007).
B.1 Normal innovations 187
 (a) Density (b) Log Density
 Figure B.16 Monthly USA CPI (1937-2007) probability plots: Empirical vs L evy.
 (a) Normal (b) L evy
 Figure B.17 Monthly US CPI QQ plots.
B.1 Normal innovations 188
 (a) GH (b) H
 (c) NIG (d) VG
 (e) Skw. Std. (f) Normal
 Figure B.18 Monthly US CPI QQ plots (individually).
B.1 Normal innovations 189
 B.1.3 Money Supply aggregate M1
 The weekly American Monetary aggregates M1 time series data is obtained from the Federal Reserve
 System [103]. The data is from the 5th January 1981 to the 21st April 2008 that is 1425 data points
 overall. That is more than enough data points for a \good" statistical study. The data is for the
 1st May 2008 and measured in billions of US dollars.
 (a) Log returns (b) Squared returns
 Figure B.19 Weekly USA Money Supply M1 (1981-2008) correlograms.
 (a) Log returns (b) Squared returns
 Figure B.20 Weekly USA Money Supply M1 (1981-2008) partial correlograms.
 The current estimated GARCH parameters might generate a highly volatile  ltered sample data
 (Figure B.21). This might be due to the high volatility of the money aggregate M1. Notice that in
 Table 6.10, M1 has the highest volatility which is about three time that of the next most volatile
 money aggregate. Therefore, for stability reasons, the other two aggregates M2 and M3 will be
B.1 Normal innovations 190
 Raw return Squared Raw return
 Lag H p Stat Crit H p Stat. Crit.
 10 1 0 2583:8 18:3070 1 0 416:0 18:3070
 15 1 0 4099:5 24:9958 1 0 892:9 24:9958
 20 1 0 5456:4 31:4104 1 0 1096:8 31:4104
 Table B.7 Ljung-Box-Pierce Q-test for USA Weekly M1 (1981-2008) raw and squared
 returns.
 Raw return Squared Raw return
 Lag H p Stat Crit H p Stat. Crit.
 10 1 0 265:9485 18:3070 1 0:0010 29:5679 18:3070
 15 1 0 498:9969 24:9958 1 0 69:2088 24:9958
 20 1 0 530:3411 31:4104 1 0 79:2994 31:4104
 Table B.8 Engle?s ARCH test results for USA M1 (1981-2008) raw and squared returns.
 (a) M1 (b) M1 Log returns
 Figure B.21 Filtered vs raw weekly USA M1 (1981-2008) data series.
B.1 Normal innovations 191
 preferred to M1 for our calibrations. In Table 6.10, M1 is also the only sample with negative
 skewness, this might also be due to its high volatility.
 (a) Log returns (b) Squared returns
 Figure B.22 Filtered weekly USA M1 (1981-2008) correlograms.
 (a) Log returns (b) Squared returns
 Figure B.23 Filtered weekly USA M1 (1981-2008) partial correlograms.
 Figures B.22 and B.23 show that the GARCH  lter successfully reduced the autocorrelation in our
 sample data. However, there is still a \slight" positive correlation in the squared returns. This might
 be what is sometime translated by a highly volatile  ltered sample data.
B.1 Normal innovations 192
 (a) Density (b) Log Density
 Figure B.24 Weekly USA M1 (1981-2008) probability plots: Empirical vs normal.
 (a) Density (b) Log Density
 Figure B.25 Weekly USA M1 (1981-2008) probability plots: Empirical vs L evy.
B.1 Normal innovations 193
 Model      LLH
 NIG 18:71749 0:86203 0:00567 0:00121  0:5
 H 105:69644 2:12448 6:61  10 6 0:00108 1 4222:08
 GH 10:33677 0:79936 0:00682 0:00120  0:69369 4340:19
 VG 70:50968 4:08735 0 0:00041 0:55041 4261:61
 Sk.Std. 0:02403 0:02403 N/A 0:00122 0:49993 4332:68
 Table B.9 Estimated parameters for weekly USA M1 (1981-2008).
 (a) Normal (b) L evy
 Figure B.26 Weekly US M1 QQ plots: Empirical vs normal.
B.1 Normal innovations 194
 (a) GH (b) H
 (c) NIG (d) VG
 (e) Skw. Std. (f) Normal
 Figure B.27 Weekly US M1 QQ plots.
B.1 Normal innovations 195
 Money Supply aggregate M1: Seasonally adjusted
 The weekly seasonally adjusted American Monetary aggregates M1 time series data is obtained from
 the Federal Reserve System [103]. The data is from the 5th January 1981 to the 21st April 2008 that
 is 1425 data points overall. That is more than enough data points for a \good" statistical study.
 The data is for the 1st May 2008 and measured in billions of US dollars.
 (a) Log returns (b) Squared returns
 Figure B.28 Seasonally adjusted weekly USA Money Supply M1 (1981-2008) correlo-
 grams.
 (a) Log returns (b) Squared returns
 Figure B.29 Seasonally adjusted weekly USA Money Supply M1 (1981-2008) partial
 correlograms.
 Figures B.31 and B.32 show that the GARCH  lter was not that successful this time in reducing
 the sample autocorrelation.
 The normal distribution perform better with this sample (see Figure B.33) than with the previous
B.1 Normal innovations 196
 Raw return Squared Raw return
 Lag H p Stat Crit H p Stat. Crit.
 10 1 0 210:6978 18:3070 1 0 242:8464 18:3070
 15 1 0 343:8959 24:9958 1 0 242:8661 24:9958
 20 1 0 392:7793 31:4104 1 0 242:9056 31:4104
 Table B.10 Ljung-Box-Pierce Q-test for seasonally adjusted USA Weekly M2 (1981-2008)
 raw and squared returns.
 Raw return Squared Raw return
 Lag H p Stat Crit H p Stat. Crit.
 10 1 0 243:5764 18:3070 1 0 67:6032 18:3070
 15 1 0 242:7911 24:9958 1 0 67:3666 24:9958
 20 1 0 242:0485 31:4104 1 0 67:1300 31:4104
 Table B.11 Engle?s ARCH test results for seasonally adjusted USA M1 (1981-2008) raw
 and squared returns.
 (a) M1 Adj. (b) M1 Adj. Log returns
 Figure B.30 Filtered vs raw weekly USA M1 Adj. (1981-2008) data series.
B.1 Normal innovations 197
 (a) Log returns (b) Squared returns
 Figure B.31 Filtered weekly USA M1 Adj. (1981-2008) correlograms.
 (a) Log returns (b) Squared returns
 Figure B.32 Filtered weekly USA M1 Adj. (1981-2008) partial correlograms.
 samples. For the  rst time, the  t under normality assumption is better than with one of the L evy
 distribution (see Figure B.34) that is the GH distribution.
 The QQ plot in Figure B.35(a) con rms the fact that the match under normality assumption is
 \good".
B.1 Normal innovations 198
 (a) Density (b) Log Density
 Figure B.33 Weekly USA Adj. M1 (1981-2008) probability plots: Empirical vs normal.
 (a) Density (b) Log Density
 Figure B.34 Weekly USA Adj. M1 (1981-2008) probability plots: Empirical vs L evy.
B.1 Normal innovations 199
 Model      LLH
 NIG 13:87633  0:34898 0:75585 0:03012 N/A 52:65
 H 14:76969  0:35021 0:70075 0:03019 N/A 52:65
 GH 19:80023  0:36612 0:00371 0:03105 10:67644 52:66
 VG 19:8000  0:36612 N/A 0:03105 0:04067 52:66
 Sk.Std. 0:32700  0:32700 N/A 0:02892 0:04067 52:62
 Table B.12 Estimated parameters for USA Adj. M1 (1981-2008).
 (a) Normal (b) L evy
 Figure B.35 Weekly US M1 Adj. QQ plots: Empirical vs normal.
B.1 Normal innovations 200
 (a) GH (b) H
 (c) NIG (d) VG
 (e) Skw. Std. (f) Normal
 Figure B.36 Weekly US M1 Adj. QQ plots.
B.2 Student t innovations 201
 B.2 Student t innovations
B.2 Student t innovations 202
 Money Supply aggregate M1
 (a) M1 (b) M1 Log returns
 Figure B.37 Filtered vs raw weekly USA M1 (1981-2008) data series.
 (a) Density (b) Log Density
 Figure B.38 Weekly USA M1 (1981-2008) probability plots: Empirical vs L evy.
B.3 Yield Curves 203
 (a) M1 Adj. (b) M1 Adj. Log returns
 Figure B.39 Filtered vs raw weekly USA M1 Adj. (1981-2008) data series.
 (a) Density (b) Log Density
 Figure B.40 Weekly USA Adj. M1 (1981-2008) probability plots: Empirical vs normal.
 Money Supply aggregate M1: Seasonally adjusted
 Money Supply aggregate M2
 B.3 Yield Curves
 Nominal Yield Curve
 The nominal and real daily historical yield curves were downloaded from the U.S. treasury website
 [107] from January 2003 to September 2008. The sample data has 1430 data points, with only the
B.3 Yield Curves 204
 (a) Density (b) Log Density
 Figure B.41 Weekly USA Adj. M1 (1981-2008) probability plots: Empirical vs L evy.
 (a) M2 (b) M2 Log returns
 Figure B.42 Filtered vs raw weekly USA M2 (1981-2008) data series.
 trading days considered.
 AIC best  t Student-t.
B.3 Yield Curves 205
 (a) Probability plot (b) QQ Plot
 Figure B.43 Estimated L1: Empirical vs normal.
 (a) Density (b) Log Density
 Figure B.44 Estimated L1 probability plots: Empirical vs L evy.
B.3 Yield Curves 206
 (a) GH (b) H
 (c) NIG (d) VG
 (e) Skw. Std. (f) Normal
 Figure B.45 Estimated L1 QQ plots.
B.3 Yield Curves 207
 Model    ( 105)   LLH
 NIG 2728:92 24:64427 53:96559  4:83289  10 6  0:5 9056:900
 H 3938:503 26:19406 34:55043  5:20755  10 6 1 9050:542
 GH 73:83758 70:64118 91:40627  1:40600  10 5  3:19922 9072:226
 VG 4437:195 3:61602 0  6:79980  10 7 35 9046:039
 Sk.Std. 67:62844 67:62844 91:2076  1:36021  10 5  3:18674 9072:230
 Table B.13 Estimated parameters for empirical L1 for SA nominal forward rate.
B.3 Yield Curves 208
 Real Yield Curve
 (a) Probability plot (b) QQ Plot
 Figure B.46 Estimated L1: Empirical vs normal.
 (a) Density (b) Log Density
 Figure B.47 Estimated L1 probability plots: Empirical vs L evy.
 AIC best  t Student-t.
B.3 Yield Curves 209
 (a) GH (b) H
 (c) NIG (d) VG
 (e) Skw. Std. (f) Normal
 Figure B.48 US real curve: Estimated L1 QQ plots.
B.3 Yield Curves 210
 Model    ( 105)  ( 105)  LLH
 NIG 1890:162  51:50919 60:56153 1:65484  0:5 8725:085
 H 2909:199  54:34723 32:85109 1:68966 1 8720:764
 GH 52:25032  44:33243 93:34401 1:46063  2:33771 8729:895
 VG 3304:674  45:19182 0 1:42259 1:72004 8717:872
 Sk.Std. 39:62266  39:62266 94:60124 1:28412  2:38186 8729:890
 Table B.14 Estimated parameters for empirical L1 for US real forward rate.
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