Forecasting models for the dollar/rand spot rates.
Date
1998
Authors
Gcilitshana, Lungelo.
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Abstract
Owing to the complexity of hedging against the unfavourable price movements, derivatives came
into being to solve this problem if used in an effective and appropriate manner. Movements in
share or stock prices, foreign exchange rates, interest rates, etc., make it difficult to anticipate or
guess the next price or exchange rate or interest rates. Hence hedging ones'self against these
movements becomes a hurdle that is difficult to overcome. Coming to the fore of the derivatives
markets made a relief to many traders, but still then, no one could be certain about the move of
the market which he is trading in. Forecasting appeared as an educated guess as to which
direction and by how much the market will move.
This research report focusses on how to forecast the foreign exchange rates using the
Dollar/Rand as an example. I have gathered the historical daily data for the DoIIar/Rand spot rates
which includes the mayhem period that happened in February 1996. The data was obtained from
one of the biggest banks of South Africa; it was drawn from the Reuters historical data giving the
open, high, low and close prices of the Dollar/Rand (USD/ZAR) spot rates. The data was then
downloaded and copied to the spreadsheet for the calculation of the historical volatilities for
different periods. To have a genuine comparison with the implied volatilities, a data of historical
implied volatilities tor approximately the same period was gathered from the SAIMB (South
African International Money Brokers). The only snag with the data was that it only catered for
specific traded periods, like 1 month, 2 months, 3 months, 6 months, 9 months and 12 months
only. Most financial institutinns are using these implied volatilities for their pricing and end-of-day
or -month or -year revaluation. By the same token the data was downloaded to the spreadsheet
for further analysis and arrangement.
Chapter 1 gives the purpose and the meaning of'forecasting, together with different methods that
this process can be achieved. Views from Makridakis et al., (1983) are used to beautify the world
of forecasting and its importance. In Chapter 2 the concept of volatility and its causes, is
discussed in detail. Besides the implied and historical volatility discussions, volatility 'smile'
concept is discussed and expanded. Volatility slope trading strategies and constraints on the slope
of the volatility term structure are discussed in detail.
Chapter 3 discusses different models used to calculate both the historical and the implied
volatility. This includes models by Kawaller et al., (1994) and Figlewski et al., ( 1990). The
Newton-Raphson method is among of the methods that can be used to get a good estimate of the
implied volatility. For a lot accurate estimates the Method of Bisection can be used in place of the
Newton-Raphson method. Mayhew (1995) even suggest a method, which involves the use of
more weighting with higher vegas (Latane and Rendleman 1976) or weighting not by vegas but
elasticity (Chiras and Manaster 1978).
Chapter 4 dwells on different forecasting models for foreign exchange markets. This includes
models by Engle (1993), who is one of the pioneers of the autoregression theory, He discusses the
ARCH, GARCH and EGARCH models; Heynen et al., (1994,1995) discusses the models for the
term structure of volatility implied by foreign exchange. In the 1995 article he dwells on the
specifications of the different autoregressive conditional heteroskedastic models. U.A. Muller et
al., (1990,1993) discusses some of the models for the changing time scale for short-term
forecasting in financial markets. This includes discussion of some statistical properties of FX rates
time-series. Xu and Taylor (1994) also discuss the term structure of volatility as implied, in
particular, by FX options. Regression is used in computation of implied volatility
Chapter 5 dwells on the empirical evidence and the market practice. This includes the statistical
analysis of the data; applying the scaling law; proprietary model which depicts the edge between
the historical volatility and implied volatility; empirical tests and the volatility forecast evaluation
applied to historical USD/ZAR daily data, using different models.
In the statistical analysis, using U.A. Muller et al., (1993) theory, the scaling law, which involves
the absolute price changes, which are directly related to the interval At, is discussed. Using my
GSD/ZAR data Imanaged to calculate the parameters described by the scaling law, using At as
one day since my data is a daily data Icould not calculate the activity model function, which
calculates the intra-day and intra-hour trading using tick-by-tick data, because of the nature of my
data. Had it not been the case, f would have been able to calculate the intra-day and intra-hour
volatilities. These statistics would have been able to depict the daily volatility, more especially on
volatile days, like the day when the Rand took its first knock in February 1996.
In the second section of the chapter the proprietary model is discussed, where an edge between
the actual volatility and implied volatility was identified. There is a positive correlation between
the actual and implied volatility although the latter is always higher than the former; hence traders
can play with this situation for arbitrage purposes. To get the estimates of historical volatility, I
used the Well-known formula of using the log-relatives of the returns of any two consecutive days.
Annnalised standard deviation of these log-relatives resulted into the required historical volatility
estimates. Moving averages were used to get estimates of different periods, as can be seen in the
text.
The main theme of the research report is to expose forecasting models that can be used in foreign
exchange currencies using DolIar/Rand as an example. Random walk model was used as
benchmark to other models like stochastic volatility, ARCH, GARCH( 1,1), and EGARCH (1,1).
Due to the complexity of the specifications of these models, I used the SHAZAM 7.0 econometric
program to generate the necessary parameters. Complex formulas of these models are given in the
Appendices at the end of the report, together with the program itself.
The significance of the forecasted volatility estimates was checked using the p-value correlation
statistic and the Akaike Information Criterion (AIC). The p-value gives us the significance of the
parameters and the AlC gives us an indication of the goodness-of-fit of the model. The formulas
used to calculate these statistics are given at the end of the report as part of the Appendices. An
account of where and how shese results can be of help in the practical situation is given under the
section of market practice. One of the areas worth mentioning is in risk management, where
estimates of the historical volatility can be used together with correlation in risk-metrics to
calculate VArt (value-at-risk). VAR is defined in simple terms as the 5thpercentile (quantile) of
the distribution of value changes. The beau.y of working with the percentile rather than, say the
variance of a distribution, is that a percentile corresponds to both a magnitude e.g., dollar amount
at risk, and exact probability e.g., the probability that the magnitude will not be exceeded. This
roughly the gist of the research report.
Description
A research report submitted to the Faculty of Science, University of the Witwatersrand,
Johannesburg, South Africa, in partial fulfillment of the requirements for the Degree of Masters of Science.
Keywords
Foreign exchange -- Mathematical models.