Stochastic volatility models: calibration, pricing and hedging
Date
2012-10-01
Authors
Poklewski-Koziell, Warrick
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Abstract
Stochastic volatility models have long provided a popular alternative to the Black-
Scholes-Merton framework. They provide, in a self-consistent way, an explanation
for the presence of implied volatility smiles/skews seen in practice. Incorporating
jumps into the stochastic volatility framework gives further freedom to nancial
mathematicians to t both the short and long end of the implied volatility surface.
We present three stochastic volatility models here - the Heston model, the Bates
model and the SVJJ model. The latter two models incorporate jumps in the stock
price process and, in the case of the SVJJ model, jumps in the volatility process. We
analyse the e ects that the di erent model parameters have on the implied volatility
surface as well as the returns distribution. We also present pricing techniques for
determining vanilla European option prices under the dynamics of the three models.
These include the fast Fourier transform (FFT) framework of Carr and Madan as
well as two Monte Carlo pricing methods. Making use of the FFT pricing framework,
we present calibration techniques for tting the models to option data. Speci cally,
we examine the use of the genetic algorithm, adaptive simulated annealing and a
MATLAB optimisation routine for tting the models to option data via a leastsquares
calibration routine. We favour the genetic algorithm and make use of it in
tting the three models to ALSI and S&P 500 option data. The last section of the
dissertation provides hedging techniques for the models via the calculation of option
price sensitivities. We nd that a delta, vega and gamma hedging scheme provides
the best results for the Heston model. The inclusion of jumps in the stock price and
volatility processes, however, worsens the performance of this scheme. MATLAB
code for some of the routines implemented is provided in the appendix.