Option Pricing models with Stochastic Volatility and Jumps - Kalsheker, Farhan (Department of Mathematics and Applied Mathematics, 2014)
Exotic equity options are specialized instruments which are typically traded over the counter. Their prices are primarily determined by option pricing models which should be able to price exotic options consistently with the market prices of corresponding vanilla options. Additionally, option pricing models should have intuitive dynamics which are able to capture real world behavior (such as stochastic volatility effects and jumps in the price of the underlying). This dissertation tackles the question of which option pricing model to use; it compares diffusion, pure jump and jump-diffusion models. All models are fitted to one-day price data on S&P500 European vanilla options; the models with the best fit exhibit the smallest error in pricing between model prices and market prices. The stochastic volatility with jumps (SVJ) models are found to perform the best. The SVJ-DE model, a new variant of this type of model (which is based on Heston-type stochastic volatility and Kou-type double exponential jumps in the log price), is presented and tested. The Heston SV model is ranked third best. There is a significant performance gap between the SV/SVJ models and the remaining models. The variance-gamma model with stochastic time is found to be the best performing model from the pure jump and simple jump-diffusion categories. The Kou jump-diffusion model with double exponential jumps and constant diffusion volatility ranks next, followed by the Merton jump-diffusion model and the variance-gamma pure jump model. On comparison of model and market implied volatility surfaces, the pure jump and simple jump-diffusion models are found to be efficient at generating volatility smile effects, but not volatility skew effects. The converse holds for the Heston SV model. The SVJ models exploit this behavior in an attempt to use the jump component to generate the smile effects on the short end of the volatility surface and the stochastic volatility diffusion component to generate the skew effects on the long end of the volatility surface. The application of the SV and SVJ models is demonstrated by computing the prices of barrier options via Monte Carlo simulation. Both of the SVJ models give similar barrier option prices. Diffusion processes and jump processes are the two main building blocks of any option pricing model. This research finds that simple jump-diffusion models and pure jump models are unable to demonstrate good performance when fitting to a complete grid of market option prices. The Heston stochastic volatility pure diffusion model gives better performance compared to these jump models. The SVJ models which have both a stochastic volatility diffusion component and a jump component are found to give the best performance. The SVJ-DE model has the added advantage of being able to generate upward and downward jumps from different exponential distributions, versus the Bates model which generates jumps from a normal distribution.