Modeling volatility in option pricing with applications

The focus of this dissertation is modeling volatility in option pricing by the Black-Scholes formula. A major drawback of the formula is that the returns from assets are assumed to have constant volatility over time. The empirical evidence is overwhelmingly against it. In this dissertation, we allow random volatility for estimating call option prices by Black-Scholes formula and by Monte Carlo simulation.;The Black-Scholes formula follows from an assumption that assets evolve according to a Geometric Brownian Motion with constant volatility. This dissertation allows time-varying random volatility in the Geometric Brownian Motion to outline a proof of the formula, thus addressing this drawback. To estimate option prices with the Black-Scholes, the dissertation considers its expectation with respect to two potential probability models of random volatility. Unfortunately, a closed form expression of the expectation of the formula for computing the option prices is intractable. Then the dissertation settles with using an approximation which to its credit incorporates in it the kurtosis of the probability model of random volatility. To our knowledge, option pricing methods in literature do not incorporate kurtosis information.;The option pricing with random volatility is pursued for two stochastic volatility models. One model is a member of generalized auto regressive conditional heteroscedasticity (GARCH). The second is a member of Stochastic Volatility models. For each model, estimation of their parameters is outlined. Two real financial series data are then used to illustrate estimation of the option prices, and compared them with those from the Black-Scholes formula with constant volatility.;Motivated by a Monte Carlo procedure in the literature for option pricing when the volatility follows a GARCH model, this dissertation lays a foundation for future research to simulate option prices when the random volatility is assumed to follow a Stochastic Volatility model instead of GARCH.;Chapter 1 reviews the basic concepts of the log returns from assets and the corresponding probability model. It uses a financial series to illustrate some crucial aspects of the log returns by descriptive summary statistics. It continues to review the GARCH and Stochastic Volatility models for modeling the log returns. Their relevant statistical aspects are pointed out.;Chapter 2 outlines a proof of the Black-Scholes formula allowing the volatility of assets to be random varying with time as assets evolve following a Geometric Brownian Motion. For computing option prices approximate expressions for the expectation of the Black-Scholes formula are derived when volatility follows a GARCH model and next when it follows a Stochastic Volatility model. These approximate expressions are computed in Chapters 3 and 4 for option pricing. Both these chapters use two financial series for illustration.;Chapter 5 reviews briefly a Monte Carlo procedure for option pricing when volatility follows a GARCH model. Motivated by it, it lays a foundation for future research to simulate option prices when the random volatility is assumed to follow a Stochastic Volatility model instead of the GARCH. Chapter 6 outlines a few ideas for future research.