We will look closely at what derivatives in general and options in particular are in chapter 1. This should provide the reader with sufficient knowledge of finance for this thesis. We will also look at how traders use derivatives. Finally, we will introduce the principles of no arbitrage and risk-neutral valuation which are the central principles in derivatives valuation.
We describe stochastic processes that are important in theoretical finance in some detail in chapter 2. We will first look at simple random walks in one or more dimensions. We will then define what is by far the most important stochastic process in theoretical finance, the Wiener process. We will also introduce white noise which is the time derivative of the Wiener process.
In chapter 3, we will have a look at the Black-Scholes equation which is of fundamental importance in option pricing. We will present the theory behind the equation and its derivation in quite some detail. We will also present a derivation of the equation using a quantum mechanical formalism.
We consider the theory behind option pricing with stochastic volatility and look at the use of path integrals in this regard in chapter 4. We will first consider the theoretical considerations behind the choice of the stochastic process followed by the volatility. We will present a derivation of the Merton-Garman equation which is of fundamental importance in option pricing with stochastic volatility. We will also present a straightforward solution for the case of zero correlation between the stochastic processes followed by the stock price and the volatility. We then consider the quantum mechanical formulation of the problem, deriving the Hamiltonian, Lagrangian and the action. We then extend the straightforward solution for zero correlation to include non-zero correlation.
Our algorithm to calculate option prices with stochastic volatility is described in detail in chapter 5. We compare its efficiency with other standard Monte Carlo techniques. We also discuss the sources of numerical errors in the program. We also present a few of the checks on the errors that we performed.
In chapter 6, we present the computational results achieved using this algorithm. We look at the various effects of the parameters on the option prices. We also calibrate our model with market data and show that it is in reasonable agreement.