Topics in volatility models

In this thesis I will present my PhD research work, focusing mainly on financial modelling of asset’s volatility and the pricing of contingent claims (financial derivatives), which consists of four topics: 1. Several changing volatility models are introduced and the pricing of European options is derived under these models; 2. A general local stochastic volatility model with stochastic interest rates (IR) is studied in the modelling of foreign exchange (FX) rates. The pricing of FX options under this model is examined through the use of an asymptotic expansion method, based on Watanabe-Yoshida theory. The perfect/partial hedging issues of FX options in the presence of local stochastic volatility and stochastic IRs are also considered. Finally, the impact of stochastic volatility on the pricing of FX-IR structured products (PRDCs) is examined; 3. A new method of non-biased Monte Carlo simulation for a stochastic volatility model (Heston Model) is proposed; 4. The LIBOR/swap market model with stochastic volatility and jump processes is studied, as well as the pricing of interest rate options under that model. In conclusion, some future research topics are suggested. Key words: Changing Volatility Models, Stochastic Volatility Models, Local Stochastic Volatility Models, Hedging Greeks, Jump Diffusion Models, Implied Volatility, Fourier Transform, Asymptotic Expansion, LIBOR Market Model, Monte Carlo Simulation, Saddle Point Approximation