The thesis studies the martingale properties, probabilistic methods and efficient unbiased Monte Carlo simulation methods for various time-homogeneous diffusion models commonly used in mathematical finance. Some of the popular stochastic volatility models such as the Heston model, the Hull-White model and the 3/2 model are special cases. The thesis consists of the following three parts: Part I: Martingale properties in time-homogeneous diffusion models: Part I of the thesis studies martingale properties of stock prices in stochastic volatility models driven by time-homogeneous diffusions. We find necessary and sufficient conditions for the martingale properties. The conditions are based on the local integrability of certain deterministic test functions. Part II: Analytical pricing methods in time-homogeneous diffusion models: Part II of the thesis studies probabilistic methods for determining the Laplace transform of the first hitting time of an integral functional of a time-homogeneous diffusion, and pricing an arithmetic Asian option when the stock price is modeled by a time-homogeneous diffusion. We also consider the pricing of discrete variance swaps and discrete gamma swaps in stochastic volatility models based on time-homogeneous diffusions. Part III: Nearly Unbiased Monte Carlo Simulation: Part III of the thesis studies the unbiased Monte Carlo simulation of option prices when the characteristic function of the stock price is known but its density function is unknown or complicated.

The thesis studies the martingale properties, probabilistic methods and efficient unbiased Monte Carlo simulation methods for various time-homogeneous diffusion models commonly used in mathematical finance. Some of the popular stochastic volatility models such as the Heston model, the Hull-White model and the 3/2 model are special cases. The thesis consists of the following three parts: Part I: Martingale properties in time-homogeneous diffusion models: Part I of the thesis studies martingale properties of stock prices in stochastic volatility models driven by time-homogeneous diffusions. We find necessary and sufficient conditions for the martingale properties. The conditions are based on the local integrability of certain deterministic test functions. Part II: Analytical pricing methods in time-homogeneous diffusion models: Part II of the thesis studies probabilistic methods for determining the Laplace transform of the first hitting time of an integral functional of a time-homogeneous diffusion, and pricing an arithmetic Asian option when the stock price is modeled by a time-homogeneous diffusion. We also consider the pricing of discrete variance swaps and discrete gamma swaps in stochastic volatility models based on time-homogeneous diffusions. Part III: Nearly Unbiased Monte Carlo Simulation: Part III of the thesis studies the unbiased Monte Carlo simulation of option prices when the characteristic function of the stock price is known but its density function is unknown or complicated.