Degree Name

Doctor of Philosophy


School of Mathematics and Applied Statistics


Due to the development of the pricing theory, options, as one of the most important financial derivatives, have become increasingly important in financial markets. The breakthrough of the theory was made by Fischer Black and Myron Scholes in 1973, who proposed a diffusion model with a mathematical formula for pricing European call options. The formula, however, was proven to be not perfect because of the constant volatility assumption made by the authors. Hence the \stochastic volatility models" were established to provide a better fit to the random feature of volatilities. The stochastic volatility models, though, are much more complex and analytical solutions are either unavailable or very difficult to obtain. Besides, since the calculation of a large number of prices is usually required in short time, fast and accurate numerical approaches are in great need.

Among all the numerical techniques, Monte Carlo simulations, tree approaches (binomial and trinomial trees) and finite-difference methods are the three key methods that can be applied to almost all option contracts under all existing models. However, each of the three methods has its own advantage and shortcomings. The motivation of this thesis is to explore the three key methods under the more adaptive stochastic volatility model and propose faster algorithms to fit the need from both academia and industry.