This thesis covers two topics related to the computational methods in financial engineering.

The stochastic alpha-beta-rho (SABR) model is popular in the financial industry because it is capable of providing good fits to various types of implied volatility curves. However, no analytical solution to the SABR model exists that can be simulated directly. In the first topic, we explore the possibility of exact simulation for the SABR model. Our contribution is threefold. (i) We propose an exact simulation method in two special but practically interesting cases. Primary difficulties involved are how to simulate two random variables whose distributions can be expressed in terms of the Hartman-Watson and the non-central chi-squared distribution functions, respectively. Two novel simu...[ Read more ]

This thesis covers two topics related to the computational methods in financial engineering.

The stochastic alpha-beta-rho (SABR) model is popular in the financial industry because it is capable of providing good fits to various types of implied volatility curves. However, no analytical solution to the SABR model exists that can be simulated directly. In the first topic, we explore the possibility of exact simulation for the SABR model. Our contribution is threefold. (i) We propose an exact simulation method in two special but practically interesting cases. Primary difficulties involved are how to simulate two random variables whose distributions can be expressed in terms of the Hartman-Watson and the non-central chi-squared distribution functions, respectively. Two novel simulation schemes are proposed to achieve numerical accuracy, efficiency, and stability. (ii) In general cases, we propose a semi-exact simulation scheme, which turns out to be accurate when the time horizon is not long, e.g., no longer than 1 year. When the time horizon is long, a piecewise semi-exact simulation scheme is developed that reduces the biases substantially. (iii) For European option pricing, we propose a conditional simulation method, which reduces the variance of plain simulation significantly by, e.g., 99%.

The time-inhomogeneous diffusion models, such as the local volatility models and term structure models for interest rates, are widely used in financial applications. However, analytical pricing formulas are usually not available even for vanilla options. In the second topic, we derive a series representation for the European-type option price with a general payoff under time-inhomogeneous diffusions. Its convergence is proved rigorously under some regularity conditions. Our series representation provides a unified framework for analytically approximating European-type option prices with general payoffs through different parametrization expansion schemes. Besides, a systematic method based on Wiener-Itô Chaos expansion is developed to derive explicit expressions for the related analytical approximations. Numerical results demonstrate that our series representation method is accurate and efficient.