Numerical Fourier method and multi-dimensional second-order Taylor scheme for stochastic differential equations

Abstract

Asset pricing has evolved through models like the capital asset pricing model and arbitrage pricing theory. Despite their groundbreaking nature, these models have limitations in addressing complex market dynamics. This thesis explores advanced mathematical tools and models to enhance asset pricing methodologies. In Chapter 2, we provide a historical overview, from early economic theories to modern quantitative finance, highlighting key models such as the Black-Scholes model and the Feynman-Kac formula. Chapter 3 focuses on the construction and application of stochastic differential equations, Brownian motions, and Itˆo’s stochastic integrals, which are essential for modern financial mathematics. Chapter 4 introduces an innovative second-order discretization scheme based on the Fourier expansion of the Brownian bridge process. Chapter 4 also presents our original, accurate approximations of the characteristic function, which are crucial for solving forward stochastic differential equations. Chapter 5 presents a backward discretization scheme using the Fourier method for approximating function values at a given time based on previous time steps. The methodology is illustrated through its application to European option pricing, with numerical experiments confirming its accuracy. Additionally, we apply these advanced techniques to practical asset pricing problems, developing robust numerical methods that ensure both accuracy and convergence. Explicit formulas for Fourier coefficients are provided, and the method’s efficiency in pricing European options is thoroughly demonstrated. This thesis contributes to asset pricing by developing new methods to overcome limitations of existing models, offering theoretical insights and practical tools for financial analysts and researchers.