A New Numerical Method for Multi-Asset Option Pricing

Abstract

This thesis aims to present our recent research results in the development of the Generalized Finite Integration Method (GFIM) for the numerical solutions of multi-dimensional partial differential equations and its application to option pricing modeled by Black-Scholes equation. GFIM elaborates the idea of Finite Integration Method (FIM) and gives the detail theorems, formulations and computation procedures to implement FIM. FIM is a numerical method for approximating solution to the differential equations by numerical quadrature. Due to unconditional stability of numerical integrations, numerical experiences showed that FIM is a stable and accurate numerical methods compared with other popular numerical methods, such as Finite Different Method (FDM), Finite Element Method (FEM) and Radial Basis Function Method (RBF). However, The mathematical principle and formulation of FIM are not clear and not complete that makes it difficult to work on further enhancements and improvements on FIM, such as approximating three or higher dimensional differential equations or improving on the accuracy. In light of this, we are interested in studying the fundamentals and principles of FIM. The research results are presented in this thesis. We first briefly review the history and the idea of FIM. Then, we generalizes the idea of FIM and introduces GFIM which defines the approximant using in the numerical approximation, gives the theorems and detail formulations to transform differential operators of governing equation into integration matrix through unrestricted nodal point collocation and provides the detail computational procedures to approximate solution to the multi-dimensional partial differential equation by using the integration matrices. With the theorems and formulations of integration matrix provided by GFIM, we then present the Generalized Finite Integration Method with Volterra Operator (GFIM-V) to improve the accuracy of numerical approximation by transforming the multiple integrals into a single integral analytically in form of Volterra operator. Detail formulation of the integration matrix by GFIM-V and the procedures to plug-in the integration matrices back to the numerical approximation framework of GFIM are discussed in this thesis. Finally, we apply the GFIM and GFIM-V incorporated of Method of Line with θ-Scheme (GFIM-MoL) or Domain Decomposition Technique in Space-Time (GFIM-ST) to solve the different types of single asset and multi-asset option pricing problems. More stable and accurate results can be obtained by GFIM compared with other numerical methods, such as FDM, FEM and RBF. The computation efficiency and stability are discussed via the numerical experiments. Based on our research results, GFIM becomes a new numerical approximation framework based on the idea of FIM for solving multi-dimensional partial differential equations and it should be a new branch of numerical method.

Date of Award	23 Feb 2023
Original language	English
Awarding Institution	City University of Hong Kong
Supervisor	Yiu Chung Benny HON (Supervisor)