Newly Developed Meshless Numerical Methods and Their Applications in Option Pricing Problems

Abstract

In this thesis, we investigate and develop some meshless numerical methods for solving various kinds of partial differential equation (PDE) models. In particular, we apply these methods to address option pricing problems and expand the applicability of meshless computation to tackle long-term evolution scenarios under complex geometries. The first chapter of this thesis provides an overview of meshless methods along with financial derivatives pricing and the subsequent chapters introduce four newly developed meshless methods respectively.

Based on the consideration of constant volatility or stochastic volatility of an underlying asset, in Chapter 2, we propose a Generalized Finite Integration Method (GFIM) incorporated with Laplace transform technique for pricing European-style options under the Black-Scholes model and Heston model respectively. We first perform Laplace transform on the governing equation and boundary conditions to remove the temporal derivatives. The meshless GFIM is then exploited to handle the spatial differential operators in the transformed space. From numerical Laplace inversion algorithm, we restore the required time-dependent option price. For verification, exotic digital call, butterfly spread call, and vanilla put options pricing problems governed by one-dimensional Black–Scholes equation and two-dimensional extended Heston equation are investigated to demonstrate the stability and efficiency of the proposed approach.

In Chapter 3, we further improve the Generalized Finite Integration Method by combining Volterra operator (GFIM-V) with Crank–Nicolson time-stepping scheme for solving exotic barrier option pricing problems under the Black-Scholes model. The imposed additional barrier constraints are handled by updating the approximate solution obtained by the meshless GFIM-V on each time step. Several numerical experiments for the solutions of single-asset and double-asset barrier option prices with various temporal step sizes and number of spatial nodal points are constructed. Comparisons with available exact solution and existing spectral convergent method indicate the advantages of the GFIM-V method in superior accuracy and unconditional stability due to the use of numerical quadrature.

Chapter 4 investigates the numerical solutions of high-dimensional Black-Scholes PDEs for multi-asset options valuation, aiming to overcome the curse of dimensionality. We develop a physics-informed (PI) machine learning algorithm based on radial basis function neural network (RBFNN) that concurrently optimizes the network architecture and predicts the option price. The physics-informed radial basis function neural network (PIRBFNN) combines the strengths of the traditional radial basis function (RBF) collocation method and physics-informed neural network (PINN) learning technology to effectively solve PDE problems in the financial context. By employing a residual-based technique to adaptively refine the distribution of hidden neurons during the training process, the PIRBFNN facilitates accurate and efficient handling of high-dimensional option pricing problems featuring non-smooth payoff conditions. The effectiveness of the proposed meshless method is validated by a collection of examples involving a single-asset European put option, a double-asset exchange option, and a triple-asset basket call option.

To underscore the advantages of fractional models over classical integer-order model in terms of financial characteristics, we explore in Chapter 5 the numerical valuation of European options within the Black-Scholes fractional framework. Regarding a unified model form represented by an initial-boundary value problem that incorporates fractional derivatives in both time and space, we develop a meshless collocation method utilizing a polynomial-augmented radial basis function (RBF) hybrid base from the perspective of spatial and temporal scales. Moreover, effective condition number technique is employed to predict the optimal order of polynomial basis. As a result, we apply the proposed scheme to address the pricing problems of call and put options, governed by the time-, space-, and time-space fractional Black-Scholes equations, respectively. The computational results of our method are compared with those of existing solvers to numerically demonstrate the efficacy of our method.

Given that the aforementioned option pricing problems are considered on a rectangular computational domain within a fixed lifetime of the financial contract, the final Chapter extends and ends the discussion by solving long-term evolution problems on irregular domains, thereby demonstrating the robustness of our methodology in terms of mesh-independency. In using the method of approximate particular solutions (MAPS), we develop a novel meshless method based on hybrid integrated radial basis function (IRBF)-polynomial bases to solve wave equation, heat equation, and convection-diffusion equations with variable coefficients over large terminal time intervals on various irregular spatial domains. By using space–time approach, the original time-dependent problem is firstly reformulated into an elliptic boundary value problem. Integrated polyharmonic splines (PS) type RBF kernels in conjunction with multivariate polynomials are then employed to construct the approximate solution space. This superior combination enables us to stably achieve a highly accurate solution. Due to the adoption of the PS, the difficulty of determining a suitable shape parameter of RBF is alleviated. Moreover, employing the recently developed ghost point technique, the convergence and stability of approximation can be further enhanced. In numerical experiments, four examples are presented to validate the proposed scheme.

Date of Award	25 Aug 2025
Original language	English
Awarding Institution	City University of Hong Kong
Supervisor	Hon Fu Raymond CHAN (Supervisor), Hongyu LIU (Supervisor) & Yiu Chung Benny HON (Co-supervisor)