The Finite Difference Method for PDEs: Mathematical Background

• Published in: Financial Instrument Pricing Using C++ pp. 641 - 668
• Main Author: Duffy, Daniel J
• Format: Book Chapter
• Language: English
• Published: United Kingdom John Wiley & Sons, Incorporated 2018; Wiley
• Subjects: Black‐Scholes equation; C++ developers; convection‐diffusion‐reaction equations; Dirichlet boundary conditions; FINANCE & ACCOUNTING; finite difference method; initial boundary value problem; option pricing
• ISBN: 0470971193; 9780470971192
• DOI: 10.1002/9781119170518.ch20

This chapter introduces one‐factor partial differential equations (PDEs). Then, it also introduces enough mathematics to model PDEs, approximates them by finite difference method (FDM) algorithms and then maps these algorithms to C++. In particular, the chapter focuses on option pricing using the Black‐Scholes PDE and describes this PDE's generalisations. It then gives an overview of convection‐diffusion‐reaction equations in n space dimensions and details their qualitative properties including transforming them to more manageable forms. Furthermore, the chapter defines how PDEs discussed are incorporated into an initial boundary value problem (IBVP) and lists the qualitative properties of the IBVP. It also examines time‐independent second‐order two‐point boundary value problems (TPBVPs) with Dirichlet boundary conditions that defined on a bounded interval. Finally, the chapter summarizes some attention points that the developers should be acquainted with when working with FDM.