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Numerical Methods in Computational Finance
A Partial Differential Equation (PDE/FDM) Approach
DANIEL J. DUFFY
This edition first published 2022
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ISBN 9781119719670 (Hardback)
ISBN 9781119719724 (ePub)
ISBN 9781119719694 (ePDF)
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Preface
This book is a detailed introduction to the mathematical theory and foundations of ordinary and partial differential equations, their approximation by the finite difference method and applications to computational finance.
Major benefits of the book are:
Step-by-step, incremental build-up of the material.
Examples and algorithms worked out in detail. Opportunity to modify the algorithms and extend them to your own applications.
Modern, state-of-the art numerical schemes for PDEs in finance.
Guidelines on C++ coding (C++11 to C++20); the book is the ideal companion to the author's book Financial Instrument Pricing Using C++ (second edition, 2018).
The book is structured so that the material can be applied to a range of existing and new application areas.
We resolve a number of outstanding issues and improve several less-than-optimal numerical methods in finance.
We have divided the book into five parts, with each part addressing a single major issue.
Part A (Chapters 1 to 7) introduces the mathematical and numerical analysis concepts that are needed to understand the finite difference method and its application to computational finance. The main reason for writing the chapters is to make the book as self-contained as possible and to introduce and define standardised notation and results that we use in later chapters. Furthermore, the presented material can also be used as a standalone reference.
We realise that some readers will not be familiar with all of the building blocks that are needed to write finite difference schemes for the Black–Scholes PDE; for this reason, Part A was written to resolve this issue. We identify and discuss all the steps to design and implement a finite difference solver for one-factor finance PDEs. To this end, we take an incremental and single responsibility approach by focusing on one major topic in each chapter:
Initial value problems and boundary value problems and their numerical approximation.
Vector spaces, matrix theory and numerical linear algebra.
My first Crank–Nicolson and Alternating Direction Explicit (ADE) methods for the one-factor Black–Scholes PDE.
Finally, Chapter 1 is devoted to major concepts (such as continuity and differentiability) in real analysis that permeate the book, and for this reason it is important to understand them.
Part A can be used by readers with no prior knowledge of partial differential equations or the finite difference method. It can be used as a mini-course or mini-project to learn the material.
Part B (Chapters 8 to 13) discusses a number of rigorous mathematical techniques relating to boundary value problems and initial boundary value problems for elliptic and parabolic partial differential equations in two-space variables. The goal is to identify and elaborate the underlying theory to unambiguously specify these problems before mapping them to a numerical solution, thereby filling some ‘mathematical gaps’ in current practice. In particular, we develop strategies to preprocess and modify a PDE before we approximate it by the finite difference method, thus removing ad hoc and heuristic tricks that are often used to arrive at (hopefully) robust numerical schemes.
The chapters in this part fill a major gap in the application of PDE/FDM to finance. In general, most of the finance literature glosses over the niceties of analysing PDEs mathematically before approximating them using the finite difference method. The new approach resolves many of issues and heuristic approaches.
