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# Numerical Methods in Computational Finance

MathematicsNumerical Methods in Computational Finance

Daniel J. Duffy’s new book, Numerical Methods in Computational Finance: A Partial Differential Equation (PDE/FDM) Approach is due to come out on 3 March, 2022.

This book is a detailed and step-by-step introduction to the mathematical foundations of ordinary partial differential equations, their approximation by the finite difference method and applications to computational finance. The book is structured so that it can be read by beginners, novices, and expert users.

Part A Mathematical Foundation for One-Factor Problems

Chapters 1 to 7 introduce the mathematical and numerical analysis concepts that are needed to understand the finite difference method and its application to computational finance.

Part B Mathematical Foundation for Two-Factor Problems

Chapters 8 to 13 discuss a number of rigorous mathematical techniques relating to elliptic and parabolic partial differential equations in two space variables. In particular, the author develops strategies to preprocess and modify a PDE before approximating it by the finite difference method, thus avoiding ad-hoc and heuristic tricks.

Part C The Foundations of the Finite Difference Method (FDM)

Chapters 14 to 17 introduce the mathematical background to the finite difference method for initial boundary value problems for parabolic PDEs. They encapsulates all the background information to construct stable and accurate finite difference schemes.

Part D Advanced Finite Difference Schemes for Two-Factor Problems

Chapters 18 to 22 introduce a number of modern finite difference methods to approximate the solution of two factor partial differential equations. This is the only book we know of that discusses these methods in any detail.

Part E Test Cases in Computational Finance

Chapters 23 to 26 are concerned with applications based on previous chapters. The author discusses finite difference schemes for a wide range of one-factor and two-factor problems.

This book is suitable as an entry-level introduction as well as a detailed treatment of modern methods as used by industry quants and MSc/MFE students in finance. The topics have applications to numerical analysis, science, and engineering.

## Daniel J. Duffy

Daniel J. Duffy started the company Datasim in 1987 to promote C++ as a new object-oriented language for developing applications in the roles of developer, architect and requirements analyst to help clients design and analyse software systems for Computer Aided Design (CAD), process control and hardware-software systems, logistics, holography (optical technology) and computational finance. He used a combination of top-down functional decomposition and bottom-up object-oriented programming techniques to create stable and extendible applications (for a discussion, see Duffy 2004 where we have grouped applications into domain categories). Previous to Datasim he worked on engineering applications in oil and gas and semiconductor industries using a range of numerical methods (for example, the finite element method (FEM)) on mainframe and mini-computers.

Daniel Duffy has BA (Mod), MSc and PhD degrees in pure, applied and numerical mathematics from Trinity College, Dublin. He has been active in promoting partial differential equation (PDE) and finite difference methods (FDM) for applications in computational finance. He was responsible for the introduction of the Fractional Step (Soviet Splitting) method and the Alternating Direction Explicit (ADE) method in computational finance. He is also the originator of the exponential fitting method for time-dependent partial differential equations.

He is also the originator of two very popular C++ online courses (both C++03 and C++11/14) on www.quantnet.com in cooperation with Quantnet LLC and Baruch College (CUNY), NYC. He also trains developers and designers around the world. He can be contacted dduffy@datasim.nl

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