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:

      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:

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. Some new improvements for two-factor PDEs are:

      Part B prepares the way for a seamless route from PDEs to robust and understandable finite difference schemes that approximate them. It eliminates much trial-and-error experimentation. In a sense the topics in Part B serve as a reference for the more hands-on topics in later chapters. At the very least, it is important to be aware of the main results.

      Part C (Chapters 14 to 17) introduces the mathematical background to the finite difference method for initial boundary value problems for parabolic partial differential equations. It encapsulates in one place all the background information that is needed to construct stable and accurate finite difference schemes for time-dependent problems. The schemes will be applied to one-factor and two-factor finance PDEs in later chapters. The advantage is that the chapters discuss finite difference schemes for generic PDEs that can then be applied to finance PDEs. We also devote two dedicated chapters to Sensitivity Analysis, and we propose at least five methods to compute the derivatives of solutions of initial boundary value problems with respect to underlying parameters. In finance, these are sometimes called option greeks.

      The chapters in this part discuss both new as well as established methods to gain a deep understanding of the foundations of the finite difference method and its applications to finance, and beyond:

      A good working knowledge of the topics in Part C is essential in order to proceed in an effective manner.

      Part D (Chapters 18 to 22) introduces a number of modern and popular finite difference methods (approximately six) to approximate the solution of initial boundary value problems for two-factor partial differential equations. To our knowledge, this is the only book that discusses these methods as well as their comparative strengths together