You-Lan Zhu

This book is mainly devoted to finite difference numerical methods for solving partial differential equations (PDEs) models of pricing a wide variety of financial derivative securities. With this objective, the book is divided into two main parts.In the first part, after an introduction concerning the basics on derivative securities, the authors explain how to establish the adequate PDE boundary value problems for different sets of derivative products (vanilla and exotic options, and interest rate derivatives). For many option problems, the analytic solutions are also derived with details. The second part is devoted to explaining and analyzing the application of finite differences techniques to the financial models stated in the first part of the book. For this, the authors recall some basics on finite difference methods, initial boundary value problems, and (having in view financial products with early exercise feature) linear complementarity and free boundary problems.In each chapter, the techniques related to these mathematical and numerical subjects are applied to a wide variety of financial products. This is a textbook for graduate students following a mathematical finance program as well as a valuable reference for those researchers working in numerical methods in financial derivatives. For this new edition, the book has been updated throughout with many new problems added. More details about numerical methods for some options, for example, Asian options with discrete sampling, are provided and the proof of solution-uniqueness of derivative security problems and the complete stability analysis of numerical methods for two-dimensional problems are added. Review of first edition:“…the book is highly well designed and structured as a textbook for graduate students following a mathematical finance program, which includes Black-Scholes dynamic hedging methodology to price financial derivatives. Also, it is a very valuable reference for those researchers working in numerical methods in financial derivatives, either with a more financial or mathematical background." -- MATHEMATICAL REVIEWS

Since the appearance of computers, numerical methods for discontinuous solutions of quasi-linear hyperbolic systems of partial differential equations have been among the most important research subjects in numerical analysis. The authors have developed a new difference method (named the singularity-separating method) for quasi-linear hyperbolic systems of partial differential equations. Its most important feature is that it possesses a high accuracy even for problems with singularities such as schocks, contact discontinuities, rarefaction waves and detonations. Besides the thorough description of the method itself, its mathematical foundation (stability-convergence theory of difference schemes for initial-boundary-value hyperbolic problems) and its application to supersonic flow around bodies are discussed. Further, the method of lines and its application to blunt body problems and conical flow problems are described in detail. This book should soon be an important working basis for both graduate students and researchers in the field of partial differential equations as well as in mathematical physics.

This book is mainly devoted to finite difference numerical methods for solving partial differential equations (PDEs) models of pricing a wide variety of financial derivative securities. With this objective, the book is divided into two main parts.In the first part, after an introduction concerning the basics on derivative securities, the authors explain how to establish the adequate PDE boundary value problems for different sets of derivative products (vanilla and exotic options, and interest rate derivatives). For many option problems, the analytic solutions are also derived with details. The second part is devoted to explaining and analyzing the application of finite differences techniques to the financial models stated in the first part of the book. For this, the authors recall some basics on finite difference methods, initial boundary value problems, and (having in view financial products with early exercise feature) linear complementarity and free boundary problems.In each chapter, the techniques related to these mathematical and numerical subjects are applied to a wide variety of financial products. This is a textbook for graduate students following a mathematical finance program as well as a valuable reference for those researchers working in numerical methods in financial derivatives. For this new edition, the book has been updated throughout with many new problems added. More details about numerical methods for some options, for example, Asian options with discrete sampling, are provided and the proof of solution-uniqueness of derivative security problems and the complete stability analysis of numerical methods for two-dimensional problems are added. Review of first edition:“…the book is highly well designed and structured as a textbook for graduate students following a mathematical finance program, which includes Black-Scholes dynamic hedging methodology to price financial derivatives. Also, it is a very valuable reference for those researchers working in numerical methods in financial derivatives, either with a more financial or mathematical background." -- MATHEMATICAL REVIEWS

In the past three decades, great progress has been made in the theory and prac­ tice of financial derivative securities. Now huge volumes of financial derivative securities are traded on the market every day. This causes a big demand for experts who know how to price financial derivative securities. This book is designed as a textbook for graduate students in a mathematical finance pro­ gram and as a reference book for the people who already work in this field. We hope that a person who has studied this book and who knows how to write codes for engineering computation can handle the business of providing efficient derivative-pricing codes. In order for this book to be used by various people, the prerequisites to study the majority of this book are multivariable calculus, linear algebra, and basic probability and statistics. In this book, the determination of the prices of financial derivative secu­ rities is reduced to solving partial differential equation problems, i. e. , a PDE approach is adopted in order to find the price of a derivative security. This book is divided into two parts. In the first part, we discuss how to establish the corresponding partial differential equations and find the final and nec­ essary boundary conditions for a specific derivative product. If possible, we derive its explicit solution and describe some properties of the solution. In many cases, no explicit solution has been found so far.