Book Image

Advanced Quantitative Finance with C++

By : Alonso Peña, Ph.D.
Book Image

Advanced Quantitative Finance with C++

By: Alonso Peña, Ph.D.

Overview of this book

<p>This book will introduce you to the key mathematical models used to price financial derivatives, as well as the implementation of main numerical models used to solve them. In particular, equity, currency, interest rates, and credit derivatives are discussed. In the first part of the book, the main mathematical models used in the world of financial derivatives are discussed. Next, the numerical methods used to solve the mathematical models are presented. Finally, both the mathematical models and the numerical methods are used to solve some concrete problems in equity, forex, interest rate, and credit derivatives.</p> <p>The models used include the Black-Scholes and Garman-Kohlhagen models, the LIBOR market model, structural and intensity credit models. The numerical methods described are Monte Carlo simulation (for single and multiple assets), Binomial Trees, and Finite Difference Methods. You will find implementation of concrete problems including European Call, Equity Basket, Currency European Call, FX Barrier Option, Interest Rate Swap, Bankruptcy, and Credit Default Swap in C++.</p>
Table of Contents (17 chapters)
Advanced Quantitative Finance with C++
Credits
About the Author
Acknowledgments
About the Reviewer
www.PacktPub.com
Preface
Index

The Finite Difference method


The Finite Difference (FD) method is a numerical technique that focuses directly on the approximate solution of a differential equation. As shown by (Black and Scholes 1973) for equity financial derivatives (contingent claims), the problem is expressed in terms of a Partial Differential Equation (PDE).

The basic idea of FDM is to discretize a differential equation. The method transforms the derivatives in the differential equation into quantities or ratios that approximate the derivatives. These quantities are not any more infinitesimal but finite, that is, they have a finite length. This is the origin of the name of finite differences. For more details, the reader can refer to The Mathematics of Financial Derivatives: A Student Introduction.

Consider the following illustration where a continuous function f(X) and the first derivative of the function is defined as follows:

The preceding function is also known as the slope, which is the ratio between the growth...