## Finite Difference Methods in Financial Engineering: A Partial Differential Equation Approach

### توضیحات

The world of quantitative finance (QF) is one of the fastest growing areas of research and its practical applications to derivatives pricing problem. Since the discovery of the famous Black-Scholes equation in the 1970’s we have seen a surge in the number of models for a wide range of products such as plain and exotic options, interest rate derivatives, real options and many others. Gone are the days when it was possible to price these derivatives analytically. For most problems we must resort to some kind of approximate method.

In this book we employ partial differential equations (PDE) to describe a range of one-factor and multi-factor derivatives products such as plain European and American options, multi-asset options, Asian options, interest rate options and real options. PDE techniques allow us to create a framework for modeling complex and interesting derivatives products. Having defined the PDE problem we then approximate it using the Finite Difference Method (FDM). This method has been used for many application areas such as fluid dynamics, heat transfer, semiconductor simulation and astrophysics, to name just a few. In this book we apply the same techniques to pricing real-life derivative products. We use both traditional (or well-known) methods as well as a number of advanced schemes that are making their way into the QF literature:

• Crank-Nicolson, exponentially fitted and higher-order schemes for one-factor and multi-factor options
• Early exercise features and approximation using front-fixing, penalty and variational methods
• Modelling stochastic volatility models using Splitting methods
• Critique of ADI and Crank-Nicolson schemes; when they work and when they don’t work
• Modelling jumps using Partial Integro Differential Equations (PIDE)
• Free and moving boundary value problems in QF

Included with the book is a CD containing information on how to set up FDM algorithms, how to map these algorithms to C++ as well as several working programs for one-factor and two-factor models. We also provide source code so that you can customize the applications to suit your own needs.

Daniel Duffy is a numerical analyst who has been working in the IT business since 1979. He has been involved in the analysis, design and implementation of systems using object-oriented, component and (more recently) intelligent agent technologies to large industrial and financial applications. As early as 1993 he was involved in C++ projects for risk management and options applications with a large Dutch bank. His main interest is in finding robust and scalable numerical schemes that approximate the partial differential equations that model financial derivatives products. He has an M.Sc. in the Finite Element Method first-order hyperbolic systems and a Ph.D. in robust finite difference methods for convection-diffusion partial differential equations. Both degrees are from Trinity College, Dublin, Ireland.
Daniel Duffy is founder of Datasim Education and Datasim Component Technology, two companies involved in training, consultancy and software development.

0 Goals of this Book and Global Overview 1

PART I THE CONTINUOUS THEORY OF PARTIAL DIFFERENTIAL EQUATIONS 5

1 An Introduction to Ordinary Differential Equations 7

2 An Introduction to Partial Differential Equations 13

3 Second-Order Parabolic Differential Equations 25

4 An Introduction to the Heat Equation in One Dimension 37

5 An Introduction to the Method of Characteristics 47

PART II FINITE DIFFERENCE METHODS: THE FUNDAMENTALS 61

6 An Introduction to the Finite Difference Method 63

7 An Introduction to the Method of Lines 79

8 General Theory of the Finite Difference Method 91

9 Finite Difference Schemes for First-Order Partial Differential Equations 103

10 FDM for the One-Dimensional Convection-Diffusion Equation 117

11 Exponentially Fitted Finite Difference Schemes 123

PART III APPLYING FDM TO ONE-FACTOR INSTRUMENT PRICING 135

12 Exact Solutions and Explicit Finite Difference Method for One-Factor Models 137

13 An Introduction to the Trinomial Method 147

14 Exponentially Fitted Difference Schemes for Barrier Options 153

15 Advanced Issues in Barrier and Lookback Option Modelling 165

16 The Meshless (Meshfree) Method in Financial Engineering 175

17 Extending the Black-Scholes Model: Jump Processes 183

PART IV FDM FOR MULTIDIMENSIONAL PROBLEMS 193

18 Finite Difference Schemes for Multidimensional Problems 195

19 An Introduction to Alternating Direction Implicit and Splitting Methods 209

20 Advanced Operator Splitting Methods: Fractional Steps 223

21 Modern Splitting Methods 229

PART V APPLYING FDM TO MULTI-FACTOR INSTRUMENT PRICING 237

22 Options with Stochastic Volatility: The Heston Model 239

23 Finite Difference Methods for Asian Options and Other ‘Mixed’ Problems 249

24 Multi-Asset Options 257

25 Finite Difference Methods for Fixed-Income Problems 273

PART VI FREE AND MOVING BOUNDARY VALUE PROBLEMS 285

26 Background to Free and Moving Boundary Value Problems 287

28 Viscosity Solutions and Penalty Methods for American Option Problems 307

29 Variational Formulation of American Option Problems 315

PART VII DESIGN AND IMPLEMENTATION IN C++ 325

30 Finding the Appropriate Finite Difference Schemes for your Financial Engineering Problem 327

31 Design and Implementation of First-Order Problems 337

32 Moving to Black-Scholes 353

33 C++ Class Hierarchies for One-Factor and Two-Factor Payoffs 363

33.1 Introduction and objectives 363

Appendices 375

A1 An introduction to integral and partial integro-differential equations 375

A2 An introduction to the finite element method 393

Bibliography 409

Index 417