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

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

ISBN: 978-1-118-67344-7

Apr 2013

442 pages

## Description

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.

**0 Goals of this Book and Global Overview 1**

0.1 What is this book? 1

0.2 Why has this book been written? 2

0.3 For whom is this book intended? 2

0.4 Why should I read this book? 2

0.5 The structure of this book 3

0.6 What this book does not cover 4

0.7 Contact, feedback and more information 4

**PART I THE CONTINUOUS THEORY OF PARTIAL DIFFERENTIAL EQUATIONS 5**

**1 An Introduction to Ordinary Differential Equations 7**

1.1 Introduction and objectives 7

1.2 Two-point boundary value problem 8

1.3 Linear boundary value problems 9

1.4 Initial value problems 10

1.5 Some special cases 10

1.6 Summary and conclusions 11

**2 An Introduction to Partial Differential Equations 13**

2.1 Introduction and objectives 13

2.2 Partial differential equations 13

2.3 Specialisations 15

2.4 Parabolic partial differential equations 18

2.5 Hyperbolic equations 20

2.6 Systems of equations 22

2.7 Equations containing integrals 23

2.8 Summary and conclusions 24

**3 Second-Order Parabolic Differential Equations 25**

3.1 Introduction and objectives 25

3.2 Linear parabolic equations 25

3.3 The continuous problem 26

3.4 The maximum principle for parabolic equations 28

3.5 A special case: one-factor generalised Black–Scholes models 29

3.6 Fundamental solution and the Green’s function 30

3.7 Integral representation of the solution of parabolic PDEs 31

3.8 Parabolic equations in one space dimension 33

3.9 Summary and conclusions 35

**4 An Introduction to the Heat Equation in One Dimension 37**

4.1 Introduction and objectives 37

4.2 Motivation and background 38

4.3 The heat equation and financial engineering 39

4.4 The separation of variables technique 40

4.5 Transformation techniques for the heat equation 44

4.6 Summary and conclusions 46

**5 An Introduction to the Method of Characteristics 47**

5.1 Introduction and objectives 47

5.2 First-order hyperbolic equations 47

5.3 Second-order hyperbolic equations 50

5.4 Applications to financial engineering 53

5.5 Systems of equations 55

5.6 Propagation of discontinuities 57

5.7 Summary and conclusions 59

**PART II FINITE DIFFERENCE METHODS: THE FUNDAMENTALS 61**

**6 An Introduction to the Finite Difference Method 63**

6.1 Introduction and objectives 63

6.2 Fundamentals of numerical differentiation 63

6.3 Caveat: accuracy and round-off errors 65

6.4 Where are divided differences used in instrument pricing? 67

6.5 Initial value problems 67

6.6 Nonlinear initial value problems 72

6.7 Scalar initial value problems 75

6.8 Summary and conclusions 76

**7 An Introduction to the Method of Lines 79**

7.1 Introduction and objectives 79

7.2 Classifying semi-discretisation methods 79

7.3 Semi-discretisation in space using FDM 80

7.4 Numerical approximation of first-order systems 85

7.5 Summary and conclusions 89

**8 General Theory of the Finite Difference Method 91**

8.1 Introduction and objectives 91

8.2 Some fundamental concepts 91

8.3 Stability and the Fourier transform 94

8.4 The discrete Fourier transform 96

8.5 Stability for initial boundary value problems 99

8.6 Summary and conclusions 101

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

9.1 Introduction and objectives 103

9.2 Scoping the problem 103

9.3 Why first-order equations are different: Essential difficulties 105

9.4 A simple explicit scheme 106

9.5 Some common schemes for initial value problems 108

9.6 Some common schemes for initial boundary value problems 110

9.7 Monotone and positive-type schemes 110

9.8 Extensions, generalisations and other applications 111

9.9 Summary and conclusions 115

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

10.1 Introduction and objectives 117

10.2 Approximation of derivatives on the boundaries 118

10.3 Time-dependent convection–diffusion equations 120

10.4 Fully discrete schemes 120

10.5 Specifying initial and boundary conditions 121

10.6 Semi-discretisation in space 121

10.7 Semi-discretisation in time 122

10.8 Summary and conclusions 122

**11 Exponentially Fitted Finite Difference Schemes 123**

11.1 Introduction and objectives 123

11.2 Motivating exponential fitting 123

11.3 Exponential fitting and time-dependent convection-diffusion 128

11.4 Stability and convergence analysis 129

11.5 Approximating the derivative of the solution 131

11.6 Special limiting cases 132

11.7 Summary and conclusions 132

**PART III APPLYING FDM TO ONE-FACTOR INSTRUMENT PRICING 135**

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

12.1 Introduction and objectives 137

12.2 Exact solutions and benchmark cases 137

12.3 Perturbation analysis and risk engines 139

12.4 The trinomial method: Preview 139

12.5 Using exponential fitting with explicit time marching 142

12.6 Approximating the Greeks 142

12.7 Summary and conclusions 144

12.8 Appendix: the formula for Vega 144

**13 An Introduction to the Trinomial Method 147**

13.1 Introduction and objectives 147

13.2 Motivating the trinomial method 147

13.3 Trinomial method: Comparisons with other methods 149

13.4 The trinomial method for barrier options 151

13.5 Summary and conclusions 152

**14 Exponentially Fitted Difference Schemes for Barrier Options 153**

14.1 Introduction and objectives 153

14.2 What are barrier options? 153

14.3 Initial boundary value problems for barrier options 154

14.4 Using exponential fitting for barrier options 154

14.5 Time-dependent volatility 156

14.6 Some other kinds of exotic options 157

14.7 Comparisons with exact solutions 159

14.8 Other schemes and approximations 162

14.9 Extensions to the model 162

14.10 Summary and conclusions 163

**15 Advanced Issues in Barrier and Lookback Option Modelling 165**

15.1 Introduction and objectives 165

15.2 Kinds of boundaries and boundary conditions 165

15.3 Discrete and continuous monitoring 168

15.4 Continuity corrections for discrete barrier options 171

15.5 Complex barrier options 171

15.6 Summary and conclusions 173

**16 The Meshless (Meshfree) Method in Financial Engineering 175**

16.1 Introduction and objectives 175

16.2 Motivating the meshless method 175

16.3 An introduction to radial basis functions 177

16.4 Semi-discretisations and convection–diffusion equations 177

16.5 Applications of the one-factor Black–Scholes equation 179

16.6 Advantages and disadvantages of meshless 180

16.7 Summary and conclusions 181

**17 Extending the Black–Scholes Model: Jump Processes 183**

17.1 Introduction and objectives 183

17.2 Jump–diffusion processes 183

17.2.1 Convolution transformations 185

17.3 Partial integro-differential equations and financial applications 186

17.4 Numerical solution of PIDE: Preliminaries 187

17.5 Techniques for the numerical solution of PIDEs 188

17.6 Implicit and explicit methods 188

17.7 Implicit–explicit Runge–Kutta methods 189

17.8 Using operator splitting 189

17.9 Splitting and predictor–corrector methods 190

17.10 Summary and conclusions 191

**PART IV FDM FOR MULTIDIMENSIONAL PROBLEMS 193**

**18 Finite Difference Schemes for Multidimensional Problems 195**

18.1 Introduction and objectives 195

18.2 Elliptic equations 195

18.3 Diffusion and heat equations 202

18.4 Advection equation in two dimensions 205

18.5 Convection–diffusion equation 207

18.6 Summary and conclusions 208

**19 An Introduction to Alternating Direction Implicit and Splitting Methods 209**

19.1 Introduction and objectives 209

19.2 What is ADI, really? 210

19.3 Improvements on the basic ADI scheme 212

19.4 ADI for first-order hyperbolic equations 215

19.5 ADI classico and three-dimensional problems 217

19.6 The Hopscotch method 218

19.7 Boundary conditions 219

19.8 Summary and conclusions 221

**20 Advanced Operator Splitting Methods: Fractional Steps 223**

20.1 Introduction and objectives 223

20.2 Initial examples 223

20.3 Problems with mixed derivatives 224

20.4 Predictor–corrector methods (approximation correctors) 226

20.5 Partial integro-differential equations 227

20.6 More general results 228

20.7 Summary and conclusions 228

**21 Modern Splitting Methods 229**

21.1 Introduction and objectives 229

21.2 Systems of equations 229

21.3 A different kind of splitting: The IMEX schemes 232

21.4 Applicability of IMEX schemes to Asian option pricing 234

21.5 Summary and conclusions 235

**PART V APPLYING FDM TO MULTI-FACTOR INSTRUMENT PRICING 237**

**22 Options with Stochastic Volatility: The Heston Model 239**

22.1 Introduction and objectives 239

22.2 An introduction to Ornstein–Uhlenbeck processes 239

22.3 Stochastic differential equations and the Heston model 240

22.4 Boundary conditions 241

22.5 Using finite difference schemes: Prologue 243

22.6 A detailed example 243

22.7 Summary and conclusions 246

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

23.1 Introduction and objectives 249

23.2 An introduction to Asian options 249

23.3 My first PDE formulation 250

23.4 Using operator splitting methods 251

23.5 Cheyette interest models 253

23.6 New developments 254

23.7 Summary and conclusions 255

**24 Multi-Asset Options 257**

24.1 Introduction and objectives 257

24.2 A taxonomy of multi-asset options 257

24.3 Common framework for multi-asset options 265

24.4 An overview of finite difference schemes for multi-asset problems 266

24.5 Numerical solution of elliptic equations 267

24.6 Solving multi-asset Black–Scholes equations 269

24.7 Special guidelines and caveats 270

24.8 Summary and conclusions 271

**25 Finite Difference Methods for Fixed-Income Problems 273**

25.1 Introduction and objectives 273

25.2 An introduction to interest rate modelling 273

25.3 Single-factor models 274

25.4 Some specific stochastic models 276

25.5 An introduction to multidimensional models 278

25.6 The thorny issue of boundary conditions 280

25.7 Introduction to approximate methods for interest rate models 282

25.8 Summary and conclusions 283

**PART VI FREE AND MOVING BOUNDARY VALUE PROBLEMS 285**

**26 Background to Free and Moving Boundary Value Problems 287**

26.1 Introduction and objectives 287

26.2 Notation and definitions 287

26.3 Some preliminary examples 288

26.4 Solutions in financial engineering: A preview 293

26.5 Summary and conclusions 294

**27 Numerical Methods for Free Boundary Value Problems: Front-Fixing Methods 295**

27.1 Introduction and objectives 295

27.2 An introduction to front-fixing methods 295

27.3 A crash course on partial derivatives 295

27.4 Functions and implicit forms 297

27.5 Front fixing for the heat equation 299

27.6 Front fixing for general problems 300

27.7 Multidimensional problems 300

27.8 Front fixing and American options 303

27.9 Other finite difference schemes 305

27.10 Summary and conclusions 306

**28 Viscosity Solutions and Penalty Methods for American Option Problems 307**

28.1 Introduction and objectives 307

28.2 Definitions and main results for parabolic problems 307

28.3 An introduction to semi-linear equations and penalty method 310

28.4 Implicit, explicit and semi-implicit schemes 311

28.5 Multi-asset American options 312

28.6 Summary and conclusions 314

**29 Variational Formulation of American Option Problems 315**

29.1 Introduction and objectives 315

29.2 A short history of variational inequalities 316

29.3 A first parabolic variational inequality 316

29.4 Functional analysis background 318

29.5 Kinds of variational inequalities 319

29.6 Variational inequalities using Rothe’s methods 323

29.7 American options and variational inequalities 324

29.8 Summary and conclusions 324

**PART VII DESIGN AND IMPLEMENTATION IN C++ 325**

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

30.1 Introduction and objectives 327

30.2 The financial model 328

30.3 The viewpoints in the continuous model 328

30.4 The viewpoints in the discrete model 332

30.5 Auxiliary numerical methods 335

30.6 New Developments 336

30.7 Summary and conclusions 336

**31 Design and Implementation of First-Order Problems 337**

31.1 Introduction and objectives 337

31.2 Software requirements 337

31.3 Modular decomposition 338

31.4 Useful C++ data structures 339

31.5 One-factor models 339

31.6 Multi-factor models 343

31.7 Generalisations and applications to quantitative finance 346

31.8 Summary and conclusions 347

31.9 Appendix: Useful data structures in C++ 348

**32 Moving to Black–Scholes 353**

32.1 Introduction and objectives 353

32.2 The PDE model 354

32.3 The FDM model 355

32.4 Algorithms and data structures 355

32.5 The C++ model 356

32.6 Test case: The two-dimensional heat equation 357

32.7 Finite difference solution 357

32.8 Moving to software and method implementation 358

32.9 Generalisations 361

32.10 Summary and conclusions 362

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

33.1 Introduction and objectives 363

33.2 Abstract and concrete payoff classes 364

33.3 Using payoff classes 367

33.4 Lightweight payoff classes 368

33.5 Super-lightweight payoff functions 369

33.6 Payoff functions for multi-asset option problems 371

33.7 Caveat: non-smooth payoff and convergence degradation 373

33.8 Summary and conclusions 374

**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

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