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

✍ Scribed by Daniel J. Duffy

• Publisher: Wiley
• Year: 2006
• Tongue: English
• Leaves: 442
• Series: The Wiley Finance Series
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

0.1 What is this Book?......Page 20
Contents......Page 8
0.4 Why Should I Read this Book?......Page 21
0.5 The Structure of this Book......Page 22
0.7 Contact, Feedback and More Information......Page 23
Part I The Continuous Theory Of Partial DifferentialI Equations......Page 24
1.1 Introduction and Objectives......Page 26
1.2.1 Special Kinds of Boundary Condition......Page 27
1.3 Linear Boundary Value Problems......Page 28
1.5 Some Special Cases......Page 29
1.6 Summary and Conclusions......Page 30
2.2 Partial Differential Equations......Page 32
2.3.1 Elliptic Equations......Page 34
2.3.2 Free Boundary Value Problems......Page 36
2.4 Parabolic Partial Differential Equations......Page 37
2.5.1 Second-Order Equations......Page 39
2.5.2 First-Order Equations......Page 40
2.6.2 First-Order Hyperbolic Systems......Page 41
2.7 Equations Containing Integrals......Page 42
2.8 Summary and Conclusions......Page 43
3.2 Linear Parabolic Equations......Page 44
3.3 The Continuous Problem......Page 45
3.4 The Maximum Principle for Parabolic Equations......Page 47
3.5 A Special Case: One-Factor Generalised Black–Scholes Models......Page 48
3.6 Fundamental Solution and the Green’s Function......Page 49
3.7 Integral Representation of the Solution of Parabolic PDEs......Page 50
3.8 Parabolic Equations in One Space Dimension......Page 52
3.9 Summary and Conclusions......Page 54
4.1 Introduction and Objectives......Page 56
4.2 Motivation and Background......Page 57
4.3 The Heat Equation and Financial Engineering......Page 58
4.4 The Separation of Variables Technique......Page 59
4.4.2 Heat Flow in a Rod Whose Ends are at a Specified Variable Temperature......Page 61
4.4.4 Eigenfunction Expansions......Page 62
4.5 Transformation Techniques for the Heat Equation......Page 63
4.5.2 Fourier Transform for the Heat Equation......Page 64
4.6 Summary and Conclusions......Page 65
5.2 First-Order Hyperbolic Equations......Page 66
5.2.1 An Example......Page 67
5.3.1 Numerical Integration Along the Characteristic Lines......Page 69
5.4 Applications to Financial Engineering......Page 72
5.5 Systems of Equations......Page 74
5.6 Propagation of Discontinuities......Page 76
5.6.1 Other Problems......Page 77
5.7 Summary and Conclusions......Page 78
Part II FiniteI DifferenceI Methods: The Fundamentals......Page 80
6.2 Fundamentals of Numerical Differentiation......Page 82
6.3 Caveat: Accuracy and Round-Off Errors......Page 84
6.5 Initial Value Problems......Page 86
6.5.1 Pad´ e Matrix Approximations......Page 87
6.5.2 Extrapolation......Page 90
6.6 Nonlinear Initial Value Problems......Page 91
6.6.1 Predictor–Corrector Methods......Page 92
6.6.2 Runge–Kutta Methods......Page 93
6.7 Scalar Initial Value Problems......Page 94
6.8 Summary and Conclusions......Page 95
7.2 Classifying Semi-Discretisation Methods......Page 98
7.3.1 A Test Case......Page 99
7.3.3 Semi-Discretisation for Convection-Diffusion Problems......Page 101
7.3.4 Essentially Positive Matrices......Page 103
7.4 Numerical Approximation of First-Order Systems......Page 104
7.4.1 Fully Discrete Schemes......Page 105
7.4.2 Semi-Linear Problems......Page 106
7.5 Summary and Conclusions......Page 108
8.2 Some Fundamental Concepts......Page 110
8.2.2 Stability......Page 112
8.3 Stability and the Fourier Transform......Page 113
8.4 The Discrete Fourier Transform......Page 115
8.4.1 Some Other Examples......Page 117
8.5 Stability for Initial Boundary Value Problems......Page 118
8.5.1 Gerschgorin’s Circle Theorem......Page 119
8.6 Summary and Conclusions......Page 120
9.2 Scoping the Problem......Page 122
9.3 Why First-Order Equations are Different: Essential difficulties......Page 124
9.4 A Simple Explicit Scheme......Page 125
9.5 Some Common Schemes for Initial Value Problems......Page 127
9.7 Monotone and Positive-Type Schemes......Page 129
9.8 Extensions, Generalisations and Other Applications......Page 130
9.8.2 Systems of Equations......Page 131
9.8.4 Several Independent Variables......Page 133
9.9 Summary and Conclusions......Page 134
10.1 Introduction and Objectives......Page 136
10.2 Approximation of Derivatives on the Boundaries......Page 137
10.4 Fully Discrete Schemes......Page 139
10.6 Semi-Discretisation in Space......Page 140
10.8 Summary and Conclusions......Page 141
11.2 Motivating Exponential Fitting......Page 142
11.2.1 ‘Continuous’ Exponential Approximation......Page 143
11.2.2 ‘Discrete’ Exponential Approximation......Page 144
11.3 Exponential Fitting and Time-Dependent Convection-Diffusion......Page 147
11.4 Stability and Convergence Analysis......Page 148
11.5 Approximating the Derivative of the Solution......Page 150
11.7 Summary and Conclusions......Page 151
Part III Applying FDM to One-Factor Instrument Pricing......Page 154
12.2 Exact Solutions and Benchmark Cases......Page 156
12.4 The Trinomial Method: Preview......Page 158
12.4.1 Stability of the Trinomial Method......Page 160
12.6 Approximating the Greeks......Page 161
12.8 Appendix: the Formula for Vega......Page 163
13.2 Motivating the Trinomial Method......Page 166
13.3 Trinomial Method: Comparisons with Other Methods......Page 168
13.3.1 A General Formulation......Page 169
13.4 The Trinomial Method for Barrier Options......Page 170
13.5 Summary and Conclusions......Page 171
14.2 What are Barrier Options?......Page 172
14.4 Using Exponential Fitting for Barrier Options......Page 173
14.5 Time-Dependent Volatility......Page 175
14.6 Some Other Kinds of Exotic Options......Page 176
14.6.2 Capped Power Call Options......Page 177
14.7 Comparisons with Exact Solutions......Page 178
14.9 Extensions to the Model......Page 181
14.10 Summary and Conclusions......Page 182
15.2 Kinds of Boundaries and Boundary Conditions......Page 184
15.3.1 What is Discrete Monitoring?......Page 187
15.3.2 Finite Difference Schemes and Jumps in Time......Page 188
15.3.3 Lookback Options and Jumps......Page 189
15.5 Complex Barrier Options......Page 190
15.6 Summary and Conclusions......Page 192
16.2 Motivating the Meshless Method......Page 194
16.4 Semi-Discretisations and Convection–Diffusion Equations......Page 196
16.5 Applications of the One-Factor Black–Scholes Equation......Page 198
16.6 Advantages and Disadvantages of Meshless......Page 199
16.7 Summary and Conclusions......Page 200
17.2 Jump–Diffusion Processes......Page 202
17.2.1 Convolution Transformations......Page 204
17.3 Partial Integro-Differential Equations and Financial Applications......Page 205
17.4 Numerical Solution of PIDE: Preliminaries......Page 206
17.6 Implicit and Explicit Methods......Page 207
17.8 Using Operator Splitting......Page 208
17.9 Splitting and Predictor–Corrector Methods......Page 209
17.10 Summary and Conclusions......Page 210
Part IV FDM For Multidimensional Problems......Page 212
18.2 Elliptic Equations......Page 214
18.2.1 A Self-Adjoint Elliptic Operator......Page 217
18.2.2 Solving the Matrix Systems......Page 218
18.2.3 Exact Solutions to Elliptic Problems......Page 219
18.3 Diffusion and Heat Equations......Page 221
18.3.1 Exact Solutions to the Heat Equation......Page 223
18.4 Advection Equation in Two Dimensions......Page 224
18.5 Convection–Diffusion Equation......Page 226
18.6 Summary and Conclusions......Page 227
19.1 Introduction and objectives......Page 228
19.2 What is ADI, Really?......Page 229
19.3.1 The D’Yakonov Scheme......Page 231
19.3.2 Approximate Factorization of Operators......Page 232
19.4 ADI for First-Order Hyperbolic Equations......Page 234
19.5 ADI Classico and Three-Dimensional Problems......Page 236
19.6 The Hopscotch Method......Page 237
19.7 Boundary Conditions......Page 238
19.8 Summary and Conclusions......Page 240
20.2 Initial Examples......Page 242
20.3 Problems with Mixed Derivatives......Page 243
20.4 Predictor–Corrector Methods (Approximation Correctors)......Page 245
20.5 Partial Integro-Differential Equations......Page 246
20.7 Summary and Conclusions......Page 247
21.2 Systems of Equations......Page 248
21.2.1 ADI and Splitting for Parabolic Systems......Page 249
21.2.2 Compound and Chooser Options......Page 250
21.3 A Different Kind of Splitting: The IMEX Schemes......Page 251
21.4 Applicability of IMEX Schemes to Asian Option Pricing......Page 253
21.5 Summary and Conclusions......Page 254
Part V Applying FDM to Multi-Factor Instrument Pricing......Page 256
22.2 An Introduction to Ornstein–Uhlenbeck Processes......Page 258
22.3 Stochastic Differential Equations and the Heston Model......Page 259
22.4 Boundary Conditions......Page 260
22.4.3 Other Kinds of Boundary Conditions......Page 261
22.6 A Detailed Example......Page 262
22.7 Summary and Conclusions......Page 265
23.2 An Introduction to Asian Options......Page 268
23.3 My First PDE Formulation......Page 269
23.4 Using Operator Splitting Methods......Page 270
23.5 Cheyette Interest Models......Page 272
23.6 New Developments......Page 273
23.7 Summary and Conclusions......Page 274
24.2 A Taxonomy of Multi-Asset Options......Page 276
24.2.1 Exchange Options......Page 279
24.2.2 Rainbow Options......Page 280
24.2.3 Basket Options......Page 281
24.2.5 Quotient Options......Page 282
24.2.8 Spread Options......Page 283
24.3 Common Framework for Multi-Asset Options......Page 284
24.4 An Overview of Finite Difference Schemes for Multi-Asset Problems......Page 285
24.5 Numerical Solution of Elliptic Equations......Page 286
24.6 Solving Multi-Asset Black–Scholes Equations......Page 288
24.7 Special Guidelines and Caveats......Page 289
24.8 Summary and Conclusions......Page 290
25.2 An Introduction to Interest Rate Modelling......Page 292
25.3 Single-Factor Models......Page 293
25.4 Some Specific Stochastic Models......Page 295
25.4.4 The Hull–White Model......Page 296
25.5 An Introduction to Multidimensional Models......Page 297
25.6.1 One-Factor Models......Page 299
25.6.2 Multi-Factor Models......Page 300
25.7.1 One-Factor Models......Page 301
25.8 Summary and Conclusions......Page 302
Part VI Free and Moving Boundary Value Problems......Page 304
26.2 Notation and Definitions......Page 306
26.3.1 Single-Phase Melting ICE......Page 307
26.3.2 One-Factor Option Modelling: American Exercise Style......Page 308
26.3.4 The Inverse Stefan Problem......Page 309
26.3.5 Two and Three Space Dimensions......Page 310
26.4.1 What Kinds of Early Exercise Features?......Page 312
26.5 Summary and Conclusions......Page 313
27.3 A Crash Course on Partial Derivatives......Page 314
27.4 Functions and Implicit forms......Page 316
27.5 Front Fixing for the Heat Equation......Page 318
27.7 Multidimensional Problems......Page 319
27.8 Front Fixing and American Options......Page 322
27.9.1 The Method of Lines and Predictor–Corrector......Page 324
27.10 Summary and Conclusions......Page 325
28.2.1 Semi-Continuity......Page 326
28.2.2 Viscosity Solutions of Nonlinear Parabolic Problems......Page 327
28.3 An Introduction to Semi-Linear Equations and Penalty Method......Page 329
28.4 Implicit, Explicit and Semi-Implicit Schemes......Page 330
28.5 Multi-Asset American Options......Page 331
28.6 Summary and Conclusions......Page 333
29.1 Introduction and Objectives......Page 334
29.3 A First Parabolic Variational Inequality......Page 335
29.4 Functional Analysis Background......Page 337
29.5.1 Diffusion with Semi-Permeable Membrane......Page 338
29.5.2 A One-Dimensional Finite Element Approximation......Page 339
29.6 Variational Inequalities Using Rothe’s Methods......Page 342
29.8 Summary and Conclusions......Page 343
Part VII Design and Implementation In C ++......Page 344
30.1 Introduction and Objectives......Page 346
30.3 The Viewpoints in the Continuous Model......Page 347
30.3.1 Payoff Functions......Page 348
30.3.2 Boundary Conditions......Page 349
30.3.3 Transformations......Page 350
30.4.1 Functional and Non-Functional requirements......Page 351
30.4.2 Approximating the Spatial Derivatives in the PDE......Page 352
30.4.4 Payoff Functions......Page 353
30.5 Auxiliary Numerical Methods......Page 354
30.7 Summary and Conclusions......Page 355
31.2 Software Requirements......Page 356
31.3 Modular Decomposition......Page 357
31.5 One-Factor Models......Page 358
31.5.1 Main Program and Output......Page 361
31.6 Multi-Factor Models......Page 362
31.7 Generalisations and Applications to Quantitative Finance......Page 365
31.8 Summary and Conclusions......Page 366
31.9 Appendix: Useful Data Structures in C ++......Page 367
32.1 Introduction and Objectives......Page 372
32.2 The PDE Model......Page 373
32.4 Algorithms and Data Structures......Page 374
32.5 The C ++Model......Page 375
32.7 Finite Difference Solution......Page 376
32.8.2 Creating a Mesh......Page 377
32.8.3 Choosing a Scheme......Page 379
32.9.3 Flexible Software Solutions......Page 380
32.10 Summary and Conclusions......Page 381
33.1 Introduction and Objectives......Page 382
33.2 Abstract and Concrete Payoff Classes......Page 383
33.3 Using Payoff Classes......Page 386
33.4 Lightweight Payoff Classes......Page 387
33.5 Super-Lightweight Payoff Functions......Page 388
33.6 Payoff Functions for Multi-Asset Option Problems......Page 390
33.7 Caveat: Non-Smooth Payoff and Convergence Degradation......Page 392
33.8 Summary and Conclusions......Page 393
A1 An Introduction to Integral and Partial Integro-Differential Equations......Page 394
A2 An Introduction to the Finite Element Method......Page 412
Bibliography......Page 428
Index......Page 436

✦ Subjects

Финансово-экономические дисциплины;Финансовая математика;