Numerical Methods in Computational Finance: A Partial Differential Equation (PDE/FDM) Approach
Daniel J. Duffy
Résumé
This book is a detailed and step-by-step introduction to the mathematical foundations of ordinary and partial differential equations, their approximation by the finite difference method and applications to computational finance. The book is structured so that it can be read by beginners, novices and expert users.
Part A Mathematical Foundation for One-Factor Problems
Chapters 1 to 7 introduce the mathematical and numerical analysis concepts that are needed to understand the finite difference method and its application to computational finance.
Part B Mathematical Foundation for Two-Factor Problems
Chapters 8 to 13 discuss a number of rigorous mathematical techniques relating to elliptic and parabolic partial differential equations in two space variables. In particular, we develop strategies to preprocess and modify a PDE before we approximate it by the finite difference method, thus avoiding ad-hoc and heuristic tricks.
Part C The Foundations of the Finite Difference Method (FDM)
Chapters 14 to 17 introduce the mathematical background to the finite difference method for initial boundary value problems for parabolic PDEs. It encapsulates all the background information to construct stable and accurate finite difference schemes.
Part D Advanced Finite Difference Schemes for Two-Factor Problems
Chapters 18 to 22 introduce a number of modern finite difference methods to approximate the solution of two factor partial differential equations. This is the only book we know of that discusses these methods in any detail.
Part E Test Cases in Computational Finance
Chapters 23 to 26 are concerned with applications based on previous chapters. We discuss finite difference schemes for a wide range of one-factor and two-factor problems.
This book is suitable as an entry-level introduction as well as a detailed treatment of modern methods as used by industry quants and MSc/MFE students in finance. The topics have applications to numerical analysis, science and engineering.
More on computational finance and the author's online courses, see www.datasim.nl.
Chapter 1 Real Analysis Foundations for this Book 1
1.1 Introduction and Objectives 1
1.2 Continuous Functions 1
1.3 Differential Calculus 5
1.4 Partial Derivatives 8
1.5 Functions and Implicit Forms 9
1.6 Metric Spaces and Cauchy Sequences 11
1.7 Summary and Conclusions 15
Chapter 2 Ordinary Differential Equations (ODEs), Part 1 17
2.1 Introduction and Objectives 17
2.2 Background and Problem Statement 17
2.3 Discretisation of Initial Value Problems: Fundamentals 20
2.4 Special Schemes 24
2.5 Foundations of Discrete Time Approximations 26
2.6 Stiff ODEs 31
2.7 Intermezzo: Explicit Solutions 33
2.8 Summary and Conclusions 34
Chapter 3 Ordinary Differential Equations (ODEs), Part 2 35
3.1 Introduction and Objectives 35
3.2 Existence and Uniqueness Results 35
3.3 Other Model Examples 37
3.4 Existence Theorems for Stochastic Differential Equations (SDEs) 39
3.5 Numerical Methods for ODEs 42
3.6 The Riccati Equation 45
3.7 Matrix Differential Equations 48
3.8 Summary and Conclusions 50
Chapter 4 An Introduction to Finite Dimensional Vector Spaces 51
4.1 Short Introduction and Objectives 51
4.2 What is a Vector Space? 52
4.3 Subspaces 55
4.4 Linear Independence and Bases 56
4.5 Linear Transformations 57
4.6 Summary and Conclusions 59
Chapter 5 Guide to Matrix Theory and Numerical Linear Algebra 61
5.1 Introduction and Objectives 61
5.2 From Vector Spaces to Matrices 61
5.3 Inner Product Spaces 62
5.4 From Vector Spaces to Matrices 63
5.5 Fundamental Matrix Properties 65
5.6 Essential Matrix Types 67
5.7 The Cayley Transform 71
5.8 Summary and Conclusions 73
Chapter 6 Numerical Solutions of Boundary Value Problems 75
6.1 Introduction and Objectives 75
6.2 An Introduction to Numerical Linear Algebra 75
6.3 Direct Methods for Linear Systems 79
6.4 Solving Tridiagonal Systems 81
6.5 Two-Point Boundary Value Problems 85
6.6 Iterative Matrix Solvers 89
6.7 Example: Iterative Solvers for Elliptic PDEs 92
6.8 Summary and Conclusions 93
Chapter 7 Black Scholes Finite Differences for the Impatient 95
7.1 Introduction and Objectives 95
7.2 The Black Scholes Equation: Fully Implicit and Crank Nicolson Methods 95
7.3 The Black Scholes Equation: Trinomial Method 99
7.4 The Heat Equation and Alternating Direction Explicit (ADE) Method 103
7.5 ADE for Black Scholes: some Test Results 104
7.6 Summary and Conclusions 108
Chapter 8 Classifying and Transforming Partial Differential Equations 109
8.1 Introduction and Objectives 109
8.2 Background and Problem Statement 109
8.3 Introduction to Elliptic Equations 109
8.4 Classification of Second-Order Equations 114
8.5 Examples of Two-Factor Models from Computational Finance 116
8.6 Summary and Conclusions 118
Chapter 9 Transforming Partial Differential Equations to a Bounded Domain 121
9.1 Introduction and Objectives 121
9.2 The Domain in which a PDE is defined: Preamble 121
9.3 Other Examples 124
9.4 Hotspots 125
9.5 What happened to Domain Truncation? 125
9.6 Another Way to remove Mixed Derivative Terms 126
9.7 Summary and Conclusions 128
Chapter 10 Boundary Value Problems for Elliptic and Parabolic Partial Differential Equations 129
10.1 Introduction and Objectives 129
10.2 Notation and Prerequisites 129
10.3 The Laplace Equation 129
10.4 Properties of The Laplace Equation 131
10.5 Some Elliptic Boundary Value Problems 134
10.6 Extended Maximum-Minimum Principles 134
10.7 Summary and Conclusions 136
Chapter 11 Fichera Theory, Energy Inequalities and Integral Relations 137
11.1 Introduction and Objectives 137
11.2 Background and Problem Statement 137
11.3 Well-posed Problems and Energy Estimates 139
11.4 The Fichera Theory: Overview 140
11.5 The Fichera Theory: The Core Business 141
11.6 The Fichera Theory: Further Examples and Applications 143
11.7 Some Useful Theorems 149
11.8 Summary and Conclusions 151
Chapter 12 An Introduction to Time-dependent Partial Differential Equations 153
12.1 Introduction and Objectives 153
12.2 Notation and Prerequisites 153
12.3 Preamble: Separation of Variables for the Heat Equation 153
12.4 Well-posed Problems 155
12.5 Variations on Initial Boundary Value Problem for the Heat Equation 159
12.6 Maximum-Minimum Principles for Parabolic PDEs 160
12.7 Parabolic Equations with Time-Dependent Boundaries 160
12.8 Uniqueness Theorems for Boundary Value Problems in Two Dimensions 162
12.9 Summary and Conclusions 164
Chapter 13 Stochastics Representations of PDEs and Applications 165
13.1 Introduction and Objectives 165
13.2 Background, Requirements and Problem Statement 165
13.3 An Overview of Stochastic Differential Equations (SDEs) 165
13.4 An Introduction to One-Dimensional Random Processes 166
13.5 An Introduction to the Numerical Approximation of SDEs 168
13.6 Path Evolution and Monte Carlo Option Pricing 172
13.7 Two-Factor Problems 177
13.8 The Ito Formula 181
13.9 Stochastics meets PDEs 182
13.10 First Exit-Time Problems 187
13.11 Summary and Conclusions 188
Chapter 14 Mathematical and Numerical Foundations of the Finite Difference Method, Part I 189
14.1 Introduction and Objectives 189
14.2 Notation and Prerequisites 189
14.3 What is the Finite Difference Method, really? 190
14.4 Fourier Analysis of Linear PDEs 190
14.5 Discrete Fourier Transform 194
14.6 Theoretical Considerations 199
4.7 First-Order Partial Differential Equations 201
14.8 Summary and Conclusions 208
Chapter 15 Mathematical and Numerical Foundations of the Finite Difference Method, Part II 209
15.1 Introduction and Objectives 209
15.2 A Short History of Numerical Methods for CDR Equations 210
15.3 Exponential Fitting and Time-dependent Convection-Diffusion 216
15.4 Stability and Convergence Analysis 217
15.5 Special limiting Cases 218
15.6 Stability for Initial Boundary Value Problems 218
15.7 Semi-Discretisation for Convection-Diffusion Problems 221
15.8 Pade Matrix Approximation 226
15.9 Time-Dependent Convection-Diffusion Equations 231
15.10 Summary and Conclusions 232
Chapter 16 Sensitivity Analysis, Option Greeks and Parameter Optimisation, Part I 233
16.1 Introduction and Objectives 233
16.2 Helicopter View of Sensitivity Analysis 233
16.3 Black-Scholes-Merton Greeks 234
16.4 Divided Differences 236
16.5 Cubic Spline Interpolation 240
16.6 Some Complex Function Theory 245
16.7 The Complex Step Method (CSM) 251
16.8 Summary and Conclusions 254
Chapter 17 Advanced Topics in Sensitivity Analysis 255
17.1 Introduction and Objectives 255
17.2 Examples of CSE 255
17.3 CSE and Black Scholes PDE 259
17.4 Using Operator Calculus to compute Greeks 262
17.5 An Introduction to Automatic Differentiation (AD) 263
17.6 Dual Numbers 265
17.7 Automatic Differentiation in C++ 266
17.8 Summary and Conclusions 267
Chapter 18 Splitting Methods, Part I 269
18.1 Introduction and Objectives 269
18.2 Background and History 269
18.3 Notation, Prerequisites and Model Problems 270
18.4 Motivation: Two-Dimensional Heat Equation 273
18.5 Other Related Schemes for the Heat Equation 277
18.6 Boundary Conditions 281
18.7 Two-Dimensional Convection PDEs 282
18.8 Three-Dimensional Problems 284
18.9 The Hopscotch Method 285
18.10 Software Design and Implementation Guidelines 286
18.11 The Future: Convection-Diffusion Equations 287
18.12 Summary and Conclusions 287
Chapter 19 The Alternating Direction Explicit (ADE) Method 289
19.1 Introduction and Objectives 289
19.2 Background and Problem Statement 290
19.3 Global Overview and Applicability of ADE 290
19.4 Motivating Examples: One-Dimensional and Two-Dimensional Diffusion Equations 291
19.5 ADE for Convection (Advection) Equation 294
19.6 Convection-Diffusion PDEs 295
19.7 Attention Points with ADE 299
19.8 Summary and Conclusions 300
Chapter 20 The Method of Lines (MOL), Splitting and the Matrix Exponential 303
20.1 Introduction and Objectives 303
20.2 Notation and Prerequisites: The Exponential Function 303
20.3 The Exponential of a Matrix: Advanced Topics 305
20.4 Motivation: One-dimensional Heat Equation 307
20.5 Semilinear Problems 309
20.6 Test Case: Double-Barrier Options 311
20.7 Summary and Conclusions 318
Chapter 21 Free and Moving Boundary Value Problems 321
21.1 Introduction and Objectives 321
21.2 Background, Problem Statement and Formulations 321
21.3 Notation and Prerequisites 321
21.4 Some Initial Examples of Free and Moving Boundary Value Problems 322
21.5 An Introduction to Parabolic Variational Inequalities 325
21.6 An Introduction to Front-Fixing 332
21.7 Python Code Example: ADE for American Option Pricing 333
21.8 Summary and Conclusions 336
Chapter 22 Splitting Methods, Part II 337
22.1 Introduction and Objectives 337
22.2 Background and Problem Statement: the Essence of Sequential Splitting 337
22.3 Notation and Mathematical Formulation 338
22.4 Mathematical Foundations of Splitting Methods 340
22.5 Some Popular Splitting Methods 343
22.6 Applications and Relationships to Computational Finance 345
22.7 Software Design and Implementation Guidelines 346
22.8 Experience Report: Comparing ADI and Splitting 347
22.9 Summary and Conclusions 348
Chapter 23 Multi-Asset Options 349
23.1 Introduction and Objectives 349
23.2 Background and Goals 349
23.3 The Bivariate Normal Distribution (BVN) and its Applications 351
23.4 PDE Models for Multi-Asset Option Problems: Requirements and Design 356
23.5 An Overview of Finite Difference Schemes for Multi-Asset Option Problems 357
23.6 American Spread Options 361
23.7 Appendices 362
23.8 Summary and Conclusions 364
Chapter 24 Asian (Average Value) Options 365
24.1 Introduction and Objectives 365
24.2 Background and Problem Statement 365
24.3 Prototype PDE Model 367
24.4 The Many Ways to handle the Convective Term 369
24.5 ADE for Asian Options 371
24.6 ADI for Asian Options 372
24.7 Summary and Conclusions 374
Chapter 25 Interest Rate Models 375
25.1 Introduction and Objectives 375
25.2 Main Use Cases 375
25.3 The CIR Model 376
25.4 Well-Posedness of the CIRPDE Model 379
25.5 Finite Difference Methods for the CIR Model 381
25.6 Heston Model and the Feller Condition 383
25.7 Summary and Conclusion 386
Chapter 26 Epilogue Models Follow-up Chapters 1 to 25 387
26.1 Introduction and Objectives 387
26.2 Mixed Derivatives and Monotone Schemes 387
26.3 The Complex Step Method (CSM) Revisited 392
26.4 Extending the Hull-White: Possible Projects 398
26.5 Summary and Conclusions 400
DANIEL DUFFY, PhD, has BA (Mod), MSc and PhD degrees in pure, applied and numerical mathematics (University of Dublin, Trinity College) and he is active in promoting partial differential equations (PDE) and the Finite Difference Method (FDM) for applications in computational finance. He was responsible for the introduction of the Fractional Step (Soviet Splitting) method and the Alternating Direction Explicit (ADE) method in computational finance. He is the originator of the exponential fitting method for convection-dominated PDEs.
Caractéristiques techniques
| PAPIER | |
| Éditeur(s) | Wiley |
| Auteur(s) | Daniel J. Duffy |
| Parution | 09/03/2022 |
| Nb. de pages | 544 |
| EAN13 | 9781119719670 |
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