Numerical Methods in Computational Finance A Partial Differential Equation (PDE/FDM) Approach

by Duffy, Daniel J.

• ISBN13: 9781119719670
• ISBN10: 1119719674
• Edition: 1st
• Format: Hardcover
• Copyright: 2022-04-25
• Publisher: Wiley

Summary

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.

Author Biography

Understand and apply ordinary and partial differential equations with this accessible, step-by-step guide

Numerical Methods in Computational Finance: A Partial Differential Equation (PDE/FDM) Approach delivers a detailed, step-by-step approach to the mathematical foundations of ordinary and partial differential equations, their approximation by the finite difference method, and their applications to computational finance.

Perfect for beginning, intermediate and expert practitioners alike, this book covers every critical aspect of the subject in an accessible format that progresses logically and gradually. It offers

• Robust mathematical foundations for one- and two-factor problems, including the mathematical and numerical concepts required to understand the finite difference method, elliptic and parabolic partial differential equations in two space variables, and more
• Discussions of the finite difference method, including initial boundary value problems for parabolic PDEs and methods to approximate the solution of two factor PDEs
• Practical applications of the included methods, with discussions of finite difference schemes for a wide range of one- and two-factor problems

Perfectly suited to anyone seeking an entry-level introduction to ordinary and partial differential equations, Numerical Methods in Computational Finance is also a must-read resource for industry quants and MSc/MFE students in finance looking for a detailed treatment of modern methods. The included topics have a wide range of applications to numerical analysis, science and engineering.

Table of Contents

Contents

Chapter 1 Real Analysis Foundations for this Book           1

1.1         Introduction and Objectives         1

1.2         Continuous Functions     1

1.2.1 Formal Definition of Continuity        2

1.2.2 An Example             3

1.2.3 Uniform Continuity              3

1.2.4 Classes of Discontinuous Functions               4

1.3         Differential Calculus        5

1.3.1 Taylor’s Theorem   6

1.3.2 Big O and Little o Notation 7

1.4         Partial Derivatives            8

1.5         Functions and Implicit Forms       9

1.6         Metric Spaces and Cauchy Sequences      11

1.6.1 Metric Spaces         11

1.6.2 Cauchy Sequences 12

1.6.3 Lipschitz Continuous Functions        13

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

2.2.1 Qualitative Properties of the Solution and Maximum Principle           18

2.3         Discretisation of Initial Value Problems: Fundamentals     20

2.3.1 Common Schemes 21

2.3.2 Discrete Maximum Principle             23

2.4         Special Schemes               24

2.4.1 Exponential Fitting               24

2.4.2 Scalar Nonlinear Problems and Predictor-Corrector Method              25

2.4.3 Extrapolation          26

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

3.2.1 An Example             36

3.3         Other Model Examples   37

3.3.1 Bernoulli ODE         37

3.3.2 Riccati ODE             37

3.3.3 Predator-Prey Models         38

3.3.4 Logistic Function    39

3.4         Existence Theorems for Stochastic Differential Equations (SDEs)  39

3.4.1 Stochastic Differential Equations (SDEs)       39

3.5         Numerical Methods for ODEs      42

3.5.1 Code Samples in Python     43

3.6         The Riccati Equation        45

3.6.1 Finite Difference Schemes  46

3.7         Matrix Differential Equations       48

3.7.1 Transition Rate Matrices and Continuous Time Markov Chains          49

3.8         Summary and Conclusions            50

Chapter 4 An Introduction to Finite Dimensional Vector Spaces 51

4.1         Short Introduction and Objectives             51

4.1.1 Notation   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

4.5.2 Rank and Nullity     58

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

5.2.1 Sums and Scalar Products of Linear Transformations             61

5.3         Inner Product Spaces      62

5.3.1 Orthonormal Basis               62

5.4         From Vector Spaces to Matrices 63

5.4.1 Some Examples      64

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

6.2.1 BLAS (Basic Linear Algebra Subprograms)    77

BLAS Level 1       77

BLAS Level 2       77

BLAS Level 3       78

6.3         Direct Methods for Linear Systems           79

6.3.2 Cholesky Decomposition    81

6.4         Solving Tridiagonal Systems         81

6.4.1 Double Sweep Method       81

Double Sweep Method   82

6.4.2 Thomas Algorithm 83

6.4.3 Block Tridiagonal Systems  84

6.5         Two-Point Boundary Value Problems       85

6.5.1 Finite Difference Approximation      87

6.5.2 Approximation of Boundary Conditions       88

6.6         Iterative Matrix Solvers  89

6.6.1 Iterative Methods  90

6.6.2 Jacobi Method        90

6.6.3 Gauss-Seidel Method          91

6.6.4 Successive Over-Relaxation (SOR)   91

6.6.5 Other Methods       91

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

7.2.1 Fully Implicit Method           96

7.2.4 Final Remarks         99

7.3         The Black Scholes Equation: Trinomial Method    99

7.4         The Heat Equation and Alternating Direction Explicit (ADE) Method           103

7.4.1 Background and Motivation             104

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

8.3.1 What is an Elliptic Operator?            110

8.3.2 Total and Principal Symbols              110

8.3.3 The Adjoint Equation           111

8.3.4 Self-Adjoint Operators and Equations           112

8.3.5 Numerical Approximation of PDEs in Adjoint Form  113

8.4         Classification of Second-Order Equations               114

8.4.1 Characteristics        114

8.4.2 Model Example      115

8.4.3 Test your Knowledge           116

8.5         Examples of Two-Factor Models from Computational Finance       116

8.5.1 Multi-Asset Options             117

8.5.2 Stochastic Dividend PDE     118

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

9.2.2 Initial Examples      123

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

Harmonic Functions and the Cauchy-Riemann Equations 130

10.4       Properties of The Laplace Equation           131

10.4.1 Maximum-Minimum Principle for Laplace’s Equation          133

10.5       Some Elliptic Boundary Value Problems  134

10.5.1 Some Motivating Examples            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

11.3.1 Time to reflect: What have we achieved and what’s next? 140

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

11.6.1 Cox-Ingersoll-Ross (CIR)    143

11.6.2 Heston Model Fundamenals           144

11.6.2.1 Standard European Call Option  145

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

12.4.1 Examples of an ill-posed Problem 156

12.4.2 The Importance of Proving that Problems are Well-Posed 158

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

12.8.1 Laplace Equation 163

12.8.2 Heat Equation      163

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

13.5.1 Euler-Maruyama Method                168

13.5.2 Milstein Method  170

13.5.3 Predictor-Corrector Method          170

13.5.4 Drift-Adjusted Predictor-Corrector Method             171

13.6       Path Evolution and Monte Carlo Option Pricing   172

13.6.1 Monte Carlo Option Pricing            173

13.6.2 Some C++ Code   174

13.7       Two-Factor Problems      177

13.8       The Ito Formula 181

13.9       Stochastics meets PDEs  182

13.9.1 A Statistics Refresher        182

13.9.2 The Feynman-Kac Formula             183

13.9.3 Kolmogorov Equations      184

13.8.4 Kolmogorov forward (Fokker-Planck (FPE)) Equation           184

13.8.5 Multi-Dimensional Problems and Boundary Conditions      185

13.9.6 Kolmogorov Backward Equation (KBE)       186

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

14.5.1 Fourier Transform for Advection Equation               192

14.5.2 Fourier Transform for Diffusion Equation  193

14.5       Discrete Fourier Transform          194

14.5.1 Finite and Infinite Dimensional Sequences and their Norms             194

14.5.4 Some more Examples        197

14.6       Theoretical Considerations           199

14.6.1    Consistency        199

14.6.2    Stability                200

14.6.3 Convergence        201

14.7       First-Order Partial Differential Equations               201

14.7.2 A Simple Explicit Scheme 204

14.7.4    Some other Schemes      207

14.7.5    General Linear Problems               208

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

15.2.1 Temporal and Spatial Stability       210

15.2.2 Motivating Exponential Fitting Methods    212

15.2.3 Eliminating Temporal and Spatial Stability Problems            213

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

15.7.1 Essentially Positive Matrices          223

15.7.2 Fully Discrete Schemes     225

15.8       Padé Matrix Approximation         226

15.9       Time-Dependent Convection-Diffusion Equations              231

15.9.1 Fully Discrete Schemes     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

16.3.1 Higher-Order and Mixed Greeks   236

16.4       Divided Differences         236

16.4.1 Approximation to First and Second Derivatives      237

16.4.2 Black Scholes Numeric Greeks and Divided Differences      239

16.5       Cubic Spline Interpolation            240

16.5.1 Caveat: Cubic Splines with Sparse Input Data         243

16.5.2 Cubic Splines for Option Greeks    243

16.5.3 Boundary Conditions         244

16.6       Some Complex Function Theory 245

16.6.1 Curves and Regions           245

16.6.2 Taylor’s Theorem and Series          247

16.6.3 Laurent’s Theorem and Series       248

16.6.5 Cauchy’s Integral Formula              249

16.6.7 Gauss’ Mean Value Theorem         251

16.7       The Complex Step Method (CSM)              251

16.7.1 Caveats   253

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

17.2.1 Simple Initial Value Problem          256

17.2.2 Population Dynamics         257

17.2.3 Comparing CSE and Complex Step Method (CSM)  258

17.2.3.1 CSM      259

17.2.3.2 CSE       259

17.3       CSE and Black Scholes PDE           259

17.3.1 Black Scholes Greeks: Algorithms and Design          260

17.3.2 Some Specific Black Scholes Greeks            261

17.4       Using Operator Calculus to compute Greeks         262

17.5       An Introduction to Automatic Differentiation (AD)             263

17.5.2 What is Automatic Differentiation: The Details       264

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

18.4.1    Alternating Direction Implicit (ADI) Method          273

18.4.2 Soviet (Operator) Splitting              275

18.4.3 Mixed Derivative and Yanenko Scheme     276

18.5       Other Related Schemes for the Heat Equation     277

18.5.1    D’Yakonov Method          277

18.5.2 Approximate Factorisation of Operators   277

18.5.3 Predictor-Corrector Methods         280

18.5.4 Partial Integro Differential Equations (PIDEs)          280

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

19.4.1 Barakat and Clark (B&C) Method  291

19.4.2 Saul’yev Method  293

19.4.3 Larkin Method      293

19.4.4 Two-Dimensional Diffusion Problems         293

19.5       ADE for Convection (Advection) Equation              294

19.6       Convection-Diffusion PDEs           295

19.7       Attention Points with ADE            299

The Consequences of Conditional Consistency     299

Call Payoff Behaviour at the Far Field       299

19.7.1 General Formulation of the ADE Method  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

20.2.1 Initial Results        304

20.2.2 The Exponential of a Matrix           305

20.3       The Exponential of a Matrix: Advanced Topics     305

20.3.1 Fundamental Theorem for Linear Systems               305

20.3.3 An Example           306

20.4       Motivation: One-dimensional Heat Equation        307

20.5       Semilinear Problems       309

20.6       Test Case: Double-Barrier Options            311

20.6.2    Using exponential Fitting of Barrier Options          313

20.6.3 Performing MOL with Boost C++ odeint     314

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

21.4.1 Single-Phase Melting Ice  322

21.4.3 American Option Pricing  324

21.4.5 Two-Phase Melting Ice     324

21.5       An Introduction to Parabolic Variational Inequalities         325

21.5.1 Formulation of Problem: Test Case             325

21.5.2 Examples of Initial Boundary Value Problems          327

21.6       An Introduction to Front-Fixing   332

21.6.1 Front-Fixing for the Heat Equation              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

22.3.2 Abstract Cauchy Problem 338

22.3.3 Examples               339

22.4       Mathematical Foundations of Splitting Methods 340

22.4.1 Lie (Trotter) Product Formula        340

22.4.2 Splitting Error       340

22.4.3 Component Splitting and Operator Splitting            342

22.4.4 Splitting as a Discretisation Method            342

22.5       Some Popular Splitting Methods               343

22.5.1 First-Order (Lie-Trotter) Splitting  343

22.5.2 Predictor-Corrector Splitting          344

22.5.3 Marchuk’s Two-Cycle (1-2-2-1) Method     344

22.5.4 Strang Splitting     345

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

23.3.1 Computing BVN by Solving a Hyperbolic PDE          352

The Finite Difference Method for the Goursat PDE            354

23.3.2 Analytical Solutions of Multi-Asset and Basket Options       355

23.4       PDE Models for Multi-Asset Option Problems: Requirements and Design 356

23.4.1 Domain Transformation   357

23.4.2 Numerical Boundary Conditions    357

23.5       An Overview of Finite Difference Schemes for Multi-Asset Option Problems           357

23.5.1 Common Design Principles              358

23.5.2 Detailed Design   359

23.5.3 Testing the Software         360

23.6       American Spread Options             361

23.7       Appendices         362

23.7.1 Traditional Approach to Numerical Boundary Conditions   362

23.7.2 Top-down Design of Monte Carlo Applications       363

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

24.2.1 Challenges             366

24.3       Prototype PDE Model     367

24.3.1 Similarity Reduction          368

24.4       The Many Ways to handle the Convective Term  369

24.4.1 Method of Lines (MOL)     369

24.4.2 Other Schemes    370

24.4.3 A Stable Monotone Upwind Scheme          371

24.5       ADE for Asian Options     371

24.6       ADI for Asian Options      372

24.6.1 Modern ADI Variations     373

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

25.3.1 Analytic Solutions               376

25.3.1 Initial Boundary Value Problem     378

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

26.2.1 The Maximum Principle and Mixed Derivatives      388

26.2.2 Some Examples    389

26.2.3 Code Sample Method of Lines (MOL) for Two-Factor Hull-White Model      390

26.3       The Complex Step Method (CSM) Revisited           392

26.3.1 Black Scholes Greeks using CSM and the Faddeeva Function            392

26.3.2 CSM and Functions of Several Complex Variables  395

26.3.3 C++ Code for extended CSM          396

26.3.4 CSM for Nonlinear Solvers              398

26.4       Extending the Hull-White: Possible Projects          398

26.5       Summary and Conclusions            400