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Numerical Methods in Computational Finance

A Partial Differential Equation (PDE/FDM) Approach
Wiley Finance 1. Aufl.

von: Daniel J. Duffy
71,99 €
Verlag:	Wiley
Format:	EPUB
Veröffentl.:	14.03.2022
ISBN/EAN:	9781119719724
Sprache:	englisch
Anzahl Seiten:	544

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Titelbeschreibung

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.

Inhaltsverzeichnis

Preface xix Who Should Read this Book? xxiii Part A : Mathematical Foundation for One-Factor Problems Chapter 1 : Real Analysis Foundations for this Book 3 1.1 Introduction and Objectives 3 1.2 Continuous Functions 4 1.2.1 Formal Definition of Continuity 5 1.2.2 An Example 6 1.2.3 Uniform Continuity 6 1.2.4 Classes of Discontinuous Functions 7 1.3 Differential Calculus 8 1.3.1 Taylor’s Theorem 9 1.3.2 Big O and Little o Notation 10 1.4 Partial Derivatives 11 1.5 Functions and Implicit Forms 13 1.6 Metric Spaces and Cauchy Sequences 14 1.6.1 Metric Spaces 15 1.6.2 Cauchy Sequences 16 1.6.3 Lipschitz Continuous Functions 17 1.7 Summary and Conclusions 19 Chapter 2 : Ordinary Differential Equations (ODEs), Part 1 21 2.1 Introduction and Objectives 21 2.2 Background and Problem Statement 22 2.2.1 Qualitative Properties of the Solution and Maximum Principle 22 2.2.2 Rationale and Generalisations 24 2.3 Discretisation of Initial Value Problems: Fundamentals 25 2.3.1 Common Schemes 26 2.3.2 Discrete Maximum Principle 28 2.4 Special Schemes 29 2.4.1 Exponential Fitting 29 2.4.2 Scalar Non-Linear Problems and Predictor-Corrector Method 31 2.4.3 Extrapolation 31 2.5 Foundations of Discrete Time Approximations 32 2.6 Stiff ODEs 37 2.7 Intermezzo: Explicit Solutions 39 2.8 Summary and Conclusions 41 Chapter 3 : Ordinary Differential Equations (ODEs), Part 2 43 3.1 Introduction and Objectives 43 3.2 Existence and Uniqueness Results 43 3.2.1 An Example 45 3.3 Other Model Examples 45 3.3.1 Bernoulli ODE 45 3.3.2 Riccati ODE 46 3.3.3 Predator-Prey Models 47 3.3.4 Logistic Function 48 3.4 Existence Theorems for Stochastic Differential Equations (SDEs) 48 3.4.1 Stochastic Differential Equations (SDEs) 49 3.5 Numerical Methods for ODEs 51 3.5.1 Code Samples in Python 52 3.6 The Riccati Equation 55 3.6.1 Finite Difference Schemes 57 3.7 Matrix Differential Equations 59 3.7.1 Transition Rate Matrices and Continuous Time Markov Chains 61 3.8 Summary and Conclusions 62 Chapter 4 : An Introduction to Finite Dimensional Vector Spaces 63 4.1 Short Introduction and Objectives 63 4.1.1 Notation 64 4.2 What Is a Vector Space? 65 4.3 Subspaces 67 4.4 Linear Independence and Bases 68 4.5 Linear Transformations 69 4.5.1 Invariant Subspaces 70 4.5.2 Rank and Nullity 71 4.6 Summary and Conclusions 72 Chapter 5 : Guide to Matrix Theory and Numerical Linear Algebra 73 5.1 Introduction and Objectives 73 5.2 From Vector Spaces to Matrices 73 5.2.1 Sums and Scalar Products of Linear Transformations 73 5.3 Inner Product Spaces 74 5.3.1 Orthonormal Basis 75 5.4 From Vector Spaces to Matrices 76 5.4.1 Some Examples 76 5.5 Fundamental Matrix Properties 77 5.6 Essential Matrix Types 80 5.6.1 Nilpotent and Related Matrices 80 5.6.2 Normal Matrices 81 5.6.3 Unitary and Orthogonal Matrices 82 5.6.4 Positive Definite Matrices 82 5.6.5 Non-Negative Matrices 83 5.6.6 Irreducible Matrices 83 5.6.7 Other Kinds of Matrices 84 5.7 The Cayley Transform 84 5.8 Summary and Conclusions 86 Chapter 6 : Numerical Solutions of Boundary Value Problems 87 6.1 Introduction and Objectives 87 6.2 An Introduction to Numerical Linear Algebra 87 6.2.1 BLAS (Basic Linear Algebra Subprograms) 90 6.3 Direct Methods for Linear Systems 92 6.3.1 LU Decomposition 92 6.3.2 Cholesky Decomposition 94 6.4 Solving Tridiagonal Systems 94 6.4.1 Double Sweep Method 94 6.4.2 Thomas Algorithm 96 6.4.3 Block Tridiagonal Systems 97 6.5 Two-Point Boundary Value Problems 99 6.5.1 Finite Difference Approximation 100 6.5.2 Approximation of Boundary Conditions 102 6.6 Iterative Matrix Solvers 103 6.6.1 Iterative Methods 103 6.6.2 Jacobi Method 104 6.6.3 Gauss–Seidel Method 104 6.6.4 Successive Over-Relaxation (SOR) 105 6.6.5 Other Methods 105 6.7 Example: Iterative Solvers for Elliptic PDEs 106 6.8 Summary and Conclusions 107 Chapter 7 : Black–Scholes Finite Differences for the Impatient 109 7.1 Introduction and Objectives 109 7.2 The Black–Scholes Equation: Fully Implicit and Crank–Nicolson Methods 110 7.2.1 Fully Implicit Method 110 7.2.2 Crank–Nicolson Method 111 7.2.3 Final Remarks 114 7.3 The Black–Scholes Equation: Trinomial Method 115 7.3.1 Comparison with Other Methods 115 7.4 The Heat Equation and Alternating Direction Explicit (ADE) Method 120 7.4.1 Background and Motivation 120 7.5 ADE for Black–Scholes: Some Test Results 121 7.6 Summary and Conclusions 126 Part B : Mathematical Foundation for Two-Factor Problems Chapter 8 : Classifying and Transforming Partial Differential Equations 129 8.1 Introduction and Objectives 129 8.2 Background and Problem Statement 129 8.3 Introduction to Elliptic Equations 130 8.3.1 What is an Elliptic Operator? 130 8.3.2 Total and Principal Symbols 131 8.3.3 The Adjoint Equation 132 8.3.4 Self-Adjoint Operators and Equations 133 8.3.5 Numerical Approximation of PDEs in Adjoint Form 134 8.3.6 Elliptic Equations with Non-Negative Characteristic Form 135 8.4 Classification of Second-Order Equations 135 8.4.1 Characteristics 136 8.4.2 Model Example 137 8.4.3 Test your Knowledge 138 8.5 Examples of Two-Factor Models from Computational Finance 139 8.5.1 Multi-Asset Options 139 8.5.2 Stochastic Dividend PDE 140 8.6 Summary and Conclusions 141 Chapter 9 : Transforming Partial Differential Equations to a Bounded Domain 143 9.1 Introduction and Objectives 143 9.2 The Domain in Which a PDE Is Defined: Preamble 143 9.2.1 Background and Specific Mappings 144 9.2.2 Initial Examples 146 9.3 Other Examples 147 9.4 Hotspots 148 9.5 What Happened to Domain Truncation? 148 9.6 Another Way to Remove Mixed Derivative Terms 149 9.7 Summary and Conclusions 151 Chapter 10 : Boundary Value Problems for Elliptic and Parabolic Partial Differential Equations 153 10.1 Introduction and Objectives 153 10.2 Notation and Prerequisites 154 10.3 The Laplace Equation 154 10.3.1 Harmonic Functions and the Cauchy–Riemann Equations 154 10.4 Properties of The Laplace Equation 156 10.4.1 Maximum-Minimum Principle for Laplace’s Equation 158 10.5 Some Elliptic Boundary Value Problems 159 10.5.1 Some Motivating Examples 159 10.6 Extended Maximum-Minimum Principles 159 10.6.1 An Example 161 10.7 Summary and Conclusions 162 Chapter 11 : Fichera Theory, Energy Inequalities and Integral Relations 163 11.1 Introduction and Objectives 163 11.2 Background and Problem Statement 163 11.2.1 The ‘Big Bang’: Cauchy–Euler Equation 163 11.3 Well-Posed Problems and Energy Estimates 165 11.3.1 Time to Reflect: What Have We Achieved and What’s Next? 167 11.4 The Fichera Theory: Overview 168 11.5 The Fichera Theory: The Core Business 168 11.6 The Fichera Theory: Further Examples and Applications 171 11.6.1 Cox–Ingersoll–Ross (CIR) 171 11.6.2 Heston Model Fundamenals 172 11.6.3 Heston Model by Fichera Theory 176 11.6.4 First-Order Hyperbolic PDE in One and Two Space Variables 177 11.7 Some Useful Theorems 178 11.7.1 Divergence (Gauss–Ostrogradsky) Theorem 179 11.7.2 Green’s Theorem/Formula 180 11.7.3 Green’s First and Second Identities 180 11.8 Summary and Conclusions 180 Chapter 12 : An Introduction to Time-Dependent Partial Differential Equations 181 12.1 Introduction and Objectives 181 12.2 Notation and Prerequisites 181 12.3 Preamble: Separation of Variables for the Heat Equation 182 12.4 Well-Posed Problems 184 12.4.1 Examples of an ill-posed Problem 185 12.4.2 The Importance of Proving that Problems Are Well-Posed 187 12.5 Variations on Initial Boundary Value Problem for the Heat Equation 188 12.5.1 Smoothness and Compatibility Conditions 188 12.6 Maximum-Minimum Principles for Parabolic PDEs 189 12.7 Parabolic Equations with Time-Dependent Boundaries 190 12.8 Uniqueness Theorems for Boundary Value Problems in Two Dimensions 192 12.8.1 Laplace Equation 192 12.8.2 Heat Equation 193 12.9 Summary and Conclusions 193 Chapter 13 : Stochastics Representations of PDEs and Applications 195 13.1 Introduction and Objectives 195 13.2 Background, Requirements and Problem Statement 196 13.3 An Overview of Stochastic Differential Equations (SDEs) 196 13.4 An Introduction to One-Dimensional Random Processes 196 13.5 An Introduction to the Numerical Approximation of SDEs 199 13.5.1 Euler–Maruyama Method 199 13.5.2 Milstein Method 201 13.5.3 Predictor-Corrector Method 201 13.5.4 Drift-Adjusted Predictor-Corrector Method 202 13.6 Path Evolution and Monte Carlo Option Pricing 203 13.6.1 Monte Carlo Option Pricing 204 13.6.2 Some C++ Code 205 13.7 Two-Factor Problems 209 13.7.1 Spread Options with Stochastic Volatility 209 13.7.2 Heston Stochastic Volatility Model 211 13.8 The Ito Formula 215 13.9 Stochastics Meets PDEs 215 13.9.1 A Statistics Refresher 215 13.9.2 The Feynman–Kac Formula 217 13.9.3 Kolmogorov Equations 218 13.9.4 Kolmogorov Forward (Fokker–Planck (FPE)) Equation 218 13.9.5 Multi-Dimensional Problems and Boundary Conditions 219 13.9.6 Kolmogorov Backward Equation (KBE) 220 13.10 First Exit-Time Problems 221 13.11 Summary and Conclusions 222 Part C : The Foundations of the Finite Difference Method (FDM) Chapter 14 : Mathematical and Numerical Foundations of the Finite Difference Method, Part I 225 14.1 Introduction and Objectives 225 14.2 Notation and Prerequisites 226 14.3 What Is the Finite Difference Method, Really? 227 14.4 Fourier Analysis of Linear PDEs 227 14.4.1 Fourier Transform for Advection Equation 229 14.4.2 Fourier Transform for Diffusion Equation 230 14.5 Discrete Fourier Transform 232 14.5.1 Finite and Infinite Dimensional Sequences and Their Norms 232 14.5.2 Discrete Fourier Transform (DFT) 233 14.5.3 Discrete von Neumann Stability Criterion 235 14.5.4 Some More Examples 235 14.6 Theoretical Considerations 237 14.6.1 Consistency 237 14.6.2 Stability 238 14.6.3 Convergence 239 14.7 First-Order Partial Differential Equations 239 14.7.1 Why First-Order Equations are Different: Essential Difficulties 242 14.7.2 A Simple Explicit Scheme 243 14.7.3 Some Common Schemes for Initial Value Problems 245 14.7.4 Some Other Schemes 246 14.7.5 General Linear Problems 248 14.8 Summary and Conclusions 248 Chapter 15: Mathematical and Numerical Foundations of the Finite Difference Method, Part II 249 15.1 Introduction and Objectives 249 15.2 A Short History of Numerical Methods for CDR Equations 250 15.2.1 Temporal and Spatial Stability 251 15.2.2 Motivating Exponential Fitting Methods 253 15.2.3 Eliminating Temporal and Spatial Stability Problems 254 15.3 Exponential Fitting and Time-Dependent Convection-Diffusion 257 15.4 Stability and Convergence Analysis 258 15.5 Special Limiting Cases 260 15.6 Stability for Initial Boundary Value Problems 260 15.6.1 Gerschgorin’s Circle Theorem 261 15.7 Semi-Discretisation for Convection-Diffusion Problems 264 15.7.1 Essentially Positive Matrices 265 15.7.2 Fully Discrete Schemes 267 15.8 Padé Matrix Approximation 269 15.8.1 Padé Matrix Approximations 270 15.9 Time-Dependent Convection-Diffusion Equations 275 15.9.1 Fully Discrete Schemes 275 15.10 Summary and Conclusions 276 Chapter 16 Sensitivity Analysis, Option Greeks and Parameter Optimisation, Part I 277 16.1 Introduction and Objectives 277 16.2 Helicopter View of Sensitivity Analysis 278 16.3 Black–Scholes–Merton Greeks 279 16.3.1 Higher-Order and Mixed Greeks 282 16.4 Divided Differences 282 16.4.1 Approximation to First and Second Derivatives 282 16.4.2 Black–Scholes Numeric Greeks and Divided Differences 285 16.5 Cubic Spline Interpolation 286 16.5.1 Caveat: Cubic Splines with Sparse Input Data 289 16.5.2 Cubic Splines for Option Greeks 290 16.5.3 Boundary Conditions 291 16.6 Some Complex Function Theory 292 16.6.1 Curves and Regions 293 16.6.2 Taylor’s Theorem and Series 294 16.6.3 Laurent’s Theorem and Series 295 16.6.4 Cauchy–Goursat Theorem 296 16.6.5 Cauchy’s Integral Formula 297 16.6.6 Cauchy’s Residue Theorem 298 16.6.7 Gauss’s Mean Value Theorem 299 16.7 The Complex Step Method (CSM) 299 16.7.1 Caveats 302 16.8 Summary and Conclusions 302 Chapter 17 Advanced Topics in Sensitivity Analysis 305 17.1 Introduction and Objectives 305 17.2 Examples of CSE 305 17.2.1 Simple Initial Value Problem 306 17.2.2 Population Dynamics 307 17.2.3 Comparing CSE and Complex Step Method (CSM) 310 17.3 CSE and Black–Scholes PDE 310 17.3.1 Black–Scholes Greeks: Algorithms and Design 311 17.3.2 Some Specific Black–Scholes Greeks 312 17.4 Using Operator Calculus to Compute Greeks 313 17.5 An Introduction to Automatic Differentiation (AD) for the Impatient 314 17.5.1 What Is Automatic Differentiation: The Details 316 17.6 Dual Numbers 317 17.7 Automatic Differentiation in C++ 318 17.8 Summary and Conclusions 319 Part D : Advanced Finite Difference Schemes for Two-Factor Problems Chapter 18 : Splitting Methods, Part I 323 18.1 Introduction and Objectives 323 18.2 Background and History 324 18.3 Notation, Prerequisites and Model Problems 325 18.4 Motivation: Two-Dimensional Heat Equation 328 18.4.1 Alternating Direction Implicit (ADI) Method 328 18.4.2 Soviet (Operator) Splitting 330 18.4.3 Mixed Derivative and Yanenko Scheme 331 18.5 Other Related Schemes for the Heat Equation 333 18.5.1 D’Yakonov Method 333 18.5.2 Approximate Factorisation of Operators 334 18.5.3 Predictor-Corrector Methods 337 18.5.4 Partial Integro Differential Equations (PIDEs) 338 18.6 Boundary Conditions 339 18.7 Two-Dimensional Convection PDEs 341 18.8 Three-Dimensional Problems 343 18.9 The Hopscotch Method 344 18.10 Software Design and Implementation Guidelines 346 18.11 The Future: Convection-Diffusion Equations 346 18.12 Summary and Conclusions 347 Chapter 19 : The Alternating Direction Explicit (ADE) Method 349 19.1 Introduction and Objectives 349 19.2 Background and Problem Statement 351 19.3 Global Overview and Applicability of ADE 351 19.4 Motivating Examples: One-Dimensional and Two-Dimensional Diffusion Equations 352 19.4.1 Barakat and Clark (B&C) Method 353 19.4.2 Saul’yev Method 354 19.4.3 Larkin Method 355 19.4.4 Two-Dimensional Diffusion Problems 355 19.5 ADE for Convection (Advection) Equation 356 19.6 Convection-Diffusion PDEs 358 19.6.1 Example: Black–Scholes PDE 359 19.6.2 Boundary Conditions 360 19.6.3 Spatial Amplification Errors 361 19.7 Attention Points with ADE 362 The Consequences of Conditional Consistency 362 Call Pay-Off Behaviour at the Far Field 362 19.7.1 General Formulation of the ADE Method 362 19.8 Summary and Conclusions 364 Chapter 20 : The Method of Lines (MOL), Splitting and the Matrix Exponential 365 20.1 Introduction and Objectives 365 20.2 Notation and Prerequisites: The Exponential Function 366 20.2.1 Initial Results 367 20.2.2 The Exponential of a Matrix 367 20.3 The Exponential of a Matrix: Advanced Topics 368 20.3.1 Fundamental Theorem for Linear Systems 368 Proof of Theorem 20.1. 369 20.3.2 An Example 369 20.4 Motivation: One-Dimensional Heat Equation 370 20.5 Semi-Linear Problems 373 20.6 Test Case: Double-Barrier Options 375 20.6.1 PDE Formulation 376 20.6.2 Using Exponential Fitting of Barrier Options 377 20.6.3 Performing MOL with Boost C++ odeint 378 20.6.4 Computing Sensitivities 381 20.6.5 American Options 384 20.7 Summary and Conclusions 384 Chapter 21 : Free and Moving Boundary Value Problems 387 21.1 Introduction and Objectives 387 21.2 Background, Problem Statement and Formulations 388 21.3 Notation and Prerequisites 388 21.4 Some Initial Examples of Free and Moving Boundary Value Problems 389 21.4.1 Single-Phase Melting Ice 389 21.4.2 Oxygen Diffusion 390 21.4.3 American Option Pricing 391 21.4.4 Two-Phase Melting Ice 392 21.5 An Introduction to Parabolic Variational Inequalities 392 21.5.1 Formulation of Problem: Test Case 392 21.5.2 Examples of Initial Boundary Value Problems 395 21.6 An Introduction to Front-Fixing 399 21.6.1 Front-Fixing for the Heat Equation 399 21.7 Python Code Example: ADE for American Option Pricing 400 21.8 Summary and Conclusions 405 Chapter 22 : Splitting Methods, Part II 407 22.1 Introduction and Objectives 407 22.2 Background and Problem Statement: The Essence of Sequential Splitting 408 22.3 Notation and Mathematical Formulation 408 22.3.1 C0 Semigroups 408 22.3.2 Abstract Cauchy Problem 409 22.3.3 Examples 410 22.4 Mathematical Foundations of Splitting Methods 411 22.4.1 Lie (Trotter) Product Formula 411 22.4.2 Splitting Error 411 22.4.3 Component Splitting and Operator Splitting 413 22.4.4 Splitting as a Discretisation Method 413 22.5 Some Popular Splitting Methods 414 22.5.1 First-Order (Lie–Trotter) Splitting 415 22.5.2 Predictor-Corrector Splitting 415 22.5.3 Marchuk’s Two-Cycle (1-2-2-1) Method 416 22.5.4 Strang Splitting 417 22.6 Applications and Relationships to Computational Finance 417 22.7 Software Design and Implementation Guidelines 418 22.8 Experience Report: Comparing ADI and Splitting 419 22.9 Summary and Conclusions 421 Part E : Test Cases in Computational Finance Chapter 23 : Multi-Asset Options 425 23.1 Introduction and Objectives 425 23.2 Background and Goals 426 23.3 The Bivariate Normal Distribution (BVN) and its Applications 427 23.3.1 Computing BVN by Solving a Hyperbolic PDE 430 23.3.2 Analytical Solutions of Multi-Asset and Basket Options 433 23.4 PDE Models for Multi-Asset Option Problems: Requirements and Design 435 23.4.1 Domain Transformation 435 23.4.2 Numerical Boundary Conditions 435 23.5 An Overview of Finite Difference Schemes for Multi-Asset Option Problems 436 23.5.1 Common Design Principles 436 23.5.2 Detailed Design 438 23.5.3 Testing the Software 440 23.6 American Spread Options 440 23.7 Appendices 442 23.7.1 Traditional Approach to Numerical Boundary Conditions 442 23.7.2 Top-Down Design of Monte Carlo Applications 443 23.8 Summary and Conclusions 444 Chapter 24 : Asian (Average Value) Options 447 24.1 Introduction and Objectives 447 24.2 Background and Problem Statement 448 24.2.1 Challenges 449 24.3 Prototype PDE Model 450 24.3.1 Similarity Reduction 451 24.4 The Many Ways to Handle the Convective Term 452 24.4.1 Method of Lines (MOL) 452 24.4.2 Other Schemes 454 24.4.3 A Stable Monotone Upwind Scheme 455 24.5 ADE for Asian Options 455 24.6 ADI for Asian Options 456 24.6.1 Modern ADI Variations 458 24.7 Summary and Conclusions 459 Chapter 25 : Interest Rate Models 461 25.1 Introduction and Objectives 461 25.2 Main Use Cases 462 25.3 The CIR Model 462 25.3.1 Analytic Solutions 463 25.3.2 Initial Boundary Value Problem 466 25.4 Well-Posedness of the CIRPDE Model 466 25.4.1 Gronwall’s Inequalities 467 25.4.2 Energy Inequalities 468 25.5 Finite Difference Methods for the CIR Model 469 25.5.1 Numerical Boundary Conditions 470 25.6 Heston Model and the Feller Condition 471 25.7 Summary and Conclusion 475 Chapter 26 : Epilogue Models Follow-Up Chapters 1 to 25 477 26.1 Introduction and Objectives 477 26.2 Mixed Derivatives and Monotone Schemes 478 26.2.1 The Maximum Principle and Mixed Derivatives 478 26.2.2 Some Examples 480 26.2.3 Code Sample Method of Lines (MOL) for Two-Factor Hull–White Model 481 26.3 The Complex Step Method (CSM) Revisited 483 26.3.1 Black–Scholes Greeks Using CSM and the Faddeeva Function 483 26.3.2 CSM and Functions of Several Complex Variables 487 26.3.3 C++ Code for Extended CSM 488 26.3.4 CSM for Non-Linear Solvers 492 26.4 Extending the Hull–White: Possible Projects 493 26.5 Summary and Conclusions 495 Bibliography 497 Index 505

Autorenportrait

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.

Back cover copy

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, this book covers every critical aspect of the subject in an accessible format that progresses logically and gradually. It offers <ul><li>Robust mathematical foundations for one-factor 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</li> <li>Discussions of the finite difference method, including initial boundary value problems for parabolic PDEs and methods to approximate the solution of two factor PDEs</li> <li>Practical applications of the included methods, with discussions of finite difference schemes for a wide range of one-factor and two-factor problems</li></ul> 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.