Оглавление
Daniel J. Duffy. Numerical Methods in Computational Finance
Table of Contents
List of Tables
List of Illustrations
Guide
Pages
Numerical Methods in Computational Finance. A Partial Differential Equation (PDE/FDM) Approach
Preface
Who Should Read this Book?
CHAPTER 1 Real Analysis Foundations for this Book
1.1 INTRODUCTION AND OBJECTIVES
1.2 CONTINUOUS FUNCTIONS
1.2.1 Formal Definition of Continuity
Definition 1.1
1.2.2 An Example
1.2.3 Uniform Continuity
1.2.4 Classes of Discontinuous Functions
1.3 DIFFERENTIAL CALCULUS
1.3.1 Taylor's Theorem
1.3.2 Big O and Little o Notation
Definition 1.2 (O-Notation)
Definition 1.3 (O-Notation)
1.4 PARTIAL DERIVATIVES
1.5 FUNCTIONS AND IMPLICIT FORMS
1.6 METRIC SPACES AND CAUCHY SEQUENCES
1.6.1 Metric Spaces
1.6.2 Cauchy Sequences
1.6.3 Lipschitz Continuous Functions
1.7 SUMMARY AND CONCLUSIONS
CHAPTER 2 Ordinary Differential Equations (ODEs), Part 1
2.1 INTRODUCTION AND OBJECTIVES
2.2 BACKGROUND AND PROBLEM STATEMENT
2.2.1 Qualitative Properties of the Solution and Maximum Principle
2.2.2 Rationale and Generalisations
2.3 DISCRETISATION OF INITIAL VALUE PROBLEMS: FUNDAMENTALS
2.3.1 Common Schemes
2.3.2 Discrete Maximum Principle
2.4 SPECIAL SCHEMES
2.4.1 Exponential Fitting
2.4.2 Scalar Non-Linear Problems and Predictor-Corrector Method
2.4.3 Extrapolation
2.5 FOUNDATIONS OF DISCRETE TIME APPROXIMATIONS
2.6 STIFF ODEs
2.7 INTERMEZZO: EXPLICIT SOLUTIONS
2.8 SUMMARY AND CONCLUSIONS
CHAPTER 3 Ordinary Differential Equations (ODEs), Part 2
3.1 INTRODUCTION AND OBJECTIVES
3.2 EXISTENCE AND UNIQUENESS RESULTS
3.2.1 An Example
3.3 OTHER MODEL EXAMPLES
3.3.1 Bernoulli ODE
3.3.2 Riccati ODE
3.3.3 Predator-Prey Models
3.3.4 Logistic Function
3.4 EXISTENCE THEOREMS FOR STOCHASTIC DIFFERENTIAL EQUATIONS (SDEs)
3.4.1 Stochastic Differential Equations (SDEs)
3.5 NUMERICAL METHODS FOR ODES
3.5.1 Code Samples in Python
3.6 THE RICCATI EQUATION
3.6.1 Finite Difference Schemes
3.7 MATRIX DIFFERENTIAL EQUATIONS
3.7.1 Transition Rate Matrices and Continuous Time Markov Chains
3.8 SUMMARY AND CONCLUSIONS
CHAPTER 4 An Introduction to Finite Dimensional Vector Spaces
4.1 SHORT INTRODUCTION AND OBJECTIVES
4.1.1 Notation
4.2 WHAT IS A VECTOR SPACE?
4.3 SUBSPACES
4.4 LINEAR INDEPENDENCE AND BASES
4.5 LINEAR TRANSFORMATIONS
4.5.1 Invariant Subspaces
4.5.2 Rank and Nullity
Eigenvalues (Characteristic Roots) and Eigenvectors (Characteristic Vectors)
4.6 SUMMARY AND CONCLUSIONS
CHAPTER 5 Guide to Matrix Theory and Numerical Linear Algebra
5.1 INTRODUCTION AND OBJECTIVES
5.2 FROM VECTOR SPACES TO MATRICES
5.2.1 Sums and Scalar Products of Linear Transformations
5.3 INNER PRODUCT SPACES
5.3.1 Orthonormal Basis
5.4 FROM VECTOR SPACES TO MATRICES
5.4.1 Some Examples
5.5 FUNDAMENTAL MATRIX PROPERTIES
5.6 ESSENTIAL MATRIX TYPES
5.6.1 Nilpotent and Related Matrices
5.6.2 Normal Matrices
5.6.3 Unitary and Orthogonal Matrices
5.6.4 Positive Definite Matrices
5.6.5 Non-Negative Matrices
5.6.6 Irreducible Matrices
5.6.7 Other Kinds of Matrices
5.7 THE CAYLEY TRANSFORM
Appendix : The Schrödinger Equation
5.8 SUMMARY AND CONCLUSIONS
CHAPTER 6 Numerical Solutions of Boundary Value Problems
6.1 INTRODUCTION AND OBJECTIVES
6.2 AN INTRODUCTION TO NUMERICAL LINEAR ALGEBRA
6.2.1 BLAS (Basic Linear Algebra Subprograms)
BLAS Level 1
BLAS Level 2
BLAS Level 3
6.3 DIRECT METHODS FOR LINEAR SYSTEMS
6.3.1 LU Decomposition
6.3.2 Cholesky Decomposition
6.4 SOLVING TRIDIAGONAL SYSTEMS. 6.4.1 Double Sweep Method
Double Sweep Method
6.4.2 Thomas Algorithm
6.4.3 Block Tridiagonal Systems
6.5 TWO-POINT BOUNDARY VALUE PROBLEMS
6.5.1 Finite Difference Approximation
6.5.2 Approximation of Boundary Conditions
6.6 ITERATIVE MATRIX SOLVERS
6.6.1 Iterative Methods
6.6.2 Jacobi Method
6.6.3 Gauss–Seidel Method
6.6.4 Successive Over-Relaxation (SOR)
6.6.5 Other Methods
6.6.5.1 Conjugate Gradient Method
6.6.5.2 The Linear Complementarity Problem (LCP) and Projected SOR (PSOR)
6.7 EXAMPLE: ITERATIVE SOLVERS FOR ELLIPTIC PDEs
6.8 SUMMARY AND CONCLUSIONS
CHAPTER 7 Black–Scholes Finite Differences for the Impatient
7.1 INTRODUCTION AND OBJECTIVES
7.2 THE BLACK–SCHOLES EQUATION: FULLY IMPLICIT AND CRANK–NICOLSON METHODS
7.2.1 Fully Implicit Method
7.2.2 Crank–Nicolson Method
7.2.3 Final Remarks
7.3 THE BLACK–SCHOLES EQUATION: TRINOMIAL METHOD
7.3.1 Comparison with Other Methods
7.4 THE HEAT EQUATION AND ALTERNATING DIRECTION EXPLICIT (ADE) METHOD
7.4.1 Background and Motivation
7.5 ADE FOR BLACK–SCHOLES: SOME TEST RESULTS
EXAMPLE 7.1
EXAMPLE 7.2
EXAMPLE 7.3
EXAMPLE 7.4
EXAMPLE 7.5
EXAMPLE 7.6
7.6 SUMMARY AND CONCLUSIONS
CHAPTER 8 Classifying and Transforming Partial Differential Equations
8.1 INTRODUCTION AND OBJECTIVES
8.2 BACKGROUND AND PROBLEM STATEMENT
8.3 INTRODUCTION TO ELLIPTIC EQUATIONS
8.3.1 What is an Elliptic Operator?
8.3.2 Total and Principal Symbols
8.3.3 The Adjoint Equation
8.3.4 Self-Adjoint Operators and Equations
8.3.5 Numerical Approximation of PDEs in Adjoint Form
8.3.6 Elliptic Equations with Non-Negative Characteristic Form
8.4 CLASSIFICATION OF SECOND-ORDER EQUATIONS
8.4.1 Characteristics
8.4.2 Model Example
8.4.3 Test your Knowledge
8.5 EXAMPLES OF TWO-FACTOR MODELS FROM COMPUTATIONAL FINANCE
8.5.1 Multi-Asset Options
8.5.2 Stochastic Dividend PDE
8.6 SUMMARY AND CONCLUSIONS
CHAPTER 9 Transforming Partial Differential Equations to a Bounded Domain
9.1 INTRODUCTION AND OBJECTIVES
9.2 THE DOMAIN IN WHICH A PDE IS DEFINED: PREAMBLE
9.2.1 Background and Specific Mappings
9.2.2 Initial Examples
9.3 OTHER EXAMPLES
9.4 HOTSPOTS
9.5 WHAT HAPPENED TO DOMAIN TRUNCATION?
9.6 ANOTHER WAY TO REMOVE MIXED DERIVATIVE TERMS
9.7 SUMMARY AND CONCLUSIONS
CHAPTER 10 Boundary Value Problems for Elliptic and Parabolic Partial Differential Equations
10.1 INTRODUCTION AND OBJECTIVES
10.2 NOTATION AND PREREQUISITES
10.3 THE LAPLACE EQUATION
10.3.1 Harmonic Functions and the Cauchy–Riemann Equations
10.4 PROPERTIES OF THE LAPLACE EQUATION
10.4.1 Maximum-Minimum Principle for Laplace's Equation
10.5 SOME ELLIPTIC BOUNDARY VALUE PROBLEMS
10.5.1 Some Motivating Examples
10.6 EXTENDED MAXIMUM-MINIMUM PRINCIPLES
10.6.1 An Example
10.7 SUMMARY AND CONCLUSIONS
CHAPTER 11 Fichera Theory, Energy Inequalities and Integral Relations
11.1 INTRODUCTION AND OBJECTIVES
11.2 BACKGROUND AND PROBLEM STATEMENT
11.2.1 The ‘Big Bang’: Cauchy–Euler Equation
11.3 WELL-POSED PROBLEMS AND ENERGY ESTIMATES
11.3.1 Time to Reflect: What Have We Achieved and What's Next?
11.4 THE FICHERA THEORY: OVERVIEW
11.5 THE FICHERA THEORY: THE CORE BUSINESS
11.6 THE FICHERA THEORY: FURTHER EXAMPLES AND APPLICATIONS
11.6.1 Cox–Ingersoll–Ross (CIR)
11.6.2 Heston Model Fundamenals
11.6.2.1 Standard European Call Option
11.6.2.2 European Put Options
11.6.2.3 Other Kinds of Boundary Conditions
11.6.3 Heston Model by Fichera Theory
11.6.4 First-Order Hyperbolic PDE in One and Two Space Variables
11.7 SOME USEFUL THEOREMS
11.7.1 Divergence (Gauss–Ostrogradsky) Theorem
11.7.2 Green's Theorem/Formula
11.7.3 Green's First and Second Identities
11.8 SUMMARY AND CONCLUSIONS
CHAPTER 12 An Introduction to Time-Dependent Partial Differential Equations
12.1 INTRODUCTION AND OBJECTIVES
12.2 NOTATION AND PREREQUISITES
12.3 PREAMBLE: SEPARATION OF VARIABLES FOR THE HEAT EQUATION
12.4 WELL-POSED PROBLEMS
12.4.1 Examples of an ill-posed Problem
12.4.2 The Importance of Proving that Problems Are Well-Posed
12.5 VARIATIONS ON INITIAL BOUNDARY VALUE PROBLEM FOR THE HEAT EQUATION
12.5.1 Smoothness and Compatibility Conditions
12.6 MAXIMUM-MINIMUM PRINCIPLES FOR PARABOLIC PDES
12.7 PARABOLIC EQUATIONS WITH TIME-DEPENDENT BOUNDARIES
12.8 UNIQUENESS THEOREMS FOR BOUNDARY VALUE PROBLEMS IN TWO DIMENSIONS
12.8.1 Laplace Equation
12.8.2 Heat Equation
12.9 SUMMARY AND CONCLUSIONS
CHAPTER 13 Stochastics Representations of PDEs and Applications
13.1 INTRODUCTION AND OBJECTIVES
13.2 BACKGROUND, REQUIREMENTS AND PROBLEM STATEMENT
13.3 AN OVERVIEW OF STOCHASTIC DIFFERENTIAL EQUATIONS (SDEs)
13.4 AN INTRODUCTION TO ONE-DIMENSIONAL RANDOM PROCESSES
13.5 AN INTRODUCTION TO THE NUMERICAL APPROXIMATION OF SDEs
13.5.1 Euler–Maruyama Method
13.5.2 Milstein Method
13.5.3 Predictor-Corrector Method
13.5.4 Drift-Adjusted Predictor-Corrector Method
13.6 PATH EVOLUTION AND MONTE CARLO OPTION PRICING
13.6.1 Monte Carlo Option Pricing
13.6.2 Some C++ Code
13.7 TWO-FACTOR PROBLEMS
13.7.1 Spread Options with Stochastic Volatility
13.7.2 Heston Stochastic Volatility Model
13.8 THE ITO FORMULA
13.9 STOCHASTICS MEETS PDEs
13.9.1 A Statistics Refresher
13.9.2 The Feynman–Kac Formula
13.9.3 Kolmogorov Equations
13.9.4 Kolmogorov Forward (Fokker–Planck (FPE)) Equation
13.9.5 Multi-Dimensional Problems and Boundary Conditions
13.9.6 Kolmogorov Backward Equation (KBE)
13.10 FIRST EXIT-TIME PROBLEMS
13.11 SUMMARY AND CONCLUSIONS
CHAPTER 14 Mathematical and Numerical Foundations of the Finite Difference Method, Part I
14.1 INTRODUCTION AND OBJECTIVES
14.2 NOTATION AND PREREQUISITES
14.3 WHAT IS THE FINITE DIFFERENCE METHOD, REALLY?
14.4 FOURIER ANALYSIS OF LINEAR PDES
14.4.1 Fourier Transform for Advection Equation
14.4.2 Fourier Transform for Diffusion Equation
14.5 DISCRETE FOURIER TRANSFORM
14.5.1 Finite and Infinite Dimensional Sequences and Their Norms
14.5.2 Discrete Fourier Transform (DFT)
14.5.3 Discrete von Neumann Stability Criterion
14.5.4 Some More Examples
14.6 THEORETICAL CONSIDERATIONS
14.6.1 Consistency
14.6.2 Stability
14.6.3 Convergence
14.7 FIRST-ORDER PARTIAL DIFFERENTIAL EQUATIONS
14.7.1 Why First-Order Equations are Different: Essential Difficulties
14.7.2 A Simple Explicit Scheme
14.7.3 Some Common Schemes for Initial Value Problems
14.7.4 Some Other Schemes
14.7.5 General Linear Problems
14.8 SUMMARY AND CONCLUSIONS
CHAPTER 15 Mathematical and Numerical Foundations of the Finite Difference Method, Part II
15.1 INTRODUCTION AND OBJECTIVES
15.2 A SHORT HISTORY OF NUMERICAL METHODS FOR CDR EQUATIONS
15.2.1 Temporal and Spatial Stability
15.2.2 Motivating Exponential Fitting Methods
15.2.3 Eliminating Temporal and Spatial Stability Problems
15.3 EXPONENTIAL FITTING AND TIME-DEPENDENT CONVECTION-DIFFUSION
15.4 STABILITY AND CONVERGENCE ANALYSIS
15.5 SPECIAL LIMITING CASES
15.6 STABILITY FOR INITIAL BOUNDARY VALUE PROBLEMS
15.6.1 Gerschgorin's Circle Theorem
15.7 SEMI-DISCRETISATION FOR CONVECTION-DIFFUSION PROBLEMS
15.7.1 Essentially Positive Matrices
15.7.2 Fully Discrete Schemes
15.8 PADÉ MATRIX APPROXIMATION
15.8.1 Padé Matrix Approximations
15.9 TIME-DEPENDENT CONVECTION-DIFFUSION EQUATIONS
15.9.1 Fully Discrete Schemes
15.10 SUMMARY AND CONCLUSIONS
CHAPTER 16 Sensitivity Analysis, Option Greeks and Parameter Optimisation, Part I
16.1 INTRODUCTION AND OBJECTIVES
16.2 HELICOPTER VIEW OF SENSITIVITY ANALYSIS
16.3 BLACK–SCHOLES–MERTON GREEKS
16.3.1 Higher-Order and Mixed Greeks
16.4 DIVIDED DIFFERENCES
16.4.1 Approximation to First and Second Derivatives
16.4.2 Black–Scholes Numeric Greeks and Divided Differences
16.5 CUBIC SPLINE INTERPOLATION
16.5.1 Caveat: Cubic Splines with Sparse Input Data
16.5.2 Cubic Splines for Option Greeks
16.5.3 Boundary Conditions
16.6 SOME COMPLEX FUNCTION THEORY
16.6.1 Curves and Regions
16.6.2 Taylor's Theorem and Series
16.6.3 Laurent's Theorem and Series
16.6.4 Cauchy–Goursat Theorem
16.6.5 Cauchy's Integral Formula
16.6.6 Cauchy's Residue Theorem
16.6.7 Gauss's Mean Value Theorem
16.7 THE COMPLEX STEP METHOD (CSM)
16.7.1 Caveats
16.8 SUMMARY AND CONCLUSIONS
CHAPTER 17 Advanced Topics in Sensitivity Analysis
17.1 INTRODUCTION AND OBJECTIVES
17.2 EXAMPLES OF CSE
17.2.1 Simple Initial Value Problem
17.2.2 Population Dynamics
17.2.3 Comparing CSE and Complex Step Method (CSM)
17.2.3.1 CSM
17.2.3.2 CSE
17.3 CSE AND BLACK–SCHOLES PDE
17.3.1 Black–Scholes Greeks: Algorithms and Design
17.3.2 Some Specific Black–Scholes Greeks
17.4 USING OPERATOR CALCULUS TO COMPUTE GREEKS
17.5 AN INTRODUCTION TO AUTOMATIC DIFFERENTIATION (AD) FOR THE IMPATIENT
17.5.1 What Is Automatic Differentiation: The Details
17.6 DUAL NUMBERS
17.7 AUTOMATIC DIFFERENTIATION IN C++
17.8 SUMMARY AND CONCLUSIONS
CHAPTER 18 Splitting Methods, Part I
18.1 INTRODUCTION AND OBJECTIVES
18.2 BACKGROUND AND HISTORY
18.3 NOTATION, PREREQUISITES AND MODEL PROBLEMS
18.4 MOTIVATION: TWO-DIMENSIONAL HEAT EQUATION. 18.4.1 Alternating Direction Implicit (ADI) Method
18.4.2 Soviet (Operator) Splitting
18.4.3 Mixed Derivative and Yanenko Scheme
18.5 OTHER RELATED SCHEMES FOR THE HEAT EQUATION
18.5.1 D'Yakonov Method
18.5.2 Approximate Factorisation of Operators
18.5.3 Predictor-Corrector Methods
18.5.4 Partial Integro Differential Equations (PIDEs)
18.6 BOUNDARY CONDITIONS
18.7 TWO-DIMENSIONAL CONVECTION PDEs
Initial Boundary Value Problems
18.8 THREE-DIMENSIONAL PROBLEMS
18.9 THE HOPSCOTCH METHOD
18.10 SOFTWARE DESIGN AND IMPLEMENTATION GUIDELINES
18.11 THE FUTURE: CONVECTION-DIFFUSION EQUATIONS
18.12 SUMMARY AND CONCLUSIONS
CHAPTER 19 The Alternating Direction Explicit (ADE) Method
19.1 INTRODUCTION AND OBJECTIVES
19.2 BACKGROUND AND PROBLEM STATEMENT
19.3 GLOBAL OVERVIEW AND APPLICABILITY OF ADE
19.4 MOTIVATING EXAMPLES: ONE-DIMENSIONAL AND TWO-DIMENSIONAL DIFFUSION EQUATIONS
19.4.1 Barakat and Clark (B&C) Method
19.4.2 Saul'yev Method
19.4.3 Larkin Method
19.4.4 Two-Dimensional Diffusion Problems
19.5 ADE FOR CONVECTION (ADVECTION) EQUATION
19.6 CONVECTION-DIFFUSION PDEs
19.6.1 Example: Black–Scholes PDE
19.6.2 Boundary Conditions
19.6.3 Spatial Amplification Errors
19.7 ATTENTION POINTS WITH ADE
The Consequences of Conditional Consistency
Call Pay-Off Behaviour at the Far Field
19.7.1 General Formulation of the ADE Method
19.8 SUMMARY AND CONCLUSIONS
CHAPTER 20 The Method of Lines (MOL), Splitting and the Matrix Exponential
20.1 INTRODUCTION AND OBJECTIVES
20.2 NOTATION AND PREREQUISITES: THE EXPONENTIAL FUNCTION
20.2.1 Initial Results
20.2.2 The Exponential of a Matrix
20.3 THE EXPONENTIAL OF A MATRIX: ADVANCED TOPICS
20.3.1 Fundamental Theorem for Linear Systems
Proof of Theorem 20.1
20.3.3 An Example
20.4 MOTIVATION: ONE-DIMENSIONAL HEAT EQUATION
20.5 SEMI-LINEAR PROBLEMS
20.6 TEST CASE: DOUBLE-BARRIER OPTIONS
20.6.1 PDE Formulation
20.6.2 Using Exponential Fitting of Barrier Options
20.6.3 Performing MOL with Boost C++ odeint
20.6.4 Computing Sensitivities
20.6.5 American Options
20.7 SUMMARY AND CONCLUSIONS
CHAPTER 21 Free and Moving Boundary Value Problems
21.1 INTRODUCTION AND OBJECTIVES
21.2 BACKGROUND, PROBLEM STATEMENT AND FORMULATIONS
21.3 NOTATION AND PREREQUISITES
21.4 SOME INITIAL EXAMPLES OF FREE AND MOVING BOUNDARY VALUE PROBLEMS
21.4.1 Single-Phase Melting Ice
21.4.2 Oxygen Diffusion
21.4.3 American Option Pricing
21.4.4 Two-Phase Melting Ice
21.5 AN INTRODUCTION TO PARABOLIC VARIATIONAL INEQUALITIES
21.5.1 Formulation of Problem: Test Case
21.5.2 Examples of Initial Boundary Value Problems
21.6 AN INTRODUCTION TO FRONT-FIXING
21.6.1 Front-Fixing for the Heat Equation
21.7 PYTHON CODE EXAMPLE: ADE FOR AMERICAN OPTION PRICING
21.8 SUMMARY AND CONCLUSIONS
CHAPTER 22 Splitting Methods, Part II
22.1 INTRODUCTION AND OBJECTIVES
22.2 BACKGROUND AND PROBLEM STATEMENT: THE ESSENCE OF SEQUENTIAL SPLITTING
22.3 NOTATION AND MATHEMATICAL FORMULATION
22.3.1 Semigroups
22.3.2 Abstract Cauchy Problem
22.3.3 Examples
22.4 MATHEMATICAL FOUNDATIONS OF SPLITTING METHODS
22.4.1 Lie (Trotter) Product Formula
22.4.2 Splitting Error
22.4.3 Component Splitting and Operator Splitting
22.4.4 Splitting as a Discretisation Method
22.5 SOME POPULAR SPLITTING METHODS
22.5.1 First-Order (Lie–Trotter) Splitting
22.5.2 Predictor-Corrector Splitting
22.5.3 Marchuk's Two-Cycle (1-2-2-1) Method
22.5.4 Strang Splitting
22.6 APPLICATIONS AND RELATIONSHIPS TO COMPUTATIONAL FINANCE
22.7 SOFTWARE DESIGN AND IMPLEMENTATION GUIDELINES
22.8 EXPERIENCE REPORT: COMPARING ADI AND SPLITTING
22.9 SUMMARY AND CONCLUSIONS
CHAPTER 23 Multi-Asset Options
23.1 INTRODUCTION AND OBJECTIVES
23.2 BACKGROUND AND GOALS
23.3 THE BIVARIATE NORMAL DISTRIBUTION (BVN) AND ITS APPLICATIONS
23.3.1 Computing BVN by Solving a Hyperbolic PDE. Computing Integrals using Partial Differential Equations (PDEs)
The Finite Difference Method for the Goursat PDE
23.3.2 Analytical Solutions of Multi-Asset and Basket Options
23.4 PDE MODELS FOR MULTI-ASSET OPTION PROBLEMS: REQUIREMENTS AND DESIGN
23.4.1 Domain Transformation
23.4.2 Numerical Boundary Conditions
23.5 AN OVERVIEW OF FINITE DIFFERENCE SCHEMES FOR MULTI-ASSET OPTION PROBLEMS
23.5.1 Common Design Principles
23.5.2 Detailed Design
23.5.3 Testing the Software
23.6 AMERICAN SPREAD OPTIONS
23.7 APPENDICES
23.7.1 Traditional Approach to Numerical Boundary Conditions
23.7.2 Top-Down Design of Monte Carlo Applications
23.8 SUMMARY AND CONCLUSIONS
CHAPTER 24 Asian (Average Value) Options
24.1 INTRODUCTION AND OBJECTIVES
24.2 BACKGROUND AND PROBLEM STATEMENT
24.2.1 Challenges
24.3 PROTOTYPE PDE MODEL
24.3.1 Similarity Reduction
24.4 THE MANY WAYS TO HANDLE THE CONVECTIVE TERM
24.4.1 Method of Lines (MOL)
24.4.2 Other Schemes
24.4.3 A Stable Monotone Upwind Scheme
24.5 ADE FOR ASIAN OPTIONS
24.6 ADI FOR ASIAN OPTIONS
24.6.1 Modern ADI Variations
24.7 SUMMARY AND CONCLUSIONS
CHAPTER 25 Interest Rate Models
25.1 INTRODUCTION AND OBJECTIVES
25.2 MAIN USE CASES
25.3 THE CIR MODEL
25.3.1 Analytic Solutions
25.3.2 Initial Boundary Value Problem
25.4 WELL-POSEDNESS OF THE CIRPDE MODEL
25.4.1 Gronwall's Inequalities
25.4.2 Energy Inequalities
25.5 FINITE DIFFERENCE METHODS FOR THE CIR MODEL
25.5.1 Numerical Boundary Conditions
25.6 HESTON MODEL AND THE FELLER CONDITION
25.7 SUMMARY AND CONCLUSION
CHAPTER 26 Epilogue Models Follow-Up Chapters 1 to 25
26.1 INTRODUCTION AND OBJECTIVES
26.2 MIXED DERIVATIVES AND MONOTONE SCHEMES
26.2.1 The Maximum Principle and Mixed Derivatives
26.2.2 Some Examples
26.2.3 Code Sample Method of Lines (MOL) for Two-Factor Hull–White Model
26.3 THE COMPLEX STEP METHOD (CSM) REVISITED
26.3.1 Black–Scholes Greeks Using CSM and the Faddeeva Function
26.3.2 CSM and Functions of Several Complex Variables
26.3.3 C++ Code for Extended CSM
26.3.4 CSM for Non-Linear Solvers
26.4 EXTENDING THE HULL–WHITE: POSSIBLE PROJECTS
26.5 SUMMARY AND CONCLUSIONS
Bibliography
Index
WILEY END USER LICENSE AGREEMENT
