Читать книгу Numerical Methods in Computational Finance - Daniel J. Duffy - Страница 2

1  Cover

2  Title Page

3  Copyright

4  Preface

5  Who Should Read this Book?

6  PART A: Mathematical Foundation for One-Factor Problems CHAPTER 1: Real Analysis Foundations for this Book 1.1 INTRODUCTION AND OBJECTIVES 1.2 CONTINUOUS FUNCTIONS 1.3 DIFFERENTIAL CALCULUS 1.4 PARTIAL DERIVATIVES 1.5 FUNCTIONS AND IMPLICIT FORMS 1.6 METRIC SPACES AND CAUCHY SEQUENCES 1.7 SUMMARY AND CONCLUSIONS CHAPTER 2: Ordinary Differential Equations (ODEs), Part 1 2.1 INTRODUCTION AND OBJECTIVES 2.2 BACKGROUND AND PROBLEM STATEMENT 2.3 DISCRETISATION OF INITIAL VALUE PROBLEMS: FUNDAMENTALS 2.4 SPECIAL SCHEMES 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 3.3 OTHER MODEL EXAMPLES 3.4 EXISTENCE THEOREMS FOR STOCHASTIC DIFFERENTIAL EQUATIONS (SDEs) 3.5 NUMERICAL METHODS FOR ODES 3.6 THE RICCATI EQUATION 3.7 MATRIX DIFFERENTIAL EQUATIONS 3.8 SUMMARY AND CONCLUSIONS CHAPTER 4: An Introduction to Finite Dimensional Vector Spaces 4.1 SHORT INTRODUCTION AND OBJECTIVES 4.2 WHAT IS A VECTOR SPACE? 4.3 SUBSPACES 4.4 LINEAR INDEPENDENCE AND BASES 4.5 LINEAR TRANSFORMATIONS 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 5.3 INNER PRODUCT SPACES 5.4 FROM VECTOR SPACES TO MATRICES 5.5 FUNDAMENTAL MATRIX PROPERTIES 5.6 ESSENTIAL MATRIX TYPES 5.7 THE CAYLEY TRANSFORM 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 6.3 DIRECT METHODS FOR LINEAR SYSTEMS 6.4 SOLVING TRIDIAGONAL SYSTEMS 6.5 TWO-POINT BOUNDARY VALUE PROBLEMS 6.6 ITERATIVE MATRIX SOLVERS 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 7.3 THE BLACK–SCHOLES EQUATION: TRINOMIAL METHOD 7.4 THE HEAT EQUATION AND ALTERNATING DIRECTION EXPLICIT (ADE) METHOD 7.5 ADE FOR BLACK–SCHOLES: SOME TEST RESULTS 7.6 SUMMARY AND CONCLUSIONS

7  PART B: Mathematical Foundation for Two-Factor Problems 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 8.4 CLASSIFICATION OF SECOND-ORDER EQUATIONS 8.5 EXAMPLES OF TWO-FACTOR MODELS FROM COMPUTATIONAL FINANCE 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 9.3 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 10.4 PROPERTIES OF THE LAPLACE EQUATION 10.5 SOME ELLIPTIC BOUNDARY VALUE PROBLEMS 10.6 EXTENDED MAXIMUM-MINIMUM PRINCIPLES 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 11.3 WELL-POSED PROBLEMS AND ENERGY ESTIMATES 11.4 THE FICHERA THEORY: OVERVIEW 11.5 THE FICHERA THEORY: THE CORE BUSINESS 11.6 THE FICHERA THEORY: FURTHER EXAMPLES AND APPLICATIONS 11.7 SOME USEFUL THEOREMS 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 12.5 VARIATIONS ON INITIAL BOUNDARY VALUE PROBLEM FOR THE HEAT EQUATION 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 12.9 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 13.6 PATH EVOLUTION AND MONTE CARLO OPTION PRICING 13.7 TWO-FACTOR PROBLEMS 13.8 THE ITO FORMULA 13.9 STOCHASTICS MEETS PDEs 13.10 FIRST EXIT-TIME PROBLEMS 13.11 SUMMARY AND CONCLUSIONS

8  PART C: The Foundations of the Finite Difference Method (FDM) 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 14.5 DISCRETE FOURIER TRANSFORM 14.6 THEORETICAL CONSIDERATIONS 14.7 FIRST-ORDER PARTIAL DIFFERENTIAL EQUATIONS 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 15.3 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 15.7 SEMI-DISCRETISATION FOR CONVECTION-DIFFUSION PROBLEMS 15.8 PADÉ MATRIX APPROXIMATION 15.9 TIME-DEPENDENT CONVECTION-DIFFUSION EQUATIONS 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 16.4 DIVIDED DIFFERENCES 16.5 CUBIC SPLINE INTERPOLATION 16.6 SOME COMPLEX FUNCTION THEORY 16.7 THE COMPLEX STEP METHOD (CSM) 16.8 SUMMARY AND CONCLUSIONS CHAPTER 17: Advanced Topics in Sensitivity Analysis 17.1 INTRODUCTION AND OBJECTIVES 17.2 EXAMPLES OF CSE 17.3 CSE AND BLACK–SCHOLES PDE 17.4 USING OPERATOR CALCULUS TO COMPUTE GREEKS 17.5 AN INTRODUCTION TO AUTOMATIC DIFFERENTIATION (AD) FOR THE IMPATIENT 17.6 DUAL NUMBERS 17.7 AUTOMATIC DIFFERENTIATION IN C++ 17.8 SUMMARY AND CONCLUSIONS

9  PART D: Advanced Finite Difference Schemes for Two-Factor Problems 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 18.5 OTHER RELATED SCHEMES FOR THE HEAT EQUATION 18.6 BOUNDARY CONDITIONS 18.7 TWO-DIMENSIONAL CONVECTION PDEs 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 19.5 ADE FOR CONVECTION (ADVECTION) EQUATION 19.6 CONVECTION-DIFFUSION PDEs 19.7 ATTENTION POINTS WITH ADE 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 20.3 THE EXPONENTIAL OF A MATRIX: ADVANCED TOPICS 20.4 MOTIVATION: ONE-DIMENSIONAL HEAT EQUATION 20.5 SEMI-LINEAR PROBLEMS 20.6 TEST CASE: DOUBLE-BARRIER 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 21.5 AN INTRODUCTION TO PARABOLIC VARIATIONAL INEQUALITIES 21.6 AN INTRODUCTION TO FRONT-FIXING 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 22.4 MATHEMATICAL FOUNDATIONS OF SPLITTING METHODS 22.5 SOME POPULAR SPLITTING METHODS 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

10  PART E: Test Cases in Computational Finance 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 23.4 PDE MODELS FOR MULTI-ASSET OPTION PROBLEMS: REQUIREMENTS AND DESIGN 23.5 AN OVERVIEW OF FINITE DIFFERENCE SCHEMES FOR MULTI-ASSET OPTION PROBLEMS 23.6 AMERICAN SPREAD OPTIONS 23.7 APPENDICES 23.8 SUMMARY AND CONCLUSIONS CHAPTER 24: Asian (Average Value) Options 24.1 INTRODUCTION AND OBJECTIVES 24.2 BACKGROUND AND PROBLEM STATEMENT 24.3 PROTOTYPE PDE MODEL 24.4 THE MANY WAYS TO HANDLE THE CONVECTIVE TERM 24.5 ADE FOR ASIAN OPTIONS 24.6 ADI FOR ASIAN OPTIONS 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 25.4 WELL-POSEDNESS OF THE CIRPDE MODEL 25.5 FINITE DIFFERENCE METHODS FOR THE CIR MODEL 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 26.3 THE COMPLEX STEP METHOD (CSM) REVISITED 26.4 EXTENDING THE HULL–WHITE: POSSIBLE PROJECTS 26.5 SUMMARY AND CONCLUSIONS

11  Bibliography

12  Index

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