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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

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