Contents
Preface xi
Acknowledgments xiii
PART I The Models 1
CHAPTER 1
Introduction to the Techniques of Derivative Modeling 3
1.1 Introduction 3
1.2 Models 3
1.2.1 What Is a Derivative? 3
1.2.2 What Is a Model? 5
1.2.3 Two Initial Methods for Modeling Derivatives 6
1.2.4 Price Processes 7
1.2.5 The Archetypal Security Process: Normal Returns 8
1.2.6 Book Outline 10
CHAPTER 2
Preliminary Mathematical Tools 11
2.1 Probability Distributions 11
2.2 n-Dimensional Jacobians and n-Form Algebra 14
2.3 Functional Analysis and Fourier Transforms 16
2.4 Normal (Central) Limit Theorem 18
2.5 Random Walks 20
2.6 Correlation 22
2.7 Functions of Two/More Variables: Path Integrals 24
2.8 Differential Forms 26
CHAPTER 3
Stochastic Calculus 27
3.1 Wiener Process 27
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viii Contents
3.2 Ito’s Lemma 30
3.3 Variable Changes to Get the Martingale 33
3.4 Other Processes: Multivariable Correlations 35
CHAPTER 4
Applications of Stochastic Calculus to Finance 37
4.1 Risk Premium Derivation 37
4.2 Analytic Formula for the Expected Payoff of a European
Option 39
CHAPTER 5
From Stochastic Processes Formalism to Differential Equation
Formalism 43
5.1 Backward and Forward Kolmogorov Equations 43
5.2 Derivation of Black-Scholes Equation, Risk-Neutral Pricing 46
5.3 Risks and Trading Strategies 48
CHAPTER 6
Understanding the Black-Scholes Equation 51
6.1 Black-Scholes Equation: A Type of Backward Kolmogorov
Equation 51
6.1.1 Forward Price 53
6.2 Black-Scholes Equation: Risk-Neutral Pricing 53
6.3 Black-Scholes Equation: Relation to Risk Premium Definition 54
6.4 Black-Scholes Equation Applies to Currency Options: Hidden
Symmetry 1 55
6.5 Black-Scholes Equation in Martingale Variables: Hidden
Symmetry 2 57
6.6 Black-Scholes Equation with Stock as a ‘‘Derivative’’ of
Option Price: Hidden Symmetry 3 59
CHAPTER 7
Interest Rate Hedging 62
7.1 Euler’s Relation 62
7.2 Interest Rate Dependence 63
7.3 Term-Structured Rates Hedging: Duration Bucketing 65
7.4 Algorithm for Deciding Which Hedging Instruments to Use 67
Contents ix
CHAPTER 8
Interest Rate Derivatives: HJM Models 68
8.1 Hull-White Model Derivation 68
8.1.1 Process and Pricing Equation 68
8.1.2 Analytic Zero-Coupon Bond Valuation 73
8.1.3 Analytic Bond Call Option 74
8.1.4 Calibration 75
8.2 Arbitrage-Free Pricing for Interest Rate Derivatives: HJM 76
CHAPTER 9
Differential Equations, Boundary Conditions, and Solutions 79
9.1 Boundary Conditions and Unique Solutions to Differential
Equations 79
9.2 Solving the Black-Scholes or Heat Equation Analytically 81
9.2.1 Green’s Functions 81
9.2.2 Separation of Variables 83
9.3 Solving the Black-Scholes Equation Numerically 84
9.3.1 Finite Difference Methods: Explicit/Implicit Methods,
Variable Choice 84
9.3.2 Gaussian Kurtosis (and Skew = 0), Faster
Convergence 90
9.3.3 Call/Put Options:Grid Point Shift Factor for Higher
Accuracy 94
9.3.4 Dividends on the Underlying Equity 96
9.3.5 American Exercise 97
9.3.6 2-D Models, Correlation and Variable Changes 100
CHAPTER 10
Credit Spreads 104
10.1 Credit Default Swaps (CDS) and the Continuous CDS
Curve 104
10.2 Valuing Bonds Using the Continuous CDS Curve 108
10.3 Equations of Motion for Bonds and Credit Default Swaps 109
CHAPTER 11
Specific Models 112
11.1 Stochastic Rates and Default 112
11.2 Convertible Bonds 114
11.3 Index Options versus SingleName Options: Trading Equity
Correlation 119
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11.4 Max of n Stocks: Trading Equity Correlation 122
11.5 Collateralized Debt Obligations (CDOs): Trading Credit
Correlation 124
11.5.1 CDO Backed by Three Bonds 126
11.5.2 CDO Backed by an Arbitrary Number of Bonds 133
PART II Exercises and Solutions 137
CHAPTER 12
Exercises 139
CHAPTER 13
Solutions 145
APPENDIX A: Central Limit Theorem-Plausibility Argument 163
APPENDIX B: Solving for the Green’s Function
of the Black-Scholes Equation 167
APPENDIX C: Expanding the von Neumann Stability
Mode for the Discretized Black-Scholes Equation 169
APPENDIX D: Multiple Bond Survival Probabilities
Given Correlated Default Probability Rates 172
References 178
Index 179