Mathematics of Derivatives: Tools for Designing Numerical Algorithms Book

Preface     xi
Acknowledgments     xiii
The Models     1
Introduction to the Techniques of Derivative Modeling     3
Introduction     3
Models     3
What Is a Derivative?     3
What Is a Model?     5
Two Initial Methods for Modeling Derivatives     6
Price Processes     7
The Archetypal Security Process: Normal Returns     8
Book Outline     10
Preliminary Mathematical Tools     11
Probability Distributions     11
n-Dimensional Jacobians and n-Form Algebra     14
Functional Analysis and Fourier Transforms     16
Normal (Central) Limit Theorem     18
Random Walks     20
Correlation     22
Functions of Two/More Variables: Path Integrals     24
Differential Forms     26
Stochastic Calculus     27
Wiener Process     27
Ito's Lemma     30
Variable Changes to Get the Martingale     33
Other Processes: Multivariable Correlations     35
Applications of Stochastic Calculus to Finance     37
Risk Premium Derivation     37
Analytic Formula for the Expected Payoff of a European Option     39
From Stochastic Processes Formalism to Differential Equation Formalism     43
Backward and Forward Kolmogorov Equations     43
Derivation of Black-Scholes Equation, Risk-Neutral Pricing     46
Risks and Trading Strategies     48
Understanding the Black-Scholes Equation     51
Black-Scholes Equation: A Type of Backward Kolmogorov Equation     51
Forward Price     53
Black-Scholes Equation: Risk-Neutral Pricing     53
Black-Scholes Equation: Relation to Risk Premium Definition     54
Black-Scholes Equation Applies to Currency Options: Hidden Symmetry 1     55
Black-Scholes Equation in Martingale Variables: Hidden Symmetry 2     57
Black-Scholes Equation with Stock as a "Derivative" of Option Price: Hidden Symmetry 3     59
Interest Rate Hedging     62
Euler's Relation     62
Interest Rate Dependence     63
Term-Structured Rates Hedging: Duration Bucketing     65
Algorithm for Deciding Which Hedging Instruments to Use     67
Interest Rate Derivatives: HJM Models     68
Hull-White Model Derivation     68
Process and Pricing Equation     68
Analytic Zero-Coupon Bond Valuation     73
Analytic Bond Call Option     74
Calibration     75
Arbitrage-Free Pricing for Interest Rate Derivatives: HJM     76
Differential Equations, Boundary Conditions, and Solutions     79
Boundary Conditions and Unique Solutions to Differential Equations     79
Solving the Black-Scholes or Heat Equation Analytically     81
Green's Functions     81
Separation of Variables     83
Solving the Black-Scholes Equation Numerically     84
Finite Difference Methods: Explicit/Implicit Methods, Variable Choice     84
Gaussian Kurtosis (and Skew = 0), Faster Convergence     90
Call/Put Options: Grid Point Shift Factor for Higher Accuracy     94
Dividends on the Underlying Equity     96
American Exercise     97
2-D Models, Correlation and Variable Changes     100
Credit Spreads     104
Credit Default Swaps (CDS) and the Continuous CDS Curve     104
Valuing Bonds Using the Continuous CDS Curve     108
Equations of Motion for Bonds and Credit Default Swaps     109
Specific Models     112
Stochastic Rates and Default     112
Convertible Bonds      114
Index Options versus Single Name Options: Trading Equity Correlation     119
Max of n Stocks: Trading Equity Correlation     122
Collateralized Debt Obligations (CDOs): Trading Credit Correlation     124
CDO Backed by Three Bonds     126
CDO Backed by an Arbitrary Number of Bonds     133
Exercises and Solutions     137
Exercises     139
Solutions     145
Central Limit Theorem-Plausibility Argument     163
Solving for the Green's Function of the Black-Scholes Equation     167
Expanding the von Neumann Stability Mode for the Discretized Black-Scholes Equation     169
Multiple Bond Survival Probabilities Given Correlated Default Probability Rates     172
References     178
Index     179