The Mathematics of Derivatives: Tools for Designing Numerical Algorithms

Acknowledgments.

PART I The Models.

CHAPTER 1. Introduction to the Techniques of Derivative Modeling.

1.1 Introduction.

1.2 Models.

1.2.1 What Is a Derivative?

1.2.2 What Is a Model?

1.2.3 Two Initial Methods for Modeling Derivatives.

1.2.4 Price Processes.

1.2.5 The Archetypal Security Process: Normal Returns.

1.2.6 Book Outline.

CHAPTER 2. Preliminary Mathematical Tools.

2.1 Probability Distributions.

2.2 n-Dimensional Jacobians and n-Form Algebra.

2.3 Functional Analysis and Fourier Transforms.

2.4 Normal (Central) Limit Theorem.

2.5 Random Walks.

2.6 Correlation.

2.7 Functions of Two/More Variables: Path Integrals.

2.8 Differential Forms.

CHAPTER 3. Stochastic Calculus.

3.1 Wiener Process.

3.2 Ito’s Lemma.

3.3 Variable Changes to Get the Martingale.

3.4 Other Processes: Multivariable Correlations.

CHAPTER 4. Applications of Stochastic Calculus to Finance.

4.1 Risk Premium Derivation.

4.2 Analytic Formula for the Expected Payoff of a European Option.

CHAPTER 5. From Stochastic Processes Formalism to Differential Equation Formalism.

5.1 Backward and Forward Kolmogorov Equations.

5.2 Derivation of Black-Scholes Equation, Risk-Neutral Pricing.

5.3 Risks and Trading Strategies.

CHAPTER 6. Understanding the Black-Scholes Equation.

6.1 Black-Scholes Equation: A Type of Backward Kolmogorov Equation.

6.1.1 Forward Price.

6.2 Black-Scholes Equation: Risk-Neutral Pricing.

6.3 Black-Scholes Equation: Relation to Risk Premium Definition.

6.4 Black-Scholes Equation Applies to Currency Options: Hidden Symmetry 1.

6.5 Black-Scholes Equation in Martingale Variables: Hidden Symmetry 2.

6.6 Black-Scholes Equation with Stock as a ‘‘Derivative’’ of Option Price: Hidden Symmetry 3.

CHAPTER 7. Interest Rate Hedging.

7.1 Euler’s Relation.

7.2 Interest Rate Dependence.

7.3 Term-Structured Rates Hedging: Duration Bucketing.

7.4 Algorithm for Deciding Which Hedging Instruments to Use.

CHAPTER 8. Interest Rate Derivatives: HJM Models.

8.1 Hull-White Model Derivation.

8.1.1 Process and Pricing Equation.

8.1.2 Analytic Zero-Coupon Bond Valuation.

8.1.3 Analytic Bond Call Option.

8.1.4 Calibration.

8.2 Arbitrage-Free Pricing for Interest Rate Derivatives: HJM.

CHAPTER 9. Differential Equations, Boundary Conditions, and Solutions.

9.1 Boundary Conditions and Unique Solutions to Differential Equations.

9.2 Solving the Black-Scholes or Heat Equation Analytically.

9.2.1 Green’s Functions.

9.2.2 Separation of Variables.

9.3 Solving the Black-Scholes Equation Numerically.

9.3.1 Finite Difference Methods: Explicit/Implicit Methods, Variable Choice.

9.3.2 Gaussian Kurtosis (and Skew = 0), Faster Convergence.

9.3.3 Call/Put Options:Grid Point Shift Factor for Higher Accuracy.

9.3.4 Dividends on the Underlying Equity.

9.3.5 American Exercise.

9.3.6 2-D Models, Correlation and Variable Changes.

CHAPTER 10. Credit Spreads.

10.1 Credit Default Swaps (CDS) and the Continuous CDS Curve.

10.2 Valuing Bonds Using the Continuous CDS Curve.

10.3 Equations of Motion for Bonds and Credit Default Swaps.

CHAPTER 11. Specific Models.

11.1 Stochastic Rates and Default.

11.2 Convertible Bonds.

11.3 Index Options versus Single Name Options: Trading Equity Correlation.

11.4 Max of n Stocks: Trading Equity Correlation.

11.5 Collateralized Debt Obligations (CDOs): Trading Credit Correlation.

11.5.1 CDO Backed by Three Bonds.

11.5.2 CDO Backed by an Arbitrary Number of Bonds.

PART II Exercises and Solutions.

CHAPTER 12. Exercises.

CHAPTER 13. Solutions.

APPENDIX A: Central Limit Theorem-Plausibility Argument.

APPENDIX B: Solving for the Green’s Function of the Black-Scholes Equation.

APPENDIX C: Expanding the von Neumann Stability Mode for the Discretized Black-Scholes Equation.

APPENDIX D: Multiple Bond Survival Probabilities Given Correlated Default Probability Rates.

References.

Index.