Interest rate modeling: theory and practice

Preface

Acknowledgments

Author

Chapter 1. The Basics of Stochastic Calculus

1.1. Brownian Motion

1.1.1. Simple Random Walks

1.1.2. Brownian Motion

1.1.3. Adaptive and Non-Adaptive Functions

1.2. Stochastic Integrals

1.2.1. Evaluation of Stochastic Integrals

1.3. Stochastic Differentials and Ito's Lemma

1.4. Multi-Factor Extensions

1.4.1. Multi-Factor Ito's Process

1.4.2. Ito's Lemma

1.4.3. Correlated Brownian Motions

1.4.4. The Multi-Factor Lognormal Model

1.5. Martingales

Chapter 2. The Martingale Representation Theorem

2.1. Changing Measures With Binomial Models

2.1.1. A Motivating Example

2.1.2. Binomial Trees and Path Probabilities

2.2. Change of Measures Under Brownian Filtration

2.2.1. The Radon-Nikodym Derivative of a Brownian Path

2.2.2. The CMG Theorem

2.3. The Martingale Representation Theorem

2.4. A Complete Market with Two Securities

2.5. Replicating And Pricing of Contingent Claims

2.6. Multi-Factor Extensions

2.7. A Complete Market With Multiple Securities

2.7.1. Existence of a Martingale Measure

2.7.2. Pricing Contingent Claims

2.8. The Black-Scholes Formula

2.9. Notes

Chapter 3. Interest Rates and Bonds

3.1. Interest Rates And Fixed-Income Instruments

3.1.1. Short Rate and Money Market Accounts

3.1.2. Term Rates and Certificates of Deposit

3.1.3. Bonds and Bond Markets

3.1.4. Quotation and Interest Accrual

3.2. Yields

3.2.1. Yield to Maturity

3.2.2. Par Bonds, Par Yields, and the Par Yield Curve

3.2.3. Yield Curves for U.S. Treasuries

3.3. Zero-Coupon Bonds And Zero-Coupon Yields

3.3.1. Zero-Coupon Bonds

3.3.2. Bootstrapping the Zero-Coupon Yields

3.3.2.1. Future Value and Present Value

3.4. Forward Rates And Forward-Rate Agreements

3.5. Yield-Based Bond Risk Management

3.5.1. Duration and Convexity

3.5.2. Portfolio Risk Management

Chapter 4. The Heath-Jarrow-Morton Model

4.1. Lognormal Model: The Starting Point

4.2. The HJM Model

4.3. Special Cases of the HJM Model

4.3.1. The Ho-Lee Model

4.3.2. The Hull-White (or Extended Vasicek) Model

4.4. Estimating The HJM Model From Yield Data

4.4.1. From a Yield Curve to a Forward-Rate Curve

4.4.2. Principal Component Analysis

4.5. A Case Study With A Two-Factor Model

4.6. Monte Carlo Implementations

4.7. Forward Prices

4.8. Forward Measure

4.9. Black's Formula For Call And Put Options

4.9.1. Equity Options under the Hull-White Model

4.9.2. Options on Coupon Bonds

4.10. Numeraires and Changes of Measure

4.11. Notes

Chapter 5. Short-Rate Models and Lattice Implementation

5.1. From Short-Rate Models To Forward-Rate Models

5.2. General Markovian Models

5.2.1. One-Factor Models

5.2.2. Monte Carlo Simulations for Options Pricing

5.3. Binomial Trees of Interest Rates

5.3.1. A Binomial Tree for the Ho-Lee Model

5.3.2. Arrow-Debreu Prices

5.3.3. A Calibrated Tree for the Ho-Lee Model

5.4. A General Tree-Building Procedure

5.4.1. A Truncated Tree for the Hull-White Model

5.4.2. Trinomial Trees with Adaptive Time Steps

5.4.3. The Black-Karasinski Model

Chapter 6. The Libor Market Model

6.1. Libor Market Instruments

6.1.1. Libor Rates

6.1.2. Forward-Rate Agreements

6.1.3. Repurchasing Agreement

6.1.4. Eurodollar Futures

6.1.5. Floating-Rate Notes

6.1.6. Swaps

6.1.7. Caps

6.1.8. Swaptions

6.1.9. Bermudan Swaptions

6.1.10. Libor Exotics

6.2. The Libor Market Model

6.3. Pricing of Caps And Floors

6.4. Pricing of Swaptions

6.5. Specifications of the Libor Market Model

6.6. Monte Carlo Simulation Method

6.6.1. The Log-Euler Scheme

6.6.2. Calculation of the Greeks

6.6.3. Early Exercise

Chapter 7. Calibration of Libor Market Model

7.1. Implied Cap And Caplet Volatilities

7.2. Calibrating the Libor Market Model to Caps

7.3. Calibration to Caps, Swaptions, And Input Correlations

7.4. Calibration Methodologies

7.4.1. Rank-Reduction Algorithm

7.4.2. The Eigenvalue Problem for Calibrating to Input Prices

7.5. Sensitivity With Respect to the Input Prices

7.6. Notes

Chapter 8. Volatility and Correlation Adjustments

8.1. Adjustment Due to Correlations

8.1.1. Futures Price versus Forward Price

8.1.2. Convexity Adjustment for Libor Rates

8.1.3. Convexity Adjustment under the Ho-Lee Model

8.1.4. An Example of Arbitrage

8.2. Adjustment Due To Convexity

8.2.1. Payment in Arrears versus Payment in Advance

8.2.2. Geometric Explanation for Convexity Adjustment

8.2.3. General Theory of Convexity Adjustment

8.2.4. Convexity Adjustment for CMS and CMT Swaps

8.3. Timing Adjustment

8.4. Quanto Derivatives

8.5. Notes

Chapter 9. Affine Term Structure Models

9.1. An Exposition with One-Factor Models

9.2. Analytical Solution Of Riccarti Equations

9.3. Pricing Options on Coupon Bonds

9.4. Distributional Properties of Square-Root Processes

9.5. Multi-Factor Models

9.5.1. Admissible ATSMs

9.5.2. Three-Factor ATSMs

9.6. Swaption Pricing Under ATSMs

9.7. Notes

References

Index