Interest rate models : theory and practice /

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Bibliographic Details
Author / Creator:	Brigo, Damiano, 1966-
Imprint:	Berlin ; New York : Springer, c2001.
Description:	xxxv, 518 p. : ill. ; 24 cm.
Language:	English
Series:	Springer finance
Subject:	Interest rates -- Mathematical models. Derivative securities -- Prices -- Mathematical models. Derivative securities -- Prices -- Mathematical models. Interest rates -- Mathematical models.
Format:	Print Book
URL for this record:	http://pi.lib.uchicago.edu/1001/cat/bib/4507979

Hidden Bibliographic Details
Other authors / contributors:	Mercurio, Fabio, 1966-
ISBN:	3540417729 (alk. paper)
Notes:	Includes bibliographical references (p. [501]-508) and index.

Table of Contents:

Preface
Motivation
Aims, Readership and Book Structure
Final Word and Acknowledgments
Description of Contents by Chapter
Abbreviations and Notation
Part I. Models: Theory And Implementation
1. Definitions and Notation
1.1. The Bank Account and the Short Rate
1.2. Zero-Coupon Bonds and Spot Interest Rates
1.3. Fundamental Interest-Rate Curves
1.4. Forward Rates
1.5. Interest-Rate Swaps and Forward Swap Rates
1.6. Interest-Rate Caps/Floors and Swaptions
2. No-Arbitrage Pricing and Numeraire Change
2.1. No-Arbitrage in Continuous Time
2.2. The Change-of-Numeraire Technique
2.3. A Change-of-Numeraire Toolkit
2.4. The Choice of a Convenient Numeraire
2.5. The Forward Measure
2.6. The Fundamental Pricing Formulas
2.6.1. The Pricing of Caps and Floors
2.7. Pricing Claims with Deferred Payoffs
2.8. Pricing Claims with Multiple Payoffs
2.9. Foreign Markets and Numeraire Change
3. One-factor short-rate models
3.1. Introduction and Guided Tour
3.2. Classical Time-Homogeneous Short-Rate Models
3.2.1. The Vasicek Model
3.2.2. The Dothan Model
3.2.3. The Cox, Ingersoll and Ross (CIR) Model
3.2.4. Affine Term-Structure Models
3.2.5. The Exponential-Vasicek (EV) Model
3.3. The Hull-White Extended Vasicek Model
3.3.1. The Short-Rate Dynamics
3.3.2. Bond and Option Pricing
3.3.3. The Construction of a Trinomial Tree
3.4. Possible Extensions of the CIR Model
3.5. The Black-Karasinski Model
3.5.1. The Short-Rate Dynamics
3.5.2. The Construction of a Trinomial Tree
3.6. Volatility Structures in One-Factor Short-Rate Models
3.7. Humped-Volatility Short-Rate Models
3.8. A General Deterministic-Shift Extension
3.8.1. The Basic Assumptions
3.8.2. Fitting the Initial Term Structure of Interest Rates
3.8.3. Explicit Formulas for European Options
3.8.4. The Vasicek Case
3.9. The CIR++ Model
3.9.1. The Construction of a Trinomial Tree
3.9.2. The Positivity of Rates and Fitting Quality
3.10. Deterministic-Shift Extension of Lognormal Models
3.11. Some Further Remarks on Derivatives Pricing
3.11.1. Pricing European Options on a Coupon-Bearing Bond
3.11.2. The Monte Carlo Simulation
3.11.3. Pricing Early-Exercise Derivatives with a Tree
3.11.4. A Fundamental Case of Early Exercise: Bermudan-Style Swaptions
3.12. Implied Cap Volatility Curves
3.12.1. The Black and Karasinski Model
3.12.2. The CIR+-I- Model
3.12.3. The Extended Exponential-Vasicek Model
3.13. Implied Swaption Volatility Surfaces
3.13.1. The Black and Karasinski Model
3.13.2. The Extended Exponential-Vasicek Model
3.14. An Example of Calibration to Real-Market Data
4. Two-Factor Short-Rate Models
4.1. Introduction and Motivation
4.2. The Two-Additive-Factor Gaussian Model G2++
4.2.1. The Short-Rate Dynamics
4.2.2. The Pricing of a Zero-Coupon Bond
4.2.3. Volatility and Correlation Structures in Two-Factor Models
4.2.4. The Pricing of a European Option on a Zero-Coupon Bond
4.2.5. The Analogy with the Hull-White Two-Factor Model
4.2.6. The Construction of an Approximating Binomial Tree
4.2.7. Examples of Calibration to Real-Market Data
4.3. The Two-Additive-Factor Extended CIR/LS Model CIR2++
4.3.1. The Basic Two-Factor CIR2 Model
4.3.2. Relationship with the Longstaff and Schwartz Model (LS)
4.3.3. Forward-Measure Dynamics and Option Pricing for CIR2
4.3.4. The CIR2++ Model and Option Pricing
5. The Heath-Jarrow-Morton (HJM) Framework
5.1. The HJM Forward-Rate Dynamics
5.2. Markovianity of the Short-Rate Process
5.3. The Ritchken and Sankarasubramanian Framework
5.4. The Mercurio and Moraleda Model
6. The LIBOR and Swap Market Models (LFM and LSM)
6.1. Introduction
6.2. Market Models: a Guided Tour
6.3. The Lognormal Forward-LIBOR Model (LFM)
6.3.1. Some Specifications of the Instantaneous Volatility of Forward Rates
6.3.2. Forward-Rate Dynamics under Different Numeraires
6.4. Calibration of the LFM to Caps and Floors Prices
6.4.1. Piecewise-Constant Instantaneous-Volatility Structures
6.4.2. Parametric Volatility Structures
6.4.3. Cap Quotes in the Market
6.5. The Term Structure of Volatility
6.5.1. Piecewise-Constant Instantaneous Volatility Structures
6.5.2. Parametric Volatility Structures
6.6. Instantaneous Correlation and Terminal Correlation
6.7. Swaptions and the Lognormal Forward-Swap Model (LSM)
6.7.1. Swaptions Hedging
6.7.2. Cash-Settled Swaptions
6.8. Incompatibility between the LFM and the LSM
6.9. The Structure of Instantaneous Correlations
6.10. Monte Carlo Pricing of Swaptions with the LFM
6.11. Rank-One Analytical Swaption Prices
6.12. Rank-r Analytical Swaption Prices
6.13. A Simpler LFM Formula for Swaptions Volatilities
6.14. A Formula for Terminal Correlations of Forward Rates
6.15. Calibration to Swaptions Prices
6.16. Connecting Caplet and 5 x 1-Swaption Volatilities
6.17. Forward and Spot Rates over Non-Standard Periods
6.17.1. Drift Interpolation
6.17.2. The Bridging Technique
6.18 Including the Caplet Smile in the LFM.
6.18.1. A Mini-tour on the Smile Problem
6.18.2. Modeling the Smile
6.18.3. The Shifted-Lognormal Case
6.18.4. The Constant Elasticity of Variance (CEV) Model
6.18.5. A Mixture-of-Lognormals Model
6.18.6. Shifting the Lognormal-Mixture Dynamics
7. Cases of Calibration of the LIBOR Market Model
7.1. The Inputs
7.2. Joint Calibration with Piecewise-Constant Volatilities as in TABLE 5
7.2.1. Instantaneous Correlations: Narrowing the Angles
7.2.2. Instantaneous Correlations: Fixing the Angles to Typ-ical Values
7.2.3. Instantaneous Correlations: Fixing the Angles to Atyp-ical Values
7.2.4. Instantaneous Correlations: Collapsing to One Factor
7.3. Joint Calibration with Parameterized Volatilities as in For-mulation 7
7.3.1. Formulation 7: Narrowing the Angles
7.3.2. Formulation 7: Calibrating only to Swaptions
7.4. Exact Swaptions Calibration with Volatilities as TABLE 1
7.4.1. Some Numerical Results
7.5. Conclusions: Where Now?
8. Monte Carlo Tests for LFM Analytical Approximations
8.1. The Specification of Rates
8.2. The "Testing Plan" for Volatilities
8.3. Test Results for Volatilities
8.3.1. Case (1): Constant Instantaneous Volatilities
8.3.2. Case (2): Volatilities as Functions of Time to Maturity
8.3.3. Case (3): Humped and Maturity-Adjusted Instanta-neous Volatilities Depending only on Time to Matu-rity, Typical Rank-Two Correlations
8.4. The "Testing Plan" for Terminal Correlations
8.5. Test Results for Terminal Correlations
8.5.1. Case (i): Humped and Maturity-Adjusted Instanta-neous Volatilities Depending only on Time to Matu-rity, Typical Rank-Two Correlations
8.5.2. Case (ü): Constant Instantaneous Volatilities, Typical Rank-Two Correlations
8.5.3. Case (üi): Humped and Maturity-Adjusted Instanta-neous Volatilities Depending only on Time to Matu-rity, Some Negative Rank-Two Correlations
8.5.4. Case (iv): Constant Instantaneous Volatilities, Some Negative Rank-Two Correlations
8.5.5. Case (v): Constant Instantaneous Volatilities, Perfect Correlations, Upwardly Shifted $$'s
8.6. Test Results: Stylized Conclusions
9. Other Interest-Rate Models
9.1. Brennan and Schwartz's Model
9.2. Balduzzi, Das, Foresi and Sundaram's Model
9.3. Flesaker and Hughston's Model
9.4. Rogers's Potential Approach
9.5. Markov Functional Models
Part II. Pricing Derivatives In Practice
10. Pricing Derivatives on a Single Interest-Rate Curve
10.1. In-Advance Swaps
10.2. In-Advance Caps
10.2.1. A First Analytical Formula (LFM)
10.2.2. A Second Analytical Formula (G2++)
10.3. Autocaps
10.4. Caps with Deferred Caplets
10.4.1. A First Analytical Formula (LFM)
10.4.2. A Second Analytical Formula (G2++)
10.5. Ratchets (One-Way Floaters)
10.6. Constant-Maturity Swaps (CMS)
10.6.1. CMS with the LFM
10.6.2. CMS with the G2++ Model
10.7. The Convexity Adjustment and Applications to CMS
10.7.1. Natural and Unnatural Time Lags
10.7.2. The Convexity-Adjustment Technique
10.7.3. Deducing a Simple Lognormal Dynamics from the Adjustment
10.7.4. Application to CMS
10.7.5. Forward Rate Resetting Unnaturally and Average-Rate Swaps
10.8. Captions and Floortions
10.9. Zero-Coupon Swaptions
10.10. Eurodollar Futures
10.10.1. The Shifted Two-Factor Vasicek G2++ Model
10.10.2. Eurodollar Futures with the LFM
10.11. LFM Pricing with "In-Between" Spot Rates
10.11.1. Accrual Swaps
10.11.2. Trigger Swaps
10.12. LFM Pricing with Early Exercise and Possible Path Depen-dence
10.13. LFM: Pricing Bermudan Swaptions
10.13.1. Longstaff and Schwartz's Approach
10.13.2. Carr and Yang's Approach
10.13.3. Andersen's Approach
11. Pricing Derivatives on Two Interest-Rate Curves
11.1. The Attractive Features of G2++ for Multi-Curve Payoffs
11.1.1. The Model
11.1.2. Interaction Between Models of the Two Curves "1" and "2"
11.1.3. The Two-Models Dynamics under a Unique Conve-nient Forward Measure
11.2. Quanto Constant-Maturity Swaps
11.2.1. Quanto CMS: The Contract
11.2.2. Quanto CMS: The G2++ Model
11.2.3. Quanto CMS: Quanto Adjustment
11.3. Differential Swaps
11.3.1. The Contract
11.3.2. Differential Swaps with the G2++ Model
11.3.3. A Market-Like Formula
11.4. Market Formulas for Basic Quanto Derivatives
11.4.1. The Pricing of Quanto Caplets/Floorlets
11.4.2. The Pricing of Quanto Caps/Floors
11.4.3. The Pricing of Differential Swaps
11.4.4. The Pricing of Quanto Swaptions
12. Pricing Equity Derivatives under Stochastic Rates
12.1. The Short Rate and Asset-Price Dynamics
12.1.1. The Dynamics under the Forward Measure
12.2. The Pricing of a European Option on the Given Asset
12.3. A More General Model
12.3.1. The Construction of an Approximating Tree for r
12.3.2. The Approximating Tree for 5
12.3.3. The Two-Dimensional Tree
Part III. Appendices
A. A Crash Introduction to Stochastic Differential Equations
A.l. From Deterministic to Stochastic Differential Equations
A.2. Ito's Formula476
A.3. Discretizing SDEs for Monte Carlo: Euler and Milstein Schemes
A.4. Examples
A.5. Two Important Theorems
B. A Useful Calculation
C. Approximating Diffusions with Trees
D. Talking to the Traders
References
Index

Interest rate models : theory and practice /

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