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Interest Rate Models - Theory and Practice With Smile, Inflation and Credit

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ISBN-10: 3540221492

ISBN-13: 9783540221494

Edition: 2nd 2006 (Revised)

Authors: Damiano Brigo, Fabio Mercurio

List price: $159.99
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Written for researchers, and graduate and undergraduate students in the disciplines of financial modelling, mathematical finance, stochastic calculus, probability, and statistics, this 2nd ed has been comprehensively revised and updated.
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Book details

List price: $159.99
Edition: 2nd
Copyright year: 2006
Publisher: Springer Berlin / Heidelberg
Publication date: 8/2/2007
Binding: Hardcover
Pages: 982
Size: 6.10" wide x 9.25" long x 0.65" tall
Weight: 3.696
Language: English

Aims, Readership and Book Structure
Final Word and Acknowledgments
Description of Contents by Chapter
Abbreviations and Notation
Basic Definitions and No Arbitrage
Definitions and Notation
The Bank Account and the Short Rate
Zero-Coupon Bonds and Spot Interest Rates
Fundamental Interest-Rate Curves
Forward Rates
Interest-Rate Swaps and Forward Swap Rates
Interest-Rate Caps/Floors and Swaptions
No-Arbitrage Pricing and Numeraire Change
No-Arbitrage in Continuous Time
The Change-of-Numeraire Technique
A Change of Numeraire Toolkit (Brigo & Mercurio 2001c)
A helpful notation: "DC"
The Choice of a Convenient Numeraire
The Forward Measure
The Fundamental Pricing Formulas
The Pricing of Caps and Floors
Pricing Claims with Deferred Payoffs
Pricing Claims with Multiple Payoffs
Foreign Markets and Numeraire Change
From Short Rate Models to HJM
One-factor short-rate models
Introduction and Guided Tour
Classical Time-Homogeneous Short-Rate Models
The Vasicek Model
The Dothan Model
The Cox, Ingersoll and Ross (CIR) Model
Affine Term-Structure Models
The Exponential-Vasicek (EV) Model
The Hull-White Extended Vasicek Model
The Short-Rate Dynamics
Bond and Option Pricing
The Construction of a Trinomial Tree
Possible Extensions of the CIR Model
The Black-Karasinski Model
The Short-Rate Dynamics
The Construction of a Trinomial Tree
Volatility Structures in One-Factor Short-Rate Models
Humped-Volatility Short-Rate Models
A General Deterministic-Shift Extension
The Basic Assumptions
Fitting the Initial Term Structure of Interest Rates
Explicit Formulas for European Options
The Vasicek Case
The CIR++ Model
The Construction of a Trinomial Tree
Early Exercise Pricing via Dynamic Programming
The Positivity of Rates and Fitting Quality
Monte Carlo Simulation
Jump Diffusion CIR and CIR++ models (JCIR, JCIR++)
Deterministic-Shift Extension of Lognormal Models
Some Further Remarks on Derivatives Pricing
Pricing European Options on a Coupon-Bearing Bond
The Monte Carlo Simulation
Pricing Early-Exercise Derivatives with a Tree
A Fundamental Case of Early Exercise: Bermudan-Style Swaptions
Implied Cap Volatility Curves
The Black and Karasinski Model
The CIR++ Model
The Extended Exponential-Vasicek Model
Implied Swaption Volatility Surfaces
The Black and Karasinski Model
The Extended Exponential-Vasicek Model
An Example of Calibration to Real-Market Data
Two-Factor Short-Rate Models
Introduction and Motivation
The Two-Additive-Factor Gaussian Model G2++
The Short-Rate Dynamics
The Pricing of a Zero-Coupon Bond
Volatility and Correlation Structures in Two-Factor Models
The Pricing of a European Option on a Zero-Coupon Bond
The Analogy with the Hull-White Two-Factor Model
The Construction of an Approximating Binomial Tree
Examples of Calibration to Real-Market Data
The Two-Additive-Factor Extended CIR/LS Model CIR2++
The Basic Two-Factor CIR2 Model
Relationship with the Longstaff and Schwartz Model (LS)
Forward-Measure Dynamics and Option Pricing for CIR2
The CIR2++ Model and Option Pricing
The Heath-Jarrow-Morton (HJM) Framework
The HJM Forward-Rate Dynamics
Markovianity of the Short-Rate Process
The Ritchken and Sankarasubramanian Framework
The Mercurio and Moraleda Model
MArket Models
The LIBOR and Swap Market Models (LFM and LSM)
Market Models: a Guided Tour
The Lognormal Forward-LIBOR Model (LFM)
Some Specifications of the Instantaneous Volatility of Forward Rates
Forward-Rate Dynamics under Different Numeraires
Calibration of the LFM to Caps and Floors Prices
Piecewise-Constant Instantaneous-Volatility Structures
Parametric Volatility Structures
Cap Quotes in the Market
The Term Structure of Volatility
Piecewise-Constant Instantaneous Volatility Structures
Parametric Volatility Structures
Instantaneous Correlation and Terminal Correlation
Swaptions and the Lognormal Forward-Swap Model (LSM)
Swaptions Hedging
Cash-Settled Swaptions
Incompatibility between the LFM and the LSM
The Structure of Instantaneous Correlations
Some convenient full rank parameterizations
Reduced-rank formulations: Rebonato's angles and eigen-values zeroing
Reducing the angles
Monte Carlo Pricing of Swaptions with the LFM
Monte Carlo Standard Error
Monte Carlo Variance Reduction: Control Variate Estimator
Rank-One Analytical Swaption Prices
Rank-r Analytical Swaption Prices
A Simpler LFM Formula for Swaptions Volatilities
A Formula for Terminal Correlations of Forward Rates
Calibration to Swaptions Prices
Instantaneous Correlations: Inputs (Historical Estimation) or Outputs (Fitting Parameters)?
The exogenous correlation matrix
Historical Estimation
Pivot matrices
Connecting Caplet and S x 1-Swaption Volatilities
Forward and Spot Rates over Non-Standard Periods
Drift Interpolation
The Bridging Technique
Cases of Calibration of the LIBOR Market Model
Inputs for the First Cases
Joint Calibration with Piecewise-Constant Volatilities as in Table 5
Joint Calibration with Parameterized Volatilities as in Formulation 7
Exact Swaptions "Cascade" Calibration with Volatilities as in Table 1
Some Numerical Results
A Pause for Thought
First summary
An automatic fast analytical calibration of LFM to swaptions. Motivations and plan
Further Numerical Studies on the Cascade Calibration Algorithm
Cascade Calibration under Various Correlations and Ranks
Cascade Calibration Diagnostics: Terminal Correlation and Evolution of Volatilities
The interpolation for the swaption matrix and its impact on the CCA
Empirically efficient Cascade Calibration
CCA with Endogenous Interpolation and Based Only on Pure Market Data
Financial Diagnostics of the RCCAEI test results
Endogenous Cascade Interpolation for missing swaptions volatilities quotes
A first partial check on the calibrated [sigma] parameters stability
Reliability: Monte Carlo tests
Cascade Calibration and the cap market
Cascade Calibration: Conclusions
Monte Carlo Tests for LFM Analytical Approximations
First Part. Tests Based on the Kullback Leibler Information (KLI)
Distance between distributions: The Kullback Leibler information
Distance of the LFM swap rate from the lognormal family of distributions
Monte Carlo tests for measuring KLI
Conclusions on the KLI-based approach
Second Part: Classical Tests
The "Testing Plan" for Volatilities
Test Results for Volatilities
Case (1): Constant Instantaneous Volatilities
Case (2): Volatilities as Functions of Time to Maturity
Case (3): Humped and Maturity-Adjusted Instantaneous Volatilities Depending only on Time to Maturity
The "Testing Plan" for Terminal Correlations
Test Results for Terminal Correlations
Case (i): Humped and Maturity-Adjusted Instantaneous Volatilities Depending only on Time to Maturity, Typical Rank-Two Correlations
Case (ii): Constant Instantaneous Volatilities, Typical Rank-Two Correlations
Case (iii): Humped and Maturity-Adjusted Instantaneous Volatilities Depending only on Time to Maturity, Some Negative Rank-Two Correlations
Case (iv): Constant Instantaneous Volatilities, Some Negative Rank-Two Correlations
Case (v): Constant Instantaneous Volatilities, Perfect Correlations, Upwardly Shifted [Phi]'s
Test Results: Stylized Conclusions
The Volatility Smile
Including the Smile in the LFM
A Mini-tour on the Smile Problem
Modeling the Smile
Local-Volatility Models
The Shifted-Lognormal Model
The Constant Elasticity of Variance Model
A Class of Analytically-Tractable Models
A Lognormal-Mixture (LM) Model
Forward Rates Dynamics under Different Measures
Decorrelation Between Underlying and Volatility
Shifting the LM Dynamics
A Lognormal-Mixture with Different Means (LMDM)
The Case of Hyperbolic-Sine Processes
Testing the Above Mixture-Models on Market Data
A Second General Class
A Particular Case: a Mixture of GBM's
An Extension of the GBM Mixture Model Allowing for Implied Volatility Skews
A General Dynamics a la Dupire (1994)
Stochastic-Volatility Models
The Andersen and Brotherton-Ratcliffe (2001) Model
The Wu and Zhang (2002) Model
The Piterbarg (2003) Model
The Hagan, Kumar, Lesniewski and Woodward (2002) Model
The Joshi and Rebonato (2003) Model
Uncertain-Parameter Models
The Shifted-Lognormal Model with Uncertain Parameters (SLMUP)
Relationship with the Lognormal-Mixture LVM
Calibration to Caplets
Swaption Pricing
Monte-Carlo Swaption Pricing
Calibration to Swaptions
Calibration to Market Data
Testing the Approximation for Swaptions Prices
Further Model Implications
Joint Calibration to Caps and Swaptions
Examples of Market Payoffs
Pricing Derivatives on a Single Interest-Rate Curve
In-Arrears Swaps
In-Arrears Caps
A First Analytical Formula (LFM)
A Second Analytical Formula (G2++)
Caps with Deferred Caplets
A First Analytical Formula (LFM)
A Second Analytical Formula (G2++)
Ratchet Caps and Floors
Analytical Approximation for Ratchet Caps with the LFM
Ratchets (One-Way Floaters)
Constant-Maturity Swaps (CMS)
CMS with the LFM
CMS with the G2++ Model
The Convexity Adjustment and Applications to CMS
Natural and Unnatural Time Lags
The Convexity-Adjustment Technique
Deducing a Simple Lognormal Dynamics from the Adjustment
Application to CMS
Forward Rate Resetting Unnaturally and Average-Rate Swaps
Average Rate Caps
Captions and Floortions
Zero-Coupon Swaptions
Eurodollar Futures
The Shifted Two-Factor Vasicek G2++ Model
Eurodollar Futures with the LFM
LFM Pricing with "In-Between" Spot Rates
Accrual Swaps
Trigger Swaps
LFM Pricing with Early Exercise and Possible Path Dependence
LFM: Pricing Bermudan Swaptions
Least Squared Monte Carlo Approach
Carr and Yang's Approach
Andersen's Approach
Numerical Example
New Generation of Contracts
Target Redemption Notes
CMS Spread Options
Pricing Derivatives on Two Interest-Rate Curves
The Attractive Features of G2++ for Multi-Curve Payoffs
The Model
Interaction Between Models of the Two Curves "1" and "2"
The Two-Models Dynamics under a Unique Convenient Forward Measure
Quanto Constant-Maturity Swaps
Quanto CMS: The Contract
Quanto CMS: The G2++ Model
Quanto CMS: Quanto Adjustment
Differential Swaps
The Contract
Differential Swaps with the G2++ Model
A Market-Like Formula
Market Formulas for Basic Quanto Derivatives
The Pricing of Quanto Caplets/Floorlets
The Pricing of Quanto Caps/Floors
The Pricing of Differential Swaps
The Pricing of Quanto Swaptions
Pricing of Options on two Currency LIBOR Rates
Spread Options
Options on the Product
Trigger Swaps
Dealing with Multiple Dates
Pricing of Inflation-Indexed Derivatives
The Foreign-Currency Analogy
Definitions and Notation
The JY Model
Inflation-Indexed Swaps
Pricing of a ZCIIS
Pricing of a YYIIS
Pricing of a YYIIS with the JY Model
Pricing of a YYIIS with a First Market Model
Pricing of a YYIIS with a Second Market Model
Inflation-Indexed Caplets/Floorlets
Pricing with the JY Model
Pricing with the Second Market Model
Inflation-Indexed Caps
Calibration to market data
Introducing Stochastic Volatility
Modeling Forward CPI's with Stochastic Volatility
Pricing Formulae
Exact Solution for the Uncorrelated Case
Approximated Dynamics for Non-zero Correlations
Example of Calibration
Pricing Hybrids with an Inflation Component
A Simple Hybrid Payoff
Introduction and Pricing under Counterparty Risk
Introduction and Guided Tour
Reduced form (Intensity) models
CDS Options Market Models
Firm Value (or Structural) Models
Further Models
The Multi-name picture: FtD, CDO and Copula Functions
First to Default (FtD) Basket
Collateralized Debt Obligation (CDO) Tranches
Where can we introduce dependence?
Copula Functions
Dynamic Loss models
What data are available in the market?
Defaultable (corporate) zero coupon bonds
Defaultable (corporate) coupon bonds
Credit Default Swaps and Defaultable Floaters
CDS payoffs: Different Formulations
CDS pricing formulas
Changing filtration: F[subscript t] without default VS complete G[subscript t]
CDS forward rates: The first definition
Market quotes, model independent implied survival probabilities and implied hazard functions
A simpler formula for calibrating intensity to a single CDS
Different Definitions of CDS Forward Rates and Analogies with the LIBOR and SWAP rates
Defaultable Floater and CDS
CDS Options and Callable Defaultable Floaters
Constant Maturity CDS
Some interesting Financial features of CMCDS
Interest-Rate Payoffs with Counterparty Risk
General Valuation of Counterparty Risk
Counterparty Risk in single Interest Rate Swaps (IRS)
Intensity Models
Introduction and Chapter Description
Poisson processes
Time homogeneous Poisson processes
Time inhomogeneous Poisson Processes
Cox Processes
CDS Calibration and Implied Hazard Rates/Intensities
Inducing dependence between Interest-rates and the default event
The Filtration Switching Formula: Pricing under partial information
Default Simulation in reduced form models
Standard error
Variance Reduction with Control Variate
Stochastic Intensity: The SSRD model
A two-factor shifted square-root diffusion model for intensity and interest rates (Brigo and Alfonsi (2003))
Calibrating the joint stochastic model to CDS: Separability
Discretization schemes for simulating ([lambda], r)
Study of the convergence of the discretization schemes for simulating CIR processes (Alfonsi (2005))
Gaussian dependence mapping: A tractable approximated SSRD
Numerical Tests: Gaussian Mapping and Correlation Impact
The impact of correlation on a few "test payoffs"
A pricing example: A Cancellable Structure
CDS Options and Jamshidian's Decomposition
Bermudan CDS Options
Stochastic diffusion intensity is not enough: Adding jumps. The JCIR(++) Model
The jump-diffusion CIR model (JCIR)
Bond (or Survival Probability) Formula
Exact calibration of CDS: The JCIR++ model
Jamshidian's Decomposition
Attaining high levels of CDS implied volatility
JCIR(++) models as a multi-name possibility
Conclusions and further research
CDS Options Market Models
CDS Options and Callable Defaultable Floaters
Once-callable defaultable floaters
A market formula for CDS options and callable defaultable floaters
Market formulas for CDS Options
Market Formula for callable DFRN
Examples of Implied Volatilities from the Market
Towards a Completely Specified Market Model
First Choice. One-period and two-period rates
Second Choice: Co-terminal and one-period CDS rates market model
Third choice. Approximation: One-period CDS rates dynamics
Hints at Smile Modeling
Constant Maturity Credit Default Swaps (CMCDS) with the market model
CDS and Constant Maturity CDS
Proof of the main result
A few numerical examples
Other Interest-Rate Models
Brennan and Schwartz's Model
Balduzzi, Das, Foresi and Sundaram's Model
Flesaker and Hughston's Model
Rogers's Potential Approach
Markov Functional Models
Pricing Equity Derivatives under Stochastic Rates
The Short Rate and Asset-Price Dynamics
The Dynamics under the Forward Measure
The Pricing of a European Option on the Given Asset
A More General Model
The Construction of an Approximating Tree for r
The Approximating Tree for S
The Two-Dimensional Tree
A Crash Intro to Stochastic Differential Equations and Poisson Processes
From Deterministic to Stochastic Differential Equations
Ito's Formula
Discretizing SDEs for Monte Carlo: Euler and Milstein Schemes
Two Important Theorems
A Crash Intro to Poisson Processes
Time inhomogeneous Poisson Processes
Doubly Stochastic Poisson Processes (or Cox Processes)
Compound Poisson processes
Jump-diffusion Processes
A Useful Calculation
A Second Useful Calculation
Approximating Diffusions with Trees
Trivia and Frequently Asked Questions
Talking to the Traders