Interest Rate Models - Theory and Practice

Transcription

1 Springer Finance Interest Rate Models - Theory and Practice With Smile, Inflation and Credit Bearbeitet von Damiano Brigo, Fabio Mercurio Neuausgabe Buch. LVI, 982 S. Hardcover ISBN Format (B x L): 15,5 x 23,5 cm Gewicht: 920 g Weitere Fachgebiete > Mathematik > Stochastik > Wahrscheinlichkeitsrechnung Zu Leseprobe schnell und portofrei erhältlich bei Die Online-Fachbuchhandlung beck-shop.de ist spezialisiert auf Fachbücher, insbesondere Recht, Steuern und Wirtschaft. Im Sortiment finden Sie alle Medien (Bücher, Zeitschriften, CDs, ebooks, etc.) aller Verlage. Ergänzt wird das Programm durch Services wie Neuerscheinungsdienst oder Zusammenstellungen von Büchern zu Sonderpreisen. Der Shop führt mehr als 8 Millionen Produkte.

2 Contents Preface... VII Motivation VII Aims, Readership and Book Structure.... XII Final Word and Acknowledgments... XIV Description of Contents by Chapter.... XIX Abbreviations and Notation... XXXV Part I. BASIC DEFINITIONS AND NO ARBITRAGE 1. 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

3 XLIV Table of Contents Part II. FROM SHORT RATE MODELS TO HJM 3. 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

4 Table of Contents XLV 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 G 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 CIR The Basic Two-Factor CIR2 Model Relationship with the Longstaff and Schwartz Model (LS) Forward-Measure Dynamics and Option Pricing for CIR 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 Part III. MARKET MODELS 6. The LIBOR and Swap Market Models (LFM and LSM) Introduction 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 228

5 XLVI Table of Contents 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 eigenvalues 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 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 Joint Calibration with Parameterized Volatilities as in Formulation Exact Swaptions Cascade Calibration with Volatilities as in TABLE 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

6 Table of Contents XLVII 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 σ 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

7 XLVIII Table of Contents Case (iv): Constant Instantaneous Volatilities, Some Negative Rank-Two Correlations Case (v): Constant Instantaneous Volatilities, Perfect Correlations, Upwardly Shifted Φ s Test Results: Stylized Conclusions Part IV. THE VOLATILITY SMILE 9. 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 ALognormal-Mixture with Different Means (LMDM) The Case of Hyperbolic-Sine Processes Testing the Above Mixture-Models on Market Data ASecond General Class AParticular Case: amixture of GBM s An Extension of the GBM Mixture Model Allowing for Implied Volatility Skews AGeneral Dynamics àla 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

8 Table of Contents XLIX 12.6 Calibration to Market Data Testing the Approximation for Swaptions Prices Further Model Implications Joint Calibration to Caps and Swaptions Part V. EXAMPLES OF MARKET PAYOFFS 13. Pricing Derivatives on a Single Interest-Rate Curve In-Arrears Swaps In-Arrears Caps A First Analytical Formula (LFM) A Second Analytical Formula (G2++) Autocaps 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...591

9 L Table of Contents 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 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 Part VI. INFLATION 15. 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

10 Table of Contents LI 16.5 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 Part VII. CREDIT 21. 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 t without default VS complete G t CDS forward rates: The first definition

11 LII Table of Contents 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 (λ, 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...830

12 Table of Contents LIII 22.8 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 Simulation 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 Part VIII. APPENDICES A. Other Interest-Rate Models A.1 Brennan and Schwartz s Model A.2 Balduzzi, Das, Foresi and Sundaram s Model A.3 Flesaker and Hughston s Model A.4 Rogers s Potential Approach A.5 Markov Functional Models

13 LIV Table of Contents B. Pricing Equity Derivatives under Stochastic Rates B.1 The Short Rate and Asset-Price Dynamics B.1.1 The Dynamics under the Forward Measure B.2 The Pricing of a European Option on the Given Asset B.3 A More General Model B.3.1 The Construction of an Approximating Tree for r B.3.2 The Approximating Tree for S B.3.3 The Two-Dimensional Tree C. A Crash Intro to Stochastic Differential Equations and Poisson Processes C.1 From Deterministic to Stochastic Differential Equations C.2 Ito s Formula C.3 Discretizing SDEs for Monte Carlo: Euler and Milstein Schemes906 C.4 Examples C.5 Two Important Theorems C.6 A Crash Intro to Poisson Processes C.6.1 Time inhomogeneous Poisson Processes C.6.2 Doubly Stochastic Poisson Processes (or Cox Processes)916 C.6.3 Compound Poisson processes C.6.4 Jump-diffusion Processes D. A Useful Calculation E. A Second Useful Calculation F. Approximating Diffusions with Trees G. Trivia and Frequently Asked Questions H. Talking to the Traders References Index