Credit Risk: Modeling, Valuation and Hedging

Transcription

1 Tomasz R. Bielecki Marek Rutkowski Credit Risk: Modeling, Valuation and Hedging Springer

2 Table of Contents Preface V Part I. Structural Approach 1. Introduction to Credit Risk Corporate Bonds Recovery Rules Safety Covenants Credit Spreads Credit Ratings Corporate Coupon Bonds Fixed and Floating Rate Notes Bank Loans and Sovereign Debt Cross Default Default Correlations Vulnerable Claims Vulnerable Claims with Unilateral Default Risk Vulnerable Claims with Bilateral Default Risk Defaultable Interest Rate Contracts Credit Derivatives Default Swaps and Options Total Rate of Return Swaps Credit Linked Notes Asset Swaps First-to-Default Contracts Credit Spread Swaps and Options Quantitative Models of Credit Risk Structural Models Reduced-Form Models Credit Risk Management Liquidity Risk Econometric Studies 30

3 XIV Table of Contents 2. Corporate Debt Defaultable Claims Risk-Neutral Valuation Formula Self-Financing Trading Strategies Martingale Measures PDE Approach PDE for the Value Function Corporate Zero-Coupon Bonds Corporate Coupon Bond Merton's Approach to Corporate Debt Merton's Model with Deterministic Interest Rates Distance-to-Default Extensions of Merton's Approach Models with Stochastic Interest Rates Discontinuous Value Process Buffet's Approach First-Passage-Time Models Properties of First Passage Times Probability Law of the First Passage Time Joint Probability Law of Y and r Black and Cox Model Corporate Zero-Coupon Bond Corporate Coupon Bond Corporate Consol Bond Optimal Capital Structure Black and Cox Approach Leland's Approach Leland and Toft Approach Further Developments Models with Stochastic Interest Rates Kim, Ramaswamy and Sundaresan Approach Longstaff and Schwartz Approach Cathcart and El-Jahel Approach Briys and de Varenne Approach Saä-Requejo and Santa-Clara Approach Further Developments Convertible Bonds Jump-Diffusion Models Incomplete Accounting Data Dependent Defaults: Structural Approach Default Correlations: J.P. Morgan's Approach Default Correlations: Zhou's Approach 117

4 Table of Contents XV Part II. Hazard Processes 4. Hazard Function of a Random Time Conditional Expectations w.r.t. Natural Filtrations Martingales Associated with a Continuous Hazard Function Martingale Representation Theorem Change of a Probability Measure Martingale Characterization of the Hazard Function Compensator of a Random Time Hazard Process of a Random Time Hazard Process F Conditional Expectations Semimartingale Representation of the Stopped Process Martingales Associated with the Hazard Process F Stochastic Intensity of a Random Time Martingale Representation Theorems General Case Case of a Brownian Filtration Change of a Probability Measure Martingale Hazard Process Martingale Hazard Process A Martingale Invariance Property Evaluation of A: Special Case Evaluation of A: General Case Uniqueness of a Martingale Hazard Process A Relationships Between Hazard Processes F and A Martingale Representation Theorem Case of the Martingale Invariance Property Valuation of Defaultable Claims Case of a Stopping Time Random Time with a Given Hazard Process Poisson Process and Conditional Poisson Process Case of Several Random Times Minimum of Several Random Times Hazard Function Martingale Hazard Process Martingale Representation Theorem Change of a Probability Measure Kusuoka's Counter-Example Validity of Condition (F.2) Validity of Condition (M.l) 218

5 XVI Table of Contents Part III. Reduced-Form Modeling 8. Intensity-Based Valuation of Defaultable Claims Defaultable Claims Risk-Neutral Valuation Formula Valuation via the Hazard Process Canonical Construction of a Default Time Integral Representation of the Value Process Case of a Deterministic Intensity Implied Probabilities of Default Exogenous Recovery Rules Valuation via the Martingale Approach Martingale Hypotheses Endogenous Recovery Rules Hedging of Defaultable Claims General Reduced-Form Approach Reduced-Form Models with State Variables Lando's Approach Duffie and Singleton Approach Hybrid Methodologies Credit Spread Models Conditionally Independent Defaults Basket Credit Derivatives Mutually Independent Default Times Conditionally Independent Default Times Valuation of the i th -to-default Contract Vanilla Default Swaps of Basket Type Default Correlations and Conditional Probabilities Default Correlations Conditional Probabilities Dependent Defaults Dependent Intensities Kusuoka's Approach Jarrow and Yu Approach Martingale Approach to Basket Credit Derivatives Valuation of the i th -to-default Claims Markov Chains Discrete-Time Markov Chains Change of a Probability Measure The Law of the Absorption Time Discrete-Time Conditionally Markov Chains 322

6 Table of Contents XVII 11.2 Continuous-Time Markov Chains Embedded Discrete-Time Markov Chain Conditional Expectations Probability Distribution of the Absorption Time Martingales Associated with Transitions Change of a Probability Measure Identification of the Intensity Matrix Continuous-Time Conditionally Markov Chains Construction of a Conditionally Markov Chain Conditional Markov Property Associated Local Martingales Forward Kolmogorov Equation Markovian Models of Credit Migrations JLT Markovian Model and its Extensions JLT Model: Discrete-Time Case JLT Model: Continuous-Time Case Kijima and Komoribayashi Model Das and Tufano Model Thomas, Allen and Morkel-Kingsbury Model Conditionally Markov Models Lando's Approach Correlated Migrations Huge and Lando Approach Heath-Jarrow-Morton Type Models HJM Model with Default Model's Assumptions Default-Free Term Structure Pre-Default Value of a Corporate Bond Dynamics of Forward Credit Spreads Default Time of a Corporate Bond Case of Zero Recovery Default-Free and Defaultable LIBOR Rates Case of a Non-Zero Recovery Rate Alternative Recovery Rules HJM Model with Credit Migrations Model's Assumption Migration Process Special Case General Case Alternative Recovery Schemes Defaultable Coupon Bonds Default Correlations Market Prices of Interest Rate and Credit Risk 417

7 XVIII Table of Contents 13.3 Applications to Credit Derivatives Valuation of Credit Derivatives Hedging of Credit Derivatives Defaultable Market Rates Interest Rate Contracts with Default Risk Default-Free LIBOR and Swap Rates Defaultable Spot LIBOR Rates Defaultable Spot Swap Rates FRAs with Unilateral Default Risk Forward Swaps with Unilateral Default Risk Multi-Period IRAs with Unilateral Default Risk Multi-Period Defaultable Forward Nominal Rates Defaultable Swaps with Unilateral Default Risk Settlement of the l st Kind Settlement of the 2 nd Kind Settlement of the 3 rd Kind Market Conventions Defaultable Swaps with Bilateral Default Risk Defaultable Forward Swap Rates Forward Swaps with Unilateral Default Risk Forward Swaps with Bilateral Default Risk Modeling of Market Rates Models of Default-Free Market Rates Modeling of Forward LIBOR Rates Modeling of Forward Swap Rates Modeling of Defaultable Forward LIBOR Rates Lotz and Schlögl Approach Schönbucher's Approach 469 References 479 Basic Notation 495 Subject Index 497