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Credit risk: modelling, valuation and hedging

Author: Bielecki, Tomasz R. ; Rutkowski, Marek Series: Springer finance Publisher: Springer, 2004.Language: EnglishDescription: 500 p. ; 24 cm.ISBN: 3540675930Type of document: BookBibliography/Index: Includes bibliographical references and index
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Item type Current location Collection Call number Status Date due Barcode Item holds
Book Europe Campus
Main Collection
Print HG4751 .B43 2004
(Browse shelf)
001212020
Available 001212020
Total holds: 0

Includes bibliographical references and index

Digitized

Credit Risk: Modeling, Valuation and Hedging Table of Contents Preface .................................................................................................................. V Part I. Structural Approach 1. Introduction to Credit Risk....................................................................... 3 1.1 Corporate Bonds ............................................................................. 4 1.1.1 Recovery Rules ...................................................................... 5 1.1.2 Safety Covenants ................................................................. 6 1.1.3 Credit Spreads .................................................................... 7 1.1.4 Credit Ratings ..................................................................... 7 1.1.5 Corporate Coupon Bonds .................................................... 8 1.1.6 Fixed and Floating Rate Notes............................................. 9 1.1.7 Bank Loans and Sovereign Debt.......................................... 11 1.1.8 Cross Default ...................................................................... 11 1.1.9 Default Correlations ............................................................ 11 1.2 Vulnerable Claims ............................................................................ 12 1.2.1 Vulnerable Claims with Unilateral Default Risk ................. 12 1.2.2 Vulnerable Claims with Bilateral Default Risk .................... 13 1.2.3 Defaultable Interest Rate Contracts.................................... 14 1.3 Credit Derivatives .............................................................................. 16 1.3.1 Default Swaps and Options ................................................ 18 1.3.2 Total Rate of Return Swaps ................................................. 21 1.3.3 Credit Linked Notes .............................................................. 22 1.3.4 Asset Swaps ........................................................................ 24 1.3.5 First-to-Default Contracts ................................................... 24 1.3.6 Credit Spread Swaps and Options ...................................... 25 1.4 Quantitative Models of Credit Risk .................................................. 26 1.4.1 Structural Models ............................................................... 26 1.4.2 Reduced-Form Models ......................................................... 27 1.4.3 Credit Risk Management ..................................................... 29 1.4.4 Liquidity Risk ........................................................................ 30 1.4.5 Econometric Studies ........................................................... 30 XIV Table of Contents 2. Corporate Debt ........................................................................................................... 31 2.1 Defaultable Claims .......................................................................................... 33 2.1.1 Risk-Neutral Valuation Formula.................................................. 34 2.1.2 Self-Financing Trading Strategies................................................ 37 2.1.3 Martingale Measures........................................................................ 38 2.2 PDE Approach.................................................................................................... 40 2.2.1 PDE for the Value Function .......................................................... 44 2.2.2 Corporate Zero-Coupon Bonds..................................................... 47 2.2.3 Corporate Coupon Bond ................................................................. 50 2.3 Merton's Approach to Corporate Debt ...................................................... 51 2.3.1 Merton's Model with Deterministic Interest Rates ................ 51 2.3.2 Distance-to-Default .......................................................................... 57 2.4 Extensions of Merton's Approach............................................................... 58 2.4.1 Models with Stochastic Interest Rates........................................ 59 2.4.2 Discontinuous Value Process ....................................................... 60 2.4.3 Buffet's Approach ............................................................................. 64 3. First-Passage-Time Models .................................................................................... 65 3.1 Properties of First Passage Times .............................................................. 66 3.1.1 Probability Law of the First Passage Time................................. 67 3.1.2 Joint Probability Law of Y and T ......................................................... 69 3.2 Black and Cox Model ...................................................................................... 71 3.2.1 Corporate Zero-Coupon Bond........................................................ 71 3.2.2 Corporate Coupon Bond ................................................................. 79 3.2.3 Corporate Consol Bond .................................................................. 81 3.3 Optimal Capital Structure............................................................................. 82 3.3.1 Black and Cox Approach ................................................................ 82 3.3.2 Leland's Approach............................................................................. 84 3.3.3 Leland and Toft Approach............................................................... 86 3.3.4 Further Developments ..................................................................... 88 3.4 Models with Stochastic Interest Rates...................................................... 90 3.4.1 Kim, Ramaswamy and Sundaresan Approach ........................ 96 3.4.2 Longstaff and Schwartz Approach .............................................. 98 3.4.3 Cathcart and El-Jahel Approach ................................................ 103 3.4.4 Briys and de Varenne Approach................................................... 104 3.4.5 Saá-Requejo and Santa-Clara Approach .................................. 107 3.5 Further Developments................................................................................... 113 3.5.1 Convertible Bonds............................................................................. 113 3.5.2 Jump-Diffusion Models................................................................... 113 3.5.3 Incomplete Accounting Data.......................................................... 113 3.6 Dependent Defaults: Structural Approach ............................................ 114 3.6.1 Default Correlations: J.P. Morgan's Approach........................ 116 3.6.2 Default Correlations: Zhou's Approach .................................... 117 Table of Contents XV Part II. Hazard Processes 4. Hazard Function of a Random Time.................................................................. 4.1 Conditional Expectations w.r.t. Natural Filtrations ........................... 4.2 Martingales Associated with a Continuous Hazard Function ........ 4.3 Martingale Representation Theorem ....................................................... 4.4 Change of a Probability Measure .............................................................. 4.5 Martingale Characterization of the Hazard Function ........................ 4.6 Compensator of a Random Time ............................................................... 123 123 127 131 133 137 140 5. Hazard Process of a Random Time ................................................................... 141 5.1 Hazard Process F ........................................................................................... 141 5.1.1 Conditional Expectations.............................................................. 143 5.1.2 Semimartingale Representation of the Stopped Process 150 5.1.3 Martingales Associated with the Hazard Process F . . . 152 5.1.4 Stochastic Intensity of a Random Time ................................... 5.2 Martingale Representation Theorems .................................................... 5.2.1 General Case ..................................................................................... 5.2.2 Case of a Brownian Filtration...................................................... 5.3 Change of a Probability Measure ............................................................. 155 156 156 159 162 6. Martingale Hazard Process ................................................................................... 165 6.1 Martingale Hazard Process A ................................................................... 165 6.1.1 Martingale Invariance Property .................................................. 166 6.1.2 Evaluation of A: Special Case ...................................................... 167 6.1.3 Evaluation of A: General Case .................................................... 169 6.1.4 Uniqueness of a Martingale Hazard Process A....................... 172 6.2 Relationships Between Hazard Processes F and A ............................ 173 6.3 Martingale Representation Theorem ...................................................... 177 6.4 Case of the Martingale Invariance Property ......................................... 179 6.4.1 Valuation of Defaultable Claims.................................................. 180 6.4.2 Case of a Stopping Time ............................................................... 182 6.5 Random Time with a Given Hazard Process ........................................ 183 6.6 Poisson Process and Conditional Poisson Process ........................... 186 7. Case of Several Random Times .......................................................................... 197 7.1 Minimum of Several Random Times ....................................................... 197 7.1.1 Hazard Function .............................................................................. 198 7.1.2 Martingale Hazard Process .......................................................... 198 7.1.3 Martingale Representation Theorem ........................................ 200 7.2 Change of a Probability Measure ............................................................. 203 7.3 Kusuoka's Counter-Example .................................................................... 209 7.3.1 Validity of Condition (F.2) ............................................................. 216 7.3.2 Validity of Condition (M.1) ........................................................... 218 XVI Table of Contents Part III. Reduced-Form Modeling 8. Intensity-Based Valuation of Defaultable Claims................................ 221 8.1 Defaultable Claims ........................................................................................ 222 8.1.1 Risk-Neutral Valuation Formula.................................................. 223 8.2 Valuation via the Hazard Process............................................................. 225 8.2.1 Canonical Construction of a Default Time................................ 227 8.2.2 Integral Representation of the Value Process ......................... 230 8.2.3 Case of a Deterministic Intensity ................................................ 232 8.2.4 Implied Probabilities of Default ................................................... 234 8.2.5 Exogenous Recovery Rules............................................................ 236 8.3 Valuation via the Martingale Approach................................................... 239 8.3.1 Martingale Hypotheses................................................................... 242 8.3.2 Endogenous Recovery Rules ........................................................ 243 8.4 Hedging of Defaultable Claims .................................................................. 246 8.5 General Reduced-Form Approach............................................................ 250 8.6 Reduced-Form Models with State Variables ......................................... 253 8.6.1 Lando's Approach ............................................................................ 253 8.6.2 Duffie and Singleton Approach ................................................... 255 8.6.3 Hybrid Methodologies...................................................................... 259 8.6.4 Credit Spread Models...................................................................... 264 9. Conditionally Independent Defaults ......................................................... 265 9.1 Basket Credit Derivatives............................................................................ 266 9.1.1 Mutually Independent Default Times ........................................ 267 9.1.2 Conditionally Independent Default Times ................................ 268 9.1.3 Valuation of the ith-to-Default Contract .................................... 274 9.1.4 Vanilla Default Swaps of Basket Type........................................ 281 9.2 Default Correlations and Conditional Probabilities ............................ 284 9.2.1 Default Correlations ........................................................................ 284 9.2.2 Conditional Probabilities ............................................................... 287 10. Dependent Defaults ......................................................................................... 10.1 Dependent Intensities ............................................................................... 10.1.1 Kusuoka's Approach .................................................................... 10.1.2 Jarrow and Yu Approach............................................................. 10.2 Martingale Approach to Basket Credit Derivatives .......................... 10.2.1 Valuation of the ith-to-Default Claims...................................... 11. Markov Chains .................................................................................................... 11.1 Discrete-Time Markov Chains................................................................. 11.1.1 Change of a Probability Measure .............................................. 11.1.2 The Law of the Absorption Time................................................ 11.1.3 Discrete-Time Conditionally Markov Chains ........................ 293 295 295 296 306 311 313 314 316 320 322 Table of Contents XVII 11.2 Continuous-Time Markov Chains ........................................................ 11.2.1 Embedded Discrete-Time Markov Chain ............................... 11.2.2 Conditional Expectations............................................................ 11.2.3 Probability Distribution of the Absorption Time ................. 11.2.4 Martingales Associated with Transitions................................ 11.2.5 Change of a Probability Measure ............................................. 11.2.6 Identification of the Intensity Matrix....................................... 11.3 Continuous-Time Conditionally Markov Chains .............................. 11.3.1 Construction of a Conditionally Markov Chain ................... 11.3.2 Conditional Markov Property .................................................... 11.3.3 Associated Local Martingales .................................................... 11.3.4 Forward Kolmogorov Equation.................................................. 12. Markovian Models of Credit Migrations ........................................................ 12.1 JLT Markovian Model and its Extensions........................................... 12.1.1 JLT Model: Discrete-Time Case ............................................... 12.1.2 JLT Model: Continuous-Time Case ......................................... 12.1.3 Kijima and Komoribayashi Model ........................................... 12.1.4 Das and Tufano Model................................................................. 12.1.5 Thomas, Allen and Morkel-Kingsbury Model........................ 12.2 Conditionally Markov Models ................................................................ 12.2.1 Lando's Approach ......................................................................... 324 329 329 332 333 334 338 340 342 346 347 350 351 352 354 362 367 369 371 373 374 12.3 Correlated Migrations................................................................................ 376 12.3.1 Huge and Lando Approach ........................................................ 380 13. Heath-Jarrow-Morton Type Models ................................................................ 385 13.1 HJM Model with Default .......................................................................... 386 13.1.1 Model's Assumptions .................................................................. 386 13.1.2 Default-Free Term Structure .................................................... 388 13.1.3 Pre-Default Value of a Corporate Bond .................................. 390 13.1.4 Dynamics of Forward Credit Spreads .................................... 392 13.1.5 Default Time of a Corporate Bond ........................................... 394 13.1.6 Case of Zero Recovery ................................................................. 397 13.1.7 Default-Free and Defaultable LIBOR Rates........................... 398 13.1.8 Case of a Non-Zero Recovery Rate............................................ 400 13.1.9 Alternative Recovery Rules ........................................................ 403 13.2 HJM Model with Credit Migrations........................................................ 405 13.2.1 Model's Assumption ..................................................................... 405 13.2.2 Migration Process ......................................................................... 407 13.2.3 Special Case ................................................................................... 408 13.2.4 General Case .................................................................................. 410 13.2.5 Alternative Recovery Schemes................................................... 413 13.2.6 Defaultable Coupon Bonds ........................................................ 415 13.2.7 Default Correlations .................................................................... 416 13.2.8 Market Prices of Interest Rate and Credit Risk.................... 417 XVIII Table of Contents 13.3 Applications to Credit Derivatives ........................................................ 421 13.3.1 Valuation of Credit Derivatives................................................. 421 13.3.2 Hedging of Credit Derivatives ................................................... 422 14. Defaultable Market Rates .................................................................................. 423 14.1 Interest Rate Contracts with Default Risk ........................................ 424 14.1.1 Default-Free LIBOR and Swap Rates .................................... 424 14.1.2 Defaultable Spot LIBOR Rates ................................................ 426 14.1.3 Defaultable Spot Swap Rates ................................................... 427 14.1.4 FRAs with Unilateral Default Risk ......................................... 428 14.1.5 Forward Swaps with Unilateral Default Risk ..................... 432 14.2 Multi-Period IRAs with Unilateral Default Risk .............................. 434 14.3 Multi-Period Defaultable Forward Nominal Rates.......................... 438 14.4 Defaultable Swaps with Unilateral Default Risk ............................ 441 14.4.1 Settlement of the 1st Kind........................................................... 442 14.4.2 Settlement of the 2nd Kind ......................................................... 444 14.4.3 Settlement of the 3rd Kind........................................................... 445 14.4.4 Market Conventions .................................................................... 446 14.5 Defaultable Swaps with Bilateral Default Risk ............................... 447 14.6 Defaultable Forward Swap Rates ......................................................... 449 14.6.1 Forward Swaps with Unilateral Default Risk ..................... 449 14.6.2 Forward Swaps with Bilateral Default Risk ........................ 450 15. Modeling of Market Rates.................................................................................. 451 15.1 Models of Default-Free Market Rates ................................................. 452 15.1.1 Modeling of Forward LIBOR Rates ......................................... 452 15.1.2 Modeling of Forward Swap Rates............................................. 458 15.2 Modeling of Defaultable Forward LIBOR Rates ............................... 465 15.2.1 Lotz and Schlögl Approach ....................................................... 465 15.2.2 Schönbucher's Approach .......................................................... 469 References........................................................................................................................ 479 Basic Notation ............................................................................................................... 495 Subject Index ................................................................................................................. 497

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