Credit Risk: Modeling, Valuation and Hedging

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Springer Science & Business Media, Jan 22, 2004 - Business & Economics - 501 pages
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Mathematical finance and financial engineering have been rapidly expanding fields of science over the past three decades. The main reason behind this phenomenon has been the success of sophisticated quantitative methodolo gies in helping professionals manage financial risks. It is expected that the newly developed credit derivatives industry will also benefit from the use of advanced mathematics. This industry has grown around the need to handle credit risk, which is one of the fundamental factors of financial risk. In recent years, we have witnessed a tremendous acceleration in research efforts aimed at better comprehending, modeling and hedging this kind of risk. Although in the first chapter we provide a brief overview of issues related to credit risk, our goal was to introduce the basic concepts and related no tation, rather than to describe the financial and economical aspects of this important sector of financial market. The interested reader may consult, for instance, Francis et al. (1999) or Nelken (1999) for a much more exhaustive description of the credit derivatives industry.
 

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Contents

Introduction to Credit Risk
3
11 Corporate Bonds
4
111 Recovery Rules
5
112 Safety Covenants
6
113 Credit Spreads
7
115 Corporate Coupon Bonds
8
116 Fixed and Floating Rate Notes
9
117 Bank Loans and Sovereign Debt
11
823 Case of a Deterministic Intensity
232
824 Imp1ied Probabilities of Default
234
825 Exogenous Recovery Rules
236
83 Valuation via the Martingale Approach
239
831 Martingale Hypotheses
242
832 Endogenous Recovery Rules
243
84 Hedging of Defaultable Claims
246
85 General ReducedForm Approach
250

12 Vulnerable Claims
12
122 Vulnerable Claims with Bilateral Default Risk
13
123 Defaultable Interest Rate Contracts
14
13 Credit Derivatives
16
131 Default Swaps and Options
18
132 Total Rate of Return Swaps
21
133 Credit Linked Notes
22
134 Asset Swaps
24
136 Credit Spread Swaps and Options
25
14 Quantitative Models of Credit Risk
26
142 ReducedForm Models
27
143 Credit Risk Management
29
144 Liquidity Risk
30
Corporate Debt
31
21 Defaultable Claims
33
211 RiskNeutral Valuation Formula
34
212 SelfFinancing Trading Strategies
37
213 Martingale Measures
38
22 PDE Approach
40
221 PDE for the Va1ue Function
44
222 Corporate ZeroCoupon Bonds
47
223 Corporate Coupon Bond
50
23 Mertons Approach to Corporate Debt
51
232 DistancetoDefault
57
24 Extensions of Mertons Approach
58
241 Models with Stochastic Interest Rates
59
242 Discontinuous Value Process
60
243 Buffets Approach
64
FirstPassageTime Models
65
31 Properties of First Passage Times
66
311 Probability Law of the First Passage Time
67
312 Joint Probability Law of Y and 𝜏
69
32 Black and Cox Model
71
322 Corporate Coupon Bond
79
323 Corporate Consol Bond
81
33 Optimal Capital Structure
82
332 Lelands Approach
84
333 Leland and Toft Approach
86
334 Further Developments
88
34 Models with Stochastic Interest Rates
90
341 Kim Ramaswamy and Sundaresan Approach
96
342 Longstaff and Schwartz Approach
98
343 Cathcart and ElJahel Approach
103
344 Briys and de Varenne Approach
104
345 SaáRequejo and SantaClara Approach
107
35 Further Developments
113
Structural Approach
114
JP Morgans Approach
116
Zhous Approach
117
Hazard Processes
121
Hazard Function of a Random Time
123
42 Martingales Associated with a Continuous Hazard Function
127
43 Martingale Representation Theorem
131
44 Change of a Probability Measure
133
45 Martingale Characterization of the Hazard Function
137
46 Compensator of a Random Time
140
Hazard Process of a Random Time
141
511 Conditional Expectations
143
512 Semimartingale Representation of the Stopped Process
150
513 Martingales Associated with the Hazard Process 𝜞
152
514 Stochastic Intensity of a Random Time
155
52 Martingale Representation Theorems
156
522 Case of a Brownian Filtration
159
53 Change of a Probability Measure
162
Martingale Hazard Process
165
611 Martinga1e Invariance Property
166
Special Case
167
General Case
169
614 Uniqueness of a Martingale Hazard Process A
172
63 Martingale Representation Theorem
177
64 Case of the Martingale Invariance Property
179
641 Valuation of Defaultable Claims
180
642 Case of a Stopping Time
182
65 Random Time with a Given Hazard Process
183
66 Poisson Process and Conditional Poisson Process
186
Case of Several Random Times
197
711 Hazard Function
198
713 Martingale Representation Theorem
200
72 Change of a Probability Measure
203
73 Kusuokas CounterExample
209
731 Validity of Condition F2
216
732 Validity of Condition M1
218
ReducedForm Modeling
219
IntensityBased Valuation of Defaultable Claims
221
81 Defaultable Claims
222
811 RiskNeutral Valuation Formula
223
82 Valuation via the Hazard Process
225
821 Canonical Construction of a Default Time
227
822 Integral Representation of the Value Process
230
86 ReducedForm Models with State Variables
253
862 Duffle and Singleton Approach
255
863 Hybrid Methodologies
259
864 Credit Spread Models
264
Conditionally Independent Defaults
265
91 Basket Credit Derivatives
266
911 Mutually Independent Default Times
267
912 Conditionally Independent Default Times
268
913 Valuation of the ithtoDefault Contract
274
914 Vanilla Default Swaps of Basket Type
281
92 Default Correlations and Conditional Probabilities
284
922 Conditional Probabilities
287
Dependent Defaults
293
101 Dependent Intensities
295
1012 Jarrow and Yu Approach
296
102 Martingale Approach to Basket Credit Derivatives
306
1021 Valuation of the ithtoDefault Claims
311
Markov Chains
313
111 DiscreteTime Markov Chains
314
1111 Change of a Probability Measure
316
1112 The Law of the Absorption Time
320
1113 DiscreteTime Conditionally Markov Chains
322
112 ContinuousTime Markov Chains
324
1121 Embedded DiscreteTime Markov Chain
329
1123 Probability Distribution of the Absorption Time
332
1124 Martingales Associated with Transitions
333
1125 Change of a Probability Measure
334
1126 Identification of the Intensity Matrix
338
113 ContinuousTime Conditionally Markov Chains
340
1131 Construction of a Conditionally Markov Chain
342
1132 Conditional Markov Property
346
1133 Associated Local Martingales
347
1134 Forward Kolmogorov Equation
350
Markovian Models of Credit Migrations
351
121 JLT Markovian Model and its Extensions
352
DiscreteTime Case
354
ContinuousTime Case
362
1213 Kijima and Komoribayashi Model
367
1214 Das and Tufano Model
369
1215 Thomas Al1en and MorkelKingsbury Model
371
122 Conditionally Markov Models
373
1221 Landos Approach
374
123 Correlated Migrations
376
1231 Huge and Lando Approach
380
HeathJarrowMorton Type Models
385
131 HJM Model with Default
386
1312 DefaultFree Term Structure
388
1313 PreDefault Value of a Corporate Bond
390
1314 Dynamics of Forward Credit Spreads
392
1315 Default Time of a Corporate Bond
394
1316 Case of Zero Recovery
397
1317 DefaultFree and Defaultable LIBOR Rates
398
1318 Case of a NonZero Recovery Rate
400
1319 Alternative Recovery Rules
403
132 HJM Model with Credit Migrations
405
1322 Migration Process
407
1323 Special Case
408
1324 General Case
410
1325 Alternative Recovery Schemes
413
1326 Defaultable Coupon Bonds
415
1327 Default Correlations
416
1328 Market Prices of Interest Rate and Credit Risk
417
133 Applications to Credit Derivatives
421
1332 Hedging of Credit Derivatives
422
Defaultable Market Rates
423
141 Interest Rate Contracts with Default Risk
424
1412 Defaultab1e Spot LIBOR Rates
426
1413 Defaultable Spot Swap Rates
427
1414 FRAs with Unilateral Default Risk
428
1415 Forward Swaps with Unilateral Default Risk
432
142 MultiPeriod IRAs with Unilateral Default Risk
434
143 MultiPeriod Defaultable Forward Nominal Rates
438
144 Defaultable Swaps with Unilateral Default Risk
441
1441 Settlement of the 1st Kind
442
1442 Settlement of the 2nd Kind
444
1443 Settlement of the 3rd Kind
445
1444 Market Conventions
446
145 Defaultable Swaps with Bilateral Default Risk
447
146 Defaultable Forward Swap Rates
449
1462 Forward Swaps with Bilateral Default Risk
450
Modeling of Market Rates
451
151 Models of DefaultFree Market Rates
452
1512 Modeling of Forward Swap Rates
458
152 Modeling of Defaultable Forward LIBOR Rates
465
1522 Schönbuchers Approach
469
References
479
Basic Notation
495
Subject Index
497
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