Contents
0.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
1 Introduction 1
1.1 A Brief Zoology of Risks . . . . . . . . . . . . . . . . . . . . . 4
1.2 Organization of Topics . . . . . . . . . . . . . . . . . . . . . . 8
2 Economic Principles of Risk Management 15
2.1 What Types of Risk CountMost? . . . . . . . . . . . . . . . . 16
2.2 Economics ofMarket Risk . . . . . . . . . . . . . . . . . . . . 19
2.2.1 Pro t-Loss Asymmetries . . . . . . . . . . . . . . . . . 19
2.2.2 Minimum Capital Requirements . . . . . . . . . . . . . 22
2.2.3 Principal-Agent E ects . . . . . . . . . . . . . . . . . . 24
2.2.4 Capital { A Scarce Resource . . . . . . . . . . . . . . . 25
2.2.5 Leverage and Risk for Financial Firms . . . . . . . . . 25
2.2.6 Allocation of Capital or Risk? . . . . . . . . . . . . . . 26
2.2.7 Risk-Adjusted Return on Capital? . . . . . . . . . . . . 28
2.2.8 Performance Incentives . . . . . . . . . . . . . . . . . . 29
2.3 Economic Principles of Credit Risk . . . . . . . . . . . . . . . 30
2.3.1 Adverse Selection and Credit Exposure . . . . . . . . . 30
2.3.2 Credit Risk Concentrations . . . . . . . . . . . . . . . 32
2.3.3 Moral Hazard . . . . . . . . . . . . . . . . . . . . . . . 33
2.4 RiskMeasurement . . . . . . . . . . . . . . . . . . . . . . . . 34
2.4.1 Value at Risk . . . . . . . . . . . . . . . . . . . . . . . 37
2.4.2 Expected Tail Loss . . . . . . . . . . . . . . . . . . . . 40
2.4.3 The Price ofMarket-Value Insurance . . . . . . . . . . 41
2.4.4 Volatility as a Measure of Market Risk . . . . . . . . . 42
2.4.5 Coherency of RiskMeasures . . . . . . . . . . . . . . . 43
2.5 Measuring Credit Risk . . . . . . . . . . . . . . . . . . . . . . 44
2.5.1 SpecializedMeasures of Credit Risk . . . . . . . . . . . 44

2.5.2 Capital Guidelines for Credit Exposures . . . . . . . . 46
3 Default Arrival: Historical Patterns and Statistical Models 51
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.1.1 The Changing Composition of Speculative Debt . . . . 55
3.1.2 Forward Default Probabilities . . . . . . . . . . . . . . 58
3.2 Structural Models of Default Probability . . . . . . . . . . . . 60
3.2.1 The Black-Scholes-Merton DefaultModel . . . . . . . . 61
3.2.2 First-PassageModels . . . . . . . . . . . . . . . . . . . 62
3.3 From Theory to Practice: Using Distance to Default to Predict
Default . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.4 Default Intensity . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.4.1 Doubly-Stochastic Default . . . . . . . . . . . . . . . . 70
3.5 Examples of IntensityModels . . . . . . . . . . . . . . . . . . 72
3.5.1 Mean-Reverting Intensities with Jumps . . . . . . . . . 73
3.5.2 CIR IntensityModels . . . . . . . . . . . . . . . . . . . 75
3.5.3 Comparison of Jump and CIR Intensities . . . . . . . . 78
3.5.4 Ane IntensityModels . . . . . . . . . . . . . . . . . . 80
3.5.5 HJMForward Default RateModels . . . . . . . . . . . 81
3.6 Default Time Simulation . . . . . . . . . . . . . . . . . . . . . 82
3.7 Statistical Prediction of Bankruptcy . . . . . . . . . . . . . . . 84
3.7.1 Comparing PredictionMethods . . . . . . . . . . . . . 88
3.7.2 Default and Aging E ects . . . . . . . . . . . . . . . . 90
4 Ratings Transitions: Historical Patterns and StatisticalModels
97
4.1 Average Transitions Frequencies . . . . . . . . . . . . . . . . . 98
4.2 Ratings Risk and the Business Cycle . . . . . . . . . . . . . . 100
4.3 Ratings Transitions and Aging . . . . . . . . . . . . . . . . . . 104
4.4 Ordered Probits of Ratings . . . . . . . . . . . . . . . . . . . . 105
4.5 Ratings asMarkov Chains . . . . . . . . . . . . . . . . . . . . 107
4.5.1 Time-Varying Transition Intensities . . . . . . . . . . . 109
4.5.2 Lando's Stochastic Transition IntensityModel . . . . . 110
5 Conceptual Approaches to Valuation of Default Risk 113
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
5.2 Risk-Neutral Versus Actual Probabilities . . . . . . . . . . . . 116
5.3 Reduced-FormPricing . . . . . . . . . . . . . . . . . . . . . . 120

5.4 StructuralModels . . . . . . . . . . . . . . . . . . . . . . . . . 125
5.4.1 The Black-Scholes-Merton Debt PricingMOdel . . . . 125
5.4.2 First-Passage Debt Pricing . . . . . . . . . . . . . . . . 126
5.5 Comparisons ofModel-Implied Spreads . . . . . . . . . . . . . 128
5.6 FromActual to Risk-Neutral Intensities . . . . . . . . . . . . . 132
5.6.1 Reduced-FormModels . . . . . . . . . . . . . . . . . . 132
5.6.2 StructuralModels . . . . . . . . . . . . . . . . . . . . . 134
6 Pricing Corporate and Sovereign Bonds 137
6.1 Uncertain Recovery . . . . . . . . . . . . . . . . . . . . . . . . 137
6.2 Reduced-FormPricing with Recovery . . . . . . . . . . . . . . 141
6.2.1 Fractional Recovery of Face Value . . . . . . . . . . . . 141
6.2.2 Conditional Expected Recovery . . . . . . . . . . . . . 145
6.2.3 Fractional Recovery ofMarket Value . . . . . . . . . . 147
6.2.4 Comparing Recovery Assumptions . . . . . . . . . . . . 152
6.3 Ratings-BasedModels of Credit Spreads . . . . . . . . . . . . 154
6.3.1 General Pricing Framework . . . . . . . . . . . . . . . 156
6.3.2 Calibrating AModel to Historical Data . . . . . . . . . 159
6.4 Pricing Sovereign Bonds . . . . . . . . . . . . . . . . . . . . . 162
6.4.1 Credit Risk in Sovereign Bonds . . . . . . . . . . . . . 164
6.4.2 ParametricModels of Sovereign Spreads . . . . . . . . 169
7 Empirical Models of Defaultable Bond Spreads 175
7.1 Credit Spreads and Economic Activity . . . . . . . . . . . . . 175
7.2 Reference Curves for Spreads . . . . . . . . . . . . . . . . . . 181
7.3 Parametric Reduced-FormModels . . . . . . . . . . . . . . . . 186
7.3.1 Square-root Di usionModels of Spreads . . . . . . . . 186
7.3.2 Jump-Di usion Spreads . . . . . . . . . . . . . . . . . 187
7.3.3 Accommodating Observable Credit Factors . . . . . . . 188
7.4 Estimating StructuralModels . . . . . . . . . . . . . . . . . . 189
7.5 ParametricModels of Sovereign Spreads . . . . . . . . . . . . 192
8 Credit Swaps 195
8.1 Other Credit Derivatives . . . . . . . . . . . . . . . . . . . . . 195
8.1.1 Total-Return Swaps . . . . . . . . . . . . . . . . . . . . 196
8.1.2 Spread Options . . . . . . . . . . . . . . . . . . . . . . 197
8.2 The Basic Credit Swap . . . . . . . . . . . . . . . . . . . . . . 198
8.2.1 Settlement Issues . . . . . . . . . . . . . . . . . . . . . 200

8.3 Simple Credit-Swap Spreads . . . . . . . . . . . . . . . . . . . 201
8.3.1 Credit-Swap Spreads: Starter Case . . . . . . . . . . . 201
8.3.2 Repo Specials and Transactions Costs . . . . . . . . . . 204
8.3.3 Payment of Accrued Credit-Swap Premium. . . . . . . 207
8.3.4 Accrued Interest on the Underlying Note . . . . . . . . 208
8.3.5 If the Underlying is a Fixed-Rate Note? . . . . . . . . 208
8.4 Model-Based CDS Rates . . . . . . . . . . . . . . . . . . . . . 210
8.4.1 The Case of Constant Intensity . . . . . . . . . . . . . 210
8.4.2 The Term Structure of Forward Default Rates . . . . . 213
8.5 The Role of Asset Swaps . . . . . . . . . . . . . . . . . . . . . 214
9 Optional Credit Pricing 219
9.1 Spread Options . . . . . . . . . . . . . . . . . . . . . . . . . . 220
9.1.1 Pricing Framework . . . . . . . . . . . . . . . . . . . . 220
9.1.2 Illustrative Numerical Example . . . . . . . . . . . . . 223
9.1.3 Numerical Option Valuation Algorithm . . . . . . . . . 224
9.1.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
9.2 Callable and Convertible Corporate Debt . . . . . . . . . . . . 228
9.2.1 Capital Structure . . . . . . . . . . . . . . . . . . . . . 229
9.2.2 Default and Equity Derivative Pricing . . . . . . . . . 233
9.2.3 Convertible Debt as an Equity Derivative . . . . . . . . 236
9.2.4 Call-Forcing Conversion . . . . . . . . . . . . . . . . . 238
9.2.5 Evidence of Delayed Calls . . . . . . . . . . . . . . . . 241
9.3 A Simple Convertible Bond PricingModel . . . . . . . . . . . 242
9.3.1 BackgroundModeling . . . . . . . . . . . . . . . . . . 243
9.3.2 Convertible BondModel Setup . . . . . . . . . . . . . 244
9.3.3 Pricing Algorithm. . . . . . . . . . . . . . . . . . . . . 246
9.3.4 Convertible Bond Hedging Strategies . . . . . . . . . . 247
9.3.5 Exposure to Equity Volatility . . . . . . . . . . . . . . 252
9.3.6 The Issuer's Propensity to Call . . . . . . . . . . . . . 252
9.3.7 Duration and Convexity . . . . . . . . . . . . . . . . . 253
10 Correlated Defaults 259
10.1 Alternative Approaches to Correlation . . . . . . . . . . . . . 259
10.2 CreditMetrics Correlated Defaults . . . . . . . . . . . . . . . . 261
10.3 Correlated Default Intensities . . . . . . . . . . . . . . . . . . 264
10.3.1 Correlated Jump Intensity Processes . . . . . . . . . . 264
10.3.2 Correlated Log-Normal Intensities . . . . . . . . . . . . 266

10.4 Copula-Based CorrelationModeling . . . . . . . . . . . . . . . 268
10.4.1 Calibration of Copula to First-PassageModel . . . . . 271
10.5 EmpiricalMethods . . . . . . . . . . . . . . . . . . . . . . . . 273
10.6 Default Time Simulation Algorithms . . . . . . . . . . . . . . 274
10.6.1 Multi-CompensatorMethod . . . . . . . . . . . . . . . 276
10.6.2 First-Defaults Simulation . . . . . . . . . . . . . . . . . 277
10.6.3 Recursive Inverse-CDF Simulation . . . . . . . . . . . 278
10.7 Joint Default Events . . . . . . . . . . . . . . . . . . . . . . . 279
10.7.1 Multivariate Exponential Event Times . . . . . . . . . 281
11 Collateralized Debt Obligations 283
11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283
11.2 Some Economics of CDOs . . . . . . . . . . . . . . . . . . . . 285
11.3 Default-RiskModel . . . . . . . . . . . . . . . . . . . . . . . . 289
11.3.1 Obligor Default Intensities . . . . . . . . . . . . . . . . 289
11.3.2 Multi-Issuer DefaultModel . . . . . . . . . . . . . . . . 290
11.3.3 Sectoral, Regional, and Global Risk . . . . . . . . . . . 291
11.3.4 Recovery Risk . . . . . . . . . . . . . . . . . . . . . . . 292
11.3.5 Collateral Credit Spreads . . . . . . . . . . . . . . . . . 292
11.3.6 Diversity Scores . . . . . . . . . . . . . . . . . . . . . . 293
11.4 Pricing Examples . . . . . . . . . . . . . . . . . . . . . . . . . 294
11.4.1 Collateral . . . . . . . . . . . . . . . . . . . . . . . . . 296
11.4.2 Sinking-Fund Tranches . . . . . . . . . . . . . . . . . . 298
11.4.3 Prioritization Schemes . . . . . . . . . . . . . . . . . . 301
11.4.4 SimulationMethodology . . . . . . . . . . . . . . . . . 303
11.4.5 Results for Par CDO Spreads . . . . . . . . . . . . . . 304
11.4.6 Risky Reinvestment . . . . . . . . . . . . . . . . . . . . 306
11.5 Default Loss Analytics . . . . . . . . . . . . . . . . . . . . . . 308
11.6 Computation of Diversity Scores . . . . . . . . . . . . . . . . . 315
12 OTC Default Risk and Valuation 323
12.1 Exposure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323
12.1.1 Potential Exposure . . . . . . . . . . . . . . . . . . . . 325
12.1.2 BIS Capital Add-Ons for Exposure . . . . . . . . . . . 326
12.1.3 Interest-Rate Swaps . . . . . . . . . . . . . . . . . . . 328
12.1.4 Mid-Market Revaluation of Swaps . . . . . . . . . . . . 331
12.1.5 Expected Swap Exposures . . . . . . . . . . . . . . . . 333
12.2 OTC Credit-Risk Value Adjustments . . . . . . . . . . . . . . 334

12.2.1 One-Sided Default Risk . . . . . . . . . . . . . . . . . . 336
12.2.2 Adjustment with Netting and Collateral . . . . . . . . 338
12.2.3 Two-Sided Default Risk . . . . . . . . . . . . . . . . . 340
12.2.4 Example: 10-Year Swap Adjustments . . . . . . . . . . 340
12.3 Additional Swap Credit Adjustments . . . . . . . . . . . . . . 344
12.3.1 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . 344
12.3.2 Base-Case Results . . . . . . . . . . . . . . . . . . . . . 345
12.3.3 O -Market Swap Rate Credit Spreads . . . . . . . . . 346
12.3.4 Dependence of Default Risk on LIBOR Rate . . . . . . 346
12.3.5 Asynchronous Swap Payments . . . . . . . . . . . . . . 347
12.3.6 The Impact of Slope of the Yield Curve . . . . . . . . . 347
12.3.7 TermStructure of Credit Spreads . . . . . . . . . . . . 348
12.3.8 When Both Counterparties are Risky . . . . . . . . . . 348
12.3.9 The Impact of Netting . . . . . . . . . . . . . . . . . . 348
12.4 Credit Spreads on Currency Swaps . . . . . . . . . . . . . . . 350
13 Integrated Market and Credit Risk Measurement 357
13.1 Market Risk Factors . . . . . . . . . . . . . . . . . . . . . . . 358
13.1.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . 359
13.1.2 Modeling Return Distributions . . . . . . . . . . . . . 361
13.1.3 Shapes of Return Distributions . . . . . . . . . . . . . 365
13.2 Delta-Gamma for Derivatives, with Jumps . . . . . . . . . . . 370
13.2.1 DeltaMeasures of Derivatives Risk . . . . . . . . . . . 370
13.2.2 Beyond Delta to Gamma . . . . . . . . . . . . . . . . . 374
13.3 Integration ofMarket and Credit Risk . . . . . . . . . . . . . 377
13.4 Examples of VaR with Credit Risk . . . . . . . . . . . . . . . 379
13.4.1 Example: Loan Portfolio VaR with Credit Risk . . . . 381
13.4.2 Example: Options Portfolio with Credit Risk . . . . . . 385
A Introduction to Ane Processes 393
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393
2 Analytical Solutions in Ane Settings . . . . . . . . . . . . . 394
2.1 Multivariate Example: Independent Coordinates . . . . 397
2.2 Multivariate Example: The HestonModel . . . . . . . 397
2.3 The Riccati Equations . . . . . . . . . . . . . . . . . . 398
2.4 Transforms of Ane Processes . . . . . . . . . . . . . . 399
3 Some Applications . . . . . . . . . . . . . . . . . . . . . . . . 401
3.1 Term-StructureModels . . . . . . . . . . . . . . . . . . 401

3.2 Default Intensities and Probabilities . . . . . . . . . . . 402
3.3 Correlated Default . . . . . . . . . . . . . . . . . . . . 402
3.4 Defaultable TermStructureModels . . . . . . . . . . . 403
3.5 Option Pricing . . . . . . . . . . . . . . . . . . . . . . 405
4 Generalized Riccati Equations . . . . . . . . . . . . . . . . . . 406
5 Solution for The Basic AneModel . . . . . . . . . . . . . . . 408
6 Intensities for Stopping Times . . . . . . . . . . . . . . . . . . 409
B Econometrics of Ane Term-Structure Models 411
C HJM Spread Curve Models 417