Reading this Book xv
Acknowledgments xvii
1 Hybrid Assets 1
1.1 Introduction 1
1.2 Hybrid Capital 1
1.3 Preferreds 3
1.4 Convertible Bonds 5
1.5 Contingent Convertibles 7
1.6 Other Types of Hybrid Debt 7
1.6.1 Hybrid Bank Capital 7
1.6.2 Hybrid Corporate Capital 13
1.6.3 Toggle Bonds 14
1.7 Regulation 15
1.7.1 Making Failures Less Likely 15
1.7.2 Making Failures Less Disruptive 15
1.8 Bail-In Capital 16
1.9 Risk and Rating 17
1.9.1 Risk 17
1.9.2 Rating 18
1.10 Conclusion 18
2 Convertible Bonds 19
2.1 Introduction 19
2.2 Anatomy of a Convertible Bond 22
2.2.1 Final Payoff 22
2.2.2 Price Graph 22
2.2.3 Quotation of a Convertible Bond 23
2.2.4 Bond Floor (B F) 25
2.2.5 Parity 27
2.2.6 Convexity 27
2.2.7 Optional Conversion 33
2.2.8 Forced Conversion 35
2.2.9 Mandatory Conversion 35
2.3 Convertible Bond Arbitrage 37
2.3.1 Components of Risk 37
2.3.2 Delta 42
2.3.3 Delta Hedging 45
2.3.4 Different Notions of Delta 45
2.3.5 Greeks 46
2.4 Standard Features 47
2.4.1 Issuer Call 47
2.4.2 Put 50
2.4.3 Coupons 53
2.4.4 Dividends 56
2.5 Additional Features 58
2.5.1 Dividend Protection 58
2.5.2 Take-Over Protection 59
2.5.3 Refixes 60
2.6 Other Convertible Bond Types 62
2.6.1 Exchangeables 62
2.6.2 Synthetic Convertibles 63
2.6.3 Cross-Currency Convertibles 64
2.6.4 Reverse Convertibles 66
2.6.5 Convertible Preferreds 67
2.6.6 Make-Whole 67
2.6.7 Contingent Conversion 67
2.6.8 Convertible Bond Option 68
2.7 Convertible Bond Terminology 68
2.7.1 144a 68
2.7.2 Fixed-Income Metrics 68
2.8 Convertible Bond Market 73
2.8.1 Market Participants 73
2.8.2 Investors 74
2.9 Conclusion 76
3 Contingent Convertibles (CoCos) 77
3.1 Introduction 77
3.2 Definition 78
3.3 Anatomy 79
3.3.1 Loss-Absorption Mechanism 79
3.3.2 Trigger 83
3.3.3 Host Instrument 86
3.4 CoCos and Convertible Bonds 87
3.4.1 Forced vs. Optional Conversion 87
3.4.2 Negative vs. Positive Convexity 88
3.4.3 Limited vs. Unlimited Upside 89
3.4.4 Similarity to Reverse Convertibles 89
3.5 CoCos and Regulations 89
3.5.1 Introduction 89
3.5.2 Basel Framework 90
3.5.3 Basel I 91
3.5.4 Basel II 92
3.5.5 Basel III 93
3.5.6 Cocos in Basel III 101
3.5.7 High and Low-Trigger CoCos 104
3.6 Ranking in the Balance Sheet 106
3.7 Alternative Structures 106
3.8 Contingent Capital: Pro and Contra 107
3.8.1 Advantages 107
3.8.2 Disadvantages 107
3.8.3 Conclusion 110
4 Corporate Hybrids 113
4.1 Introduction 113
4.2 Issuer of Hybrid Debt 113
4.3 Investing in Hybrid Debt 114
4.4 Structure of a Corporate Hybrid Bond 115
4.4.1 Coupons 115
4.4.2 Replacement Capital Covenant 118
4.4.3 Issuer Calls 119
4.5 View of Rating Agencies 121
4.6 Risk in Hybrid Bonds 122
4.6.1 Subordination Risk 122
4.6.2 Deferral Risk 122
4.6.3 Extension Risk 122
4.7 Convexity in Hybrid Bonds 122
4.7.1 Case Study: Henkel 5.375% 2104 122
4.7.2 Duration Dynamics 126
4.8 Equity Character of Hybrid Bonds 126
5 Bail-In Bonds 127
5.1 Introduction 127
5.2 Definition 128
5.3 Resolution Regime 129
5.3.1 Resolution Tools 130
5.3.2 Timetable 130
5.4 Case Studies 133
5.4.1 Bail-In of Senior Bonds 133
5.4.2 Saving Lehman Brothers 134
5.5 Consequences of Bail-In 136
5.5.1 Higher Funding Costs 136
5.5.2 Higher GDP 136
5.5.3 Availability of Bail-In Bonds 136
5.5.4 Paying Bankers in Bail-In Bonds 136
5.6 Conclusion 137
6 Modeling Hybrids: An Introduction 139
6.1 Introduction 139
6.2 Heuristic Approaches 140
6.2.1 Corporate Hybrids: Yield of a Callable Bond 140
6.2.2 Convertible Bonds: Break Even 142
6.3 Building Models 143
6.3.1 Introduction 143
6.3.2 Martingales 145
6.3.3 Model Map 146
6.3.4 Cheapness 147
6.4 How Many Factors? 149
6.5 Sensitivity Analysis 152
6.5.1 Introduction 152
6.5.2 Non-linear Model 153
7 Modeling Hybrids: Stochastic Processes 159
7.1 Introduction 159
7.2 Probability Density Functions 159
7.2.1 Introduction 159
7.2.2 Normal Distribution 160
7.2.3 Lognormal Distribution 161
7.2.4 Exponential Distribution 162
7.2.5 Poisson Distribution 163
7.3 Brownian Motion 164
7.4 Ito Process 165
7.4.1 Introduction 165
7.4.2 Ito's Lemma 166
7.4.3 Share Prices as Geometric Brownian Motion 169
7.5 Poisson Process 172
7.5.1 Definition 172
7.5.2 Advanced Poisson Processes 174
7.5.3 Conclusion 176
8 Modeling Hybrids: Risk Neutrality 177
8.1 Introduction 177
8.2 Closed-Form Solution 180
8.2.1 Introduction 180
8.2.2 Black-Scholes Solution 182
8.2.3 Solving the Black-Scholes Equation 183
8.2.4 Case Study: Reverse Convertible 184
8.3 Tree-Based Methods 186
8.3.1 Introduction 186
8.3.2 Framework 187
8.3.3 Geometry of the Trinomial Tree 189
8.3.4 Modeling Share Prices on a Trinomial Tree 193
8.3.5 European Options on a Trinomial Tree 199
8.3.6 American Options 200
8.3.7 Bermudan Options: Imposing a Particular Time Slice 203
8.4 Finite Difference Technique 204
8.5 Monte Carlo 205
8.5.1 Introduction 205
8.5.2 Generating Random Numbers 206
9 Modeling Hybrids: Advanced Issues 211
9.1 Tail Risk in Hybrids 211
9.2 Jump Diffusion 212
9.2.1 Introduction 212
9.2.2 Share Price Process with Jump to Default 214
9.2.3 Trinomial Trees with Jump to Default 217
9.2.4 Pricing Convertible Bonds with Jump Diffusion 221
9.2.5 Lost in Translation 226
9.3 Correlation 227
9.3.1 Correlation Risk in Hybrids 227
9.3.2 Definition 228
9.3.3 Correlating Wiener Processes 229
9.3.4 Cholesky Factorization 230
9.3.5 Cholesky Example 233
9.3.6 Correlating Events 234
9.3.7 Using Equity Correlation 235
9.3.8 Case Study: Correlated Defaults 237
9.3.9 Case Study: Asset Correlation vs. Default Correlation 238
9.4 Structural Models 240
9.5 Conclusion 242
10 Modeling Hybrids: Handling Credit 243
10.1 Credit Spread 243
10.1.1 Definition 243
10.1.2 Working with Credit Spreads 244
10.1.3 Option-Adjusted Spread 246
10.2 Default Intensity 246
10.2.1 Introduction 246
10.3 Credit Default Swaps 248
10.3.1 Definition 248
10.3.2 Example of a CDS Curve 250
10.3.3 Availability of CDS Data 250
10.3.4 Premium and Credit Leg 251
10.3.5 Valuation 252
10.3.6 Rule of Thumb 255
10.3.7 Market Convention 256
10.3.8 Case Study: Implied Default Probability 257
10.4 Credit Triangle 259
10.4.1 Definition 259
10.4.2 Case Study 260
10.4.3 The Big Picture 263
10.5 Stochastic Credit 263
11 Constant Elasticity of Variance 267
11.1 From Black-Scholes to CEV 267
11.1.1 Introduction 267
11.1.2 Leverage Effect 268
11.1.3 Link with Black-Scholes 269
11.2 Historical Parameter Estimation 270
11.3 Valuation: Analytical Solution 274
11.3.1 Moving Away from Black-Scholes 274
11.3.2 Semi-Closed-Form Formula 275
11.3.3 Numerical Example 276
11.4 Valuation: Trinomial Trees for CEV 277
11.4.1 American Options 277
11.4.2 Trinomial Trees for CEV 277
11.4.3 Numerical Example 279
11.5 Jump-Extended CEV Process 283
11.5.1 Introduction 283
11.5.2 JDCEV-Generated Skew 284
11.5.3 Convertible Bonds Priced under JDCEV 284
11.6 Case Study: Pricing Mandatories with CEV 286
11.6.1 Mandatory Conversion 286
11.6.2 Numerical Example 287
11.7 Case Study: Pricing Convertibles with a Reset 288
11.7.1 Refixing the Conversion Price 288
11.7.2 Involvement of CEV 291
11.7.3 Numerical Example 292
11.8 Calibration of CEV 295
11.8.1 Introduction 295
11.8.2 Local or Global Calibration 296
11.8.3 Calibrating CEV: Step by Step 296
12 Pricing Contingent Debt 301
12.1 Introduction 301
12.2 Credit Derivatives Method 302
12.2.1 Introduction 302
12.2.2 Loss 302
12.2.3 Trigger Intensity (¿ Trigger) 303
12.2.4 CoCo Spread Calculation Example 305
12.2.5 Case Study: Lloyds Contingent Convertibles 305
12.3 Equity Derivatives Method 307
12.3.1 Introduction 307
12.3.2 Step 1: Zero-Coupon CoCo 308
12.3.3 Step 2: Adding Coupons 309
12.3.4 Numerical Example 311
12.3.5 Case Study: Lloyds Contingent Convertibles 313
12.3.6 Case Study: Tier 1 and Tier 2 CoCos 316
12.4 Coupon Deferral 317
12.5 Using Lattice Models 321
12.6 Linking Credit to Equity 323
12.6.1 Introduction 323
12.6.2 Hedging Credit Through Equity 326
12.6.3 Credit Elasticity 326
12.7 CoCos with Upside: CoCoCo 329
12.7.1 Downside Balanced with Upside 329
12.7.2 Numerical Example 330
12.8 Adding Stochastic Credit 333
12.8.1 Two-Factor Model 333
12.8.2 Monte Carlo Method 335
12.8.3 Pricing CoCos in a Two-Factor Model 337
12.8.4 Case Study 338
12.9 Avoiding Death Spirals 339
12.10 Appendix: Pricing Contingent Debt on a Trinomial Tree 341
12.10.1 Generalized Procedure 341
12.10.2 Positioning Nodes on the Trigger 343
12.10.3 Solving the CoCo Price 345
13 Multi-Factor Models for Hybrids 347
13.1 Introduction 347
13.2 Early Exercise 348
13.3 American Monte Carlo 352
13.3.1 Longstaff and Schwartz (LS) Technique 352
13.3.2 Convergence 356
13.3.3 Example: Longstaff and Schwartz (LS) Step by Step 356
13.3.4 Adding Calls and Puts 362
13.4 Multi-Factor Models 364
13.4.1 Adding Stochastic Interest Rates 364
13.4.2 Equity-Interest Rate Correlation 365
13.4.3 Adapting Longstaff and Schwartz (LS) 366
13.4.4 Convertible Bond under Stochastic Interest Rates 367
13.4.5 Adding Investor Put 371
13.5 Conclusion 371
References 373
Index 381
