Table of Contents

Reading this Book xiii

Preface xv

Acknowledgements xvii

PART I THE CONVERTIBLES MARKET 1

1 Terminology 3

1.1 The Payoff 3

1.2 Advantages of Convertibles 4

1.2.1 For the Issuer 5

1.2.2 For the Investor 8

1.3 Basic Terminology 13

1.4 Advanced Terminology 17

1.5 Legal Terminology 20

1.6 Analytics and Hedge Ratios 21

2 Convertible Bond Anatomy 25

2.1 Payoff to the Investor 25

2.2 Payoff Graph 26

2.2.1 Example 30

2.3 Boundary Conditions 31

2.3.1 Bond Floor 32

2.3.2 Parity 33

2.3.3 Investment Premium 33

2.3.4 Conversion Premium 34

2.4 Effect of the Call Protection 35

2.5 Announcement Effect 35

2.5.1 Dilution 41

2.5.2 Arbitrage Activity 41

3 Convertible and Hybrid Structures 43

3.1 Preferred Shares 43

3.2 Convertible Bond Option 45

3.3 Reverse Convertible 45

3.4 Perpetuals 46

3.5 Cross-Currency 46

3.6 Mandatory 48

3.6.1 PERCS 48

3.6.2 PEPS 48

3.7 Cashout Option 51

3.8 Exchangeable 51

3.9 Dividend Entitlement 52

4 Convertible Bonds Market 55

4.1 The Convertible Universe 55

4.1.1 Credit Rating 55

4.1.2 Convertible Type 56

4.1.3 Convertible Category 56

4.1.4 Maturity 57

4.1.5 Region 57

4.1.6 144A 57

4.2 The Prospectus 58

4.3 The Investors 58

4.3.1 Outright Investors 58

4.3.2 Convertible Bond Arbitrageurs 59

4.3.3 Example 60

4.3.4 Conclusions 62

4.4 Market Participants 62

4.4.1 Lead Manager 63

4.4.2 Trustee 63

4.4.3 Paying Agent 64

4.4.4 Market Makers 64

4.5 New Issuance 64

PART II PRICING 67

5 The Road to Convexity 69

5.1 Break-Even Analysis 69

5.1.1 Dollar Method 70

5.1.2 Equity Method 70

5.2 Discounted Yield Advantage 72

5.3 Convexity 74

5.4 Jensen?s Inequality 75

5.5 Time Decay 77

5.6 Double-Signed Gamma 79

5.7 Colour 80

5.8 First Steps Using Convexity 81

5.8.1 A Fixed Income Investor 81

5.8.2 An Equity Investor 82

6 Basic Binomial Trees 85

6.1 Models 85

6.2 The Basic Ingredients 86

6.3 A Primer in Stochastic Calculus 91

6.3.1 Stochastic Equations 91

6.3.2 Ito?s Lemma 92

6.3.3 Shares as Generalized Wiener Processes 93

6.3.4 Shares as a Log Process 93

6.3.5 Linking Both 94

6.4 Elementary Credit Model 95

6.4.1 Probabilities 95

6.4.2 Recovery Rate 98

6.4.3 Credit Triangle 98

6.5 Binomial Equity Models 99

6.5.1 Introduction 99

6.5.2 Binomial Tree 100

6.5.3 Unconditional Default Risk in the Binomial Tree 109

6.5.4 Adding Conditional Default Risk 116

6.5.5 Alternative Ways to Incorporate Credit Risk 120

6.6 Pricing Convertibles Using Binomial Trees 122

6.7 Credit Spread Modelling in Binomial Trees: A Practitioner?s Approach 155

6.8 Conclusions 156

7 Multinomial Models 159

7.1 Convergence of the Binomial Model 159

7.1.1 Distribution Error 160

7.1.2 Non-linearity Error 160

7.2 Moments 161

7.3 Multinomial Models 164

7.4 Trinomial Model 166

7.4.1 Solving Moment-Matching Equations 166

7.4.2 Alternative Trinomial Models 167

7.5 Heptanomial Model 170

7.5.1 Solving Moment-Matching Equations 170

7.5.2 Calculation Time 171

7.6 Further Optimization 172

7.6.1 Smoothing 173

7.6.2 Adaptive Mesh Method 174

7.6.3 Truncation 175

7.6.4 Richardson Extrapolation 175

7.6.5 Bardhan?Derman?Kani?Ergener Correction 175

7.7 Other Refinements 179

7.7.1 Stock Borrowing 179

7.7.2 Cross-Currency 182

7.7.3 Discrete Dividends 184

7.7.4 Transaction Costs 196

7.7.5 Rational Issuers 199

7.7.6 Pricing Dilution 201

7.8 Resets in Multinomial Models 201

7.8.1 Convertible Bond Pricing: Conclusions 203

8 Ascots 207

8.1 Risk Components of a Convertible 207

8.2 Asset Swaps 208

8.2.1 Introduction 208

8.2.2 Credit Risk 211

8.2.3 Closing Out the Swap 212

8.3 Ascots 213

8.3.1 Making the Asset Swap Callable 213

8.3.2 Convertible Asset Swap Package 213

8.3.3 Ascot Features 215

8.3.4 Ascot Term Sheet 216

8.4 Advantages for the Credit Buyer 216

8.5 Advantages for the Ascot Buyer 217

8.5.1 Credit 217

8.5.2 Leverage 218

8.6 Pricing of Ascots 219

8.6.1 Intrinsic Model 219

8.6.2 Option Model 219

8.7 Ascot Greeks 222

8.7.1 Rho 222

8.7.2 Delta 223

8.7.3 Vega 225

8.8 CB Warrants 226

PART III RISK MANAGEMENT AND STRATEGIES 227

9 Measuring the Risk 229

9.1 Portfolio Risk 229

9.2 A Portfolio in Trouble 231

9.3 Risk Categories 238

9.3.1 Market Risk 238

9.3.2 Liquidity Risk 239

9.3.3 Takeover Risk 242

9.3.4 Example: Nokian Tyres 0% 2014 246

9.3.5 Example: Allergan Inc 1.5% 2026 247

9.3.6 Documentation Risk 248

9.3.7 Model Risk 248

9.3.8 Counterparty Risk 249

9.3.9 Operational Risk 249

9.3.10 Regulation Risk 250

9.3.11 Financing Risk 250

9.4 Coherent Risk Measures 251

9.5 Option Greeks 253

9.5.1 Introduction 253

9.5.2 Extended Tree Method 257

9.5.3 Delta 258

9.5.4 Gamma 260

9.5.5 Rho 261

9.5.6 Omicron 263

9.5.7 Vega 265

9.5.8 Volga 266

9.5.9 Epsilon 269

9.5.10 Theta 270

9.6 Fixed Income Measures 272

9.6.1 Duration (Modified) 272

9.6.2 Yields 273

9.7 Cross Greeks 275

9.7.1 Charm 278

9.7.2 Vanna 279

9.8 Speed and Colour 282

9.9 VaR and Beyond 283

9.9.1 VaR Approaches 284

9.9.2 VaR-Related Measures 289

9.9.3 VaR Caveats 291

9.10 Back Testing 292

9.11 Stress Testing 293

10 Dynamic Hedging 295

10.1 Hedge Instruments 295

10.2 Delta Hedging 297

10.2.1 Introduction 297

10.2.2 More than Only Delta 297

10.2.3 Delta Hedge: Neutral, Over- or Under-hedge 299

10.2.4 Delta Caveats 302

10.2.5 Delta and Volatility 302

10.3 Volatility 302

10.3.1 Estimating Historical Volatility 304

10.3.2 Volatility Cone 306

10.3.3 Volatility Surface 308

10.3.4 Term Structure of ?I 309

10.3.5 Volatility Smile of ?I 310

10.3.6 Volsurface Movements 310

10.3.7 At-the-Money Volatility 310

10.4 Gamma Trading 311

10.4.1 Rebalancing the Delta Hedge 312

10.4.2 Dynamic Hedging with Transaction Costs 314

10.4.3 Hedging at What Volatility? 317

10.5 The Variance Swap 324

10.5.1 Introduction 324

10.5.2 Volatility Convexity 326

10.5.3 Spot and Forward Start 327

10.5.4 Mark to Market of the Variance Swap 327

10.5.5 Caveats 328

11 Monte Carlo Techniques for Convertibles 329

11.1 Adding More Realism 329

11.1.1 Introduction 329

11.1.2 Deterministic Volatility 330

11.1.3 Multifactor Models 330

11.2 Monte Carlo Method 334

11.2.1 Introduction 334

11.2.2 Generating Random Paths 336

11.2.3 Errors 338

11.2.4 Variance Reduction 338

11.3 American Monte Carlo 340

11.3.1 Introduction 340

11.3.2 Longstaff and Schwartz Model 343

11.3.3 Example 346

References 363

Index 369

About the Author

Jan De Spiegeleer (Geneva, Switzerland) is Head of RiskManagement at Jabre Capital Partners, a Geneva-based hedge fund. Hedeveloped an extensive knowledge of derivatives pricing, hedgingand trading while working for KBC Financial Products in London,where he was Managing Director of the equity derivatives desk.Prior to his financial career, Jan worked for ten years as anofficer in the Belgian Army, and served in Iraq. Wim Schoutens (Leuven, Belgium) is a research professorin financial engineering in the Department of Mathematics at theCatholic University of Leuven, Belgium. He has extensive practicalexperience of model implementation and is well known for hisconsulting work to the banking industry and other institutions. Wimis the author of Levy Processes in Finance andLevy Processes in Credit Risk, and co-editor ofExotic Option Pricing and Advanced Levy Models allpublished by John Wiley and Sons. He is Managing Editor of theInternational Journal of Theoretical and Applied Finance andAssociate Editor of Mathematical Finance, QuantitativeFinance and Review of Derivatives Research.