Market Risk Analysis: Pricing, Hedging and Trading Financial Instruments (v. 3)

Market Risk Analysis Volume III......Page 4
Contents......Page 10
List of Figures......Page 16
List of Tables......Page 20
List of Examples......Page 22
Foreword......Page 24
Preface to Volume III......Page 28
III.1.1 Introduction......Page 34
III.1.2 Interest Rates......Page 35
III.1.2.1 Continuously Compounded Spot and Forward Rates......Page 36
III.1.2.2 Discretely Compounded Spot Rates......Page 37
III.1.2.4 Spot and Forward Rates with Discrete Compounding......Page 39
III.1.3 Categorization of Bonds......Page 41
III.1.3.1 Categorization by Issuer......Page 42
III.1.4 Characteristics of  Bonds and Interest Rates......Page 43
III.1.4.1 Present Value, Price and Yield......Page 44
III.1.4.2 Relationship between Price and Yield......Page 46
III.1.4.3 Yield Curves......Page 47
III.1.4.4 Behaviour of Market Interest Rates......Page 50
III.1.4.5 Characteristics of Spot and Forward Term Structures......Page 52
III.1.5 Duration and Convexity......Page 53
III.1.5.1 Macaulay Duration......Page 54
III.1.5.2 Modified Duration......Page 56
III.1.5.4 Duration and Convexity of a Bond Portfolio......Page 57
III.1.5.5 Duration–Convexity Approximations to Bond Price Change......Page 58
III.1.5.6 Immunizing Bond Portfolios......Page 60
III.1.6 Bonds with Semi-Annual and Floating Coupons......Page 61
III.1.6.1 Semi-Annual and Quarterly Coupons......Page 62
III.1.6.2 Floating Rate Notes......Page 64
III.1.7 Forward Rate Agreements and Interest Rate Swaps......Page 66
III.1.7.1 Forward Rate Agreements......Page 67
III.1.7.2 Interest Rate Swaps......Page 68
III.1.7.3 Cash Flows on Vanilla Swaps......Page 69
III.1.7.4 Cross-Currency Swaps......Page 71
III.1.7.5 Other Swaps......Page 73
III.1.8.1 PV01 and Value Duration......Page 74
III.1.8.2 Approximations to PV01......Page 77
III.1.8.3 Understanding Interest Rate Risk......Page 78
III.1.9.1 Calibration Instruments......Page 81
III.1.9.2 Bootstrapping......Page 82
III.1.9.3 Splines......Page 84
III.1.9.4 Parametric Models......Page 85
III.1.9.5 Case Study: Statistical Properties of Forward LIBOR Rates......Page 86
III.1.10 Convertible Bonds......Page 92
III.1.10.1 Characteristics of Convertible Bonds......Page 93
III.1.10.2 Survey of Pricing Models for Convertible Bonds......Page 94
III.1.11 Summary and Conclusions......Page 95
III.2.1 Introduction......Page 98
III.2.2.1 Interest Rate and Swap Futures......Page 101
III.2.2.2 Bond Futures......Page 103
III.2.2.3 Currency Futures and Forwards......Page 106
III.2.2.4 Energy and Commodity Futures......Page 107
III.2.2.5 Stock Futures and Index Futures......Page 112
III.2.2.6 Exchange Traded Funds and ETF Futures......Page 113
III.2.2.7 New Futures Markets......Page 115
III.2.3.1 No Arbitrage Pricing......Page 120
III.2.3.2 Accounting for Dividends......Page 121
III.2.3.3 Dividend Risk and Interest Rate Risk......Page 123
III.2.3.4 Currency Forwards and the Interest Rate Differential......Page 124
III.2.3.5 No Arbitrage Prices for Forwards on Bonds......Page 125
III.2.3.6 Commodity Forwards, Carry Costs and Convenience Yields......Page 126
III.2.3.7 Fair Values of Futures and Spot......Page 127
III.2.4.1 No Arbitrage Range......Page 128
III.2.4.2 Correlation between Spot and Futures Returns......Page 130
III.2.4.3 Introducing Basis Risk......Page 131
III.2.4.4 Basis Risk in Commodity Markets......Page 133
III.2.5 Hedging with Forwards and Futures......Page 134
III.2.5.1 Traditional ‘Insurance’ Approach......Page 135
III.2.5.2 Mean–Variance Approach......Page 137
III.2.5.3 Understanding the Minimum Variance Hedge Ratio......Page 139
III.2.5.4 Position Risk......Page 141
III.2.5.5 Proxy Hedging......Page 143
III.2.5.6 Basket Hedging......Page 144
III.2.5.7 Performance Measures for Hedged Portfolios......Page 145
III.2.6.1 Hedging Forex Risk......Page 146
III.2.6.2 Hedging International Stock Portfolios......Page 147
III.2.6.3 Case Study: Hedging an Energy Futures Portfolio......Page 151
III.2.6.4 Hedging Bond Portfolios......Page 157
III.2.7 Using Futures for Short Term Hedging......Page 159
III.2.7.1 Regression Based Minimum Variance Hedge Ratios......Page 160
III.2.7.2 Academic Literature on Minimum Variance Hedging......Page 162
III.2.7.3 Short Term Hedging in Liquid Markets......Page 164
III.2.8 Summary and Conclusions......Page 166
III.3.1 Introduction......Page 170
III.3.2 Foundations......Page 172
III.3.2.1 Arithmetic and Geometric Brownian Motion......Page 173
III.3.2.2 Risk Neutral Valuation......Page 175
III.3.2.3 Numeraire and Measure......Page 177
III.3.2.4 Market Prices and Model Prices......Page 179
III.3.2.5 Parameters and Calibration......Page 180
III.3.2.6 Option Pricing: Review of the Binomial Model......Page 181
III.3.3 Characteristics of Vanilla Options......Page 184
III.3.3.1 Elementary Options......Page 185
III.3.3.2 Put–Call Parity......Page 186
III.3.3.3 Moneyness......Page 187
III.3.3.4 American Options......Page 188
III.3.3.5 Early Exercise Boundary......Page 189
III.3.3.6 Pricing American Options......Page 191
III.3.4.1 Delta......Page 192
III.3.4.3 Other Greeks......Page 194
III.3.4.4 Position Greeks......Page 196
III.3.4.5 Delta–Gamma Hedging......Page 197
III.3.4.6 Delta–Gamma–Vega Hedging......Page 198
III.3.5.1 Bull Strategies......Page 200
III.3.5.2 Bear Strategies......Page 201
III.3.5.3 Other Spread Strategies......Page 202
III.3.5.4 Volatility Strategies......Page 203
III.3.5.5 Replication of P&L Profiles......Page 205
III.3.6 The Black–Scholes–Merton Model......Page 206
III.3.6.1 Assumptions......Page 207
III.3.6.2 Black–Scholes–Merton PDE......Page 208
III.3.6.3 Is the Underlying the Spot or the Futures Contract?......Page 209
III.3.6.4 Black–Scholes–Merton Pricing Formula......Page 211
III.3.6.5 Interpretation of the Black–Scholes–Merton Formula......Page 213
III.3.6.7 Adjusting BSM Prices for Stochastic Volatility......Page 216
III.3.7 The Black–Scholes–Merton Greeks......Page 219
III.3.7.1 Delta......Page 220
III.3.7.2 Theta and Rho......Page 221
III.3.7.3 Gamma......Page 222
III.3.7.4 Vega, Vanna and Volga......Page 223
III.3.7.5 Static Hedges for Standard European Options......Page 226
III.3.8 Interest Rate Options......Page 227
III.3.8.1 Caplets and Floorlets......Page 228
III.3.8.2 Caps, Floors and their Implied Volatilities......Page 229
III.3.8.3 European Swaptions......Page 231
III.3.8.4 Short Rate Models......Page 232
III.3.8.5 LIBOR Model......Page 234
III.3.8.6 Case Study: Application of PCA to LIBOR Model Calibration......Page 236
III.3.9 Pricing Exotic Options......Page 240
III.3.9.1 Pay-offs to Exotic Options......Page 241
III.3.9.2 Exchange Options and Best/Worst of Two Asset Options......Page 242
III.3.9.3 Spread Options......Page 244
III.3.9.4 Currency Protected Options......Page 246
III.3.9.6 Chooser Options and Contingent Options......Page 247
III.3.9.8 Capped Options and Ladder Options......Page 249
III.3.9.9 Look-Back and Look-Forward Options......Page 251
III.3.9.10 Barrier Options......Page 252
III.3.9.11 Asian Options......Page 254
III.3.10 Summary and Conclusions......Page 257
III.4.1 Introduction......Page 260
III.4.2.1 ‘Backing Out’ Implied Volatility from a Market Price......Page 264
III.4.2.2 Equity Index Volatility Skew......Page 266
III.4.2.3 Smiles and Skews in Other Markets......Page 269
III.4.2.4 Term Structures of Implied Volatilities......Page 271
III.4.2.5 Implied Volatility Surfaces......Page 272
III.4.2.6 Cap and Caplet Volatilities......Page 273
III.4.2.7 Swaption Volatilities......Page 275
III.4.3 Local Volatility......Page 276
III.4.3.1 Forward Volatility......Page 277
III.4.3.2 Dupire’s Equation......Page 278
III.4.3.3 Parametric Models of Local Volatility......Page 281
III.4.3.4 Lognormal Mixture Diffusion......Page 282
III.4.4.1 Sticky Models......Page 288
III.4.4.2 Case Study I: Principal Component Analysis of Implied Volatilities......Page 290
III.4.4.3 Case Study II: Modelling the ATM Volatility–Index Relationship......Page 294
III.4.4.4 Case Study III: Modelling the Skew Sensitivities......Page 297
III.4.4.5 Applications of Implied Volatility Dynamics to Hedging Options......Page 298
III.4.5 Stochastic Volatility Models......Page 301
III.4.5.1 Stochastic Volatility PDE......Page 302
III.4.5.2 Properties of Stochastic Volatility......Page 304
III.4.5.3 Model Implied Volatility Surface......Page 308
III.4.5.4 Model Local Volatility Surface......Page 310
III.4.5.5 Heston Model......Page 311
III.4.5.6 GARCH Diffusions......Page 313
III.4.5.7 CEV and SABR Models......Page 318
III.4.5.8 Jumps in Prices and in Stochastic Volatility......Page 320
III.4.6 Scale Invariance and Hedging......Page 322
III.4.6.2 Definition of Scale Invariance......Page 324
III.4.6.3 Scale Invariance and Homogeneity......Page 325
III.4.6.4 Model Free Price Hedge Ratios......Page 327
III.4.6.5 Minimum Variance Hedging......Page 330
III.4.6.6 Minimum Variance Hedge Ratios in Specific Models......Page 332
III.4.6.7 Empirical Results......Page 333
III.4.7 Trading Volatility......Page 336
III.4.7.1 Variance Swaps and Volatility Swaps......Page 337
III.4.7.2 Trading Forward Volatility......Page 339
III.4.7.3 Variance Risk Premium......Page 340
III.4.7.4 Construction of a Volatility Index......Page 341
III.4.7.6 Term Structures of Volatility Indices......Page 342
III.4.7.7 Vix and Other Volatility Indices......Page 344
III.4.7.8 Volatility Index Futures......Page 345
III.4.7.9 Options on Volatility Indices......Page 347
III.4.7.10 Using Realized Volatility Forecasts to Trade Volatility......Page 348
III.4.8 Summary and Conclusion......Page 349
III.5.1 Introduction......Page 354
III.5.2.1 Interest Rate Sensitive Portfolios......Page 356
III.5.2.2 Equity Portfolios......Page 357
III.5.2.3 International Exposures......Page 360
III.5.2.5 Options Portfolios......Page 361
III.5.2.7 Nominal versus Percentage Risk Factors and Sensitivities......Page 363
III.5.3.1 Present Value Invariant and Duration Invariant Maps......Page 365
III.5.3.2 PV01 Invariant Cash Flow Maps......Page 366
III.5.3.3 Volatility Invariant Maps......Page 367
III.5.3.4 Complex Cash Flow Maps......Page 369
III.5.4.1 Risk Management of Interest Rate Sensitive Portfolios......Page 370
III.5.4.2 Mapping Portfolios of Commodity Futures......Page 371
III.5.5 Mapping an Options Portfolio to Price Risk Factors......Page 373
III.5.5.1 Taylor Expansions......Page 374
III.5.5.2 Value Delta and Value Gamma......Page 375
III.5.5.3 Delta–Gamma Approximation: Single Underlying......Page 377
III.5.5.4 Effect of Gamma on Portfolio Risk......Page 379
III.5.5.5 Price Beta Mapping......Page 380
III.5.5.6 Delta–Gamma Approximation: Several Underlyings......Page 382
III.5.5.7 Including Time and Interest Rates Sensitivities......Page 384
III.5.6.1 Vega Risk in Options Portfolios......Page 386
III.5.6.2 Second Order Approximations: Vanna and Volga......Page 387
III.5.6.3 Vega Bucketing......Page 388
III.5.6.4 Volatility Beta Mapping......Page 389
III.5.7.1 Estimating the Volatility Betas......Page 390
III.5.7.2 Model Risk of Volatility Mapping......Page 393
III.5.7.4 Using PCA with Volatility Betas......Page 394
III.5.8 Summary and Conclusions......Page 397
References......Page 400
Index......Page 410
Plates......Page 420