List of Figures xiii List of Tables xvii List of Examples xix Foreword xxi Preface to Volume III xxv III. 1 Bonds and Swaps 1 III.1.1 Introduction 1 III.1.2 Interest Rates 2 III.1.2.1 Continuously Compounded Spot and Forward Rates 3 III.1.2.2 Discretely Compounded Spot Rates 4 III.1.2.3Translation between Discrete Rates and Continuous Rates 6 III.1.2.4 Spot and Forward Rates with Discrete Compounding 6 III.1.2.5 LIBOR 8 III.1.3 Categorization of Bonds 8 III.1.3.1 Categorization by Issuer 9 III.1.3.2 Categorization by Coupon and Maturity 10 III.1.4 Characteristics of Bonds and Interest Rates 10 III.1.4.1 Present Value, Price and Yield 11 III.1.4.2 Relationship between Price and Yield 13 III.1.4.3 Yield Curves 14 III.1.4.4 Behaviour of Market Interest Rates 17 III.1.4.5 Characteristics of Spot and Forward Term Structures 19 III.1.5 Duration and Convexity 20 III.1.5.1 Macaulay Duration 21 III.1.5.2 Modified Duration 23 III.1.5.3 Convexity 24 III.1.5.4 Duration and Convexity of a Bond Portfolio 24 III.1.5.5 Duration–Convexity Approximations to Bond Price Change 25 III.1.5.6 Immunizing Bond Portfolios 26 III.1.6 Bonds with Semi-Annual and Floating Coupons 28 III.1.6.1 Semi-Annual and Quarterly Coupons 29 III.1.6.2 Floating Rate Notes 31 III.1.6.3 Other Floaters 33 III.1.7 Forward Rate Agreements and Interest Rate Swaps 33 III.1.7.1 Forward Rate Agreements 34 III.1.7.2 Interest Rate Swaps 35 III.1.7.3 Cash Flows on Vanilla Swaps 36 III.1.7.4 Cross-Currency Swaps 38 III.1.7.5 Other Swaps 40 III.1.8 Present Value of a Basis Point 41 III.1.8.1 PV01 and Value Duration 41 III.1.8.2 Approximations to PV 01 44 III.1.8.3 Understanding Interest Rate Risk 45 III.1.9 Yield Curve Fitting 48 III.1.9.1 Calibration Instruments 48 III.1.9.2 Bootstrapping 49 III.1.9.3 Splines 51 III.1.9.4 Parametric Models 52 III.1.9.5 Case Study: Statistical Properties of Forward LIBOR Rates 53 III.1.10 Convertible Bonds 59 III.1.10.1 Characteristics of Convertible Bonds 60 III.1.10.2 Survey of Pricing Models for Convertible Bonds 61 III.1.11 Summary and Conclusions 62 III. 2 Futures and Forwards 65 III.2.1 Introduction 65 III.2.2 Characteristics of Futures and Forwards 68 III.2.2.1 Interest Rate and Swap Futures 68 III 2.2.2 Bond Futures 70 III.2.2.3 Currency Futures and Forwards 73 III.2.2.4 Energy and Commodity Futures 74 III.2.2.5 Stock Futures and Index Futures 79 III.2.2.6 Exchange Traded Funds and ETF Futures 80 III.2.2.7 New Futures Markets 82 III.2.3 Theoretical Relationships between Spot, Forward and Futures 87 III.2.3.1 No Arbitrage Pricing 87 III.2.3.2 Accounting for Dividends 88 III.2.3.3 Dividend Risk and Interest Rate Risk 90 III.2.3.4 Currency Forwards and the Interest Rate Differential 91 III.2.3.5 No Arbitrage Prices for Forwards on Bonds 92 III.2.3.6 Commodity Forwards, Carry Costs and Convenience Yields 93 III.2.3.7 Fair Values of Futures and Spot 94 III.2.4 The Basis 95 III.2.4.1 No Arbitrage Range 95 III.2.4.2 Correlation between Spot and Futures Returns 97 III.2.4.3 Introducing Basis Risk 98 III.2.4.4 Basis Risk in Commodity Markets 100 III.2.5 Hedging with Forwards and Futures 101 III.2.5.1 Traditional ‘Insurance’ Approach 102 III.2.5.2 Mean–Variance Approach 104 III.2.5.3 Understanding the Minimum Variance Hedge Ratio 106 III.2.5.4 Position Risk 108 III.2.5.5 Proxy Hedging 110 III.2.5.6 Basket Hedging 111 III.2.5.7 Performance Measures for Hedged Portfolios 112 III.2.6 Hedging in Practice 113 III.2.6.1 Hedging Forex Risk 113 III.2.6.2 Hedging International Stock Portfolios 114 III.2.6.3 Case Study: Hedging an Energy Futures Portfolio 118 III.2.6.4 Hedging Bond Portfolios 124 III.2.7 Using Futures for Short Term Hedging 126 III.2.7.1 Regression Based Minimum Variance Hedge Ratios 127 III.2.7.2 Academic Literature on Minimum Variance Hedging 129 III.2.7.3 Short Term Hedging in Liquid Markets 131 III.2.8 Summary and Conclusions 133 III. 3 Options 137 III.3.1 Introduction 137 III.3.2 Foundations 139 III.3.2.1 Arithmetic and Geometric Brownian Motion 140 III.3.2.2 Risk Neutral Valuation 142 III.3.2.3 Numeraire and Measure 144 III.3.2.4 Market Prices and Model Prices 146 III.3.2.5 Parameters and Calibration 147 III.3.2.6 Option Pricing: Review of the Binomial Model 148 III.3.3 Characteristics of Vanilla Options 151 III.3.3.1 Elementary Options 152 III.3.3.2 Put–Call Parity 153 III 3.3.3 Moneyness 154 III.3.3.4 American Options 155 III.3.3.5 Early Exercise Boundary 156 III.3.3.6 Pricing American Options 158 III.3.4 Hedging Options 159 III.3.4.1 Delta 159 III.3.4.2 Delta Hedging 161 III.3.4.3 Other Greeks 161 III.3.4.4 Position Greeks 163 III.3.4.5 Delta–Gamma Hedging 164 III.3.4.6 Delta–Gamma–Vega Hedging 165 III.3.5 Trading Options 167 III.3.5.1 Bull Strategies 167 III.3.5.2 Bear Strategies 168 III.3.5.3 Other Spread Strategies 169 III.3.5.4 Volatility Strategies 170 III.3.5.5 Replication of P&L Profiles 172 III.3.6 The Black–Scholes–Merton Model 173 III.3.6.1 Assumptions 174 III.3.6.2 Black–Scholes–Merton PDE 175 III.3.6.3 Is the Underlying the Spot or the Futures Contract? 176 III.3.6.4 Black–Scholes–Merton Pricing Formula 178 III.3.6.5 Interpretation of the Black–Scholes–Merton Formula 180 III.3.6.6 Implied Volatility 183 III.3.6.7 Adjusting BSM Prices for Stochastic Volatility 183 III.3.7 The Black–Scholes–Merton Greeks 186 III.3.7.1 Delta 187 III.3.7.2 Theta and Rho 188 III.3.7.3 Gamma 189 III.3.7.4 Vega, Vanna and Volga 190 III.3.7.5 Static Hedges for Standard European Options 193 III.3.8 Interest Rate Options 194 III.3.8.1 Caplets and Floorlets 195 III.3.8.2 Caps, Floors and their Implied Volatilities 196 III.3.8.3 European Swaptions 198 III.3.8.4 Short Rate Models 199 III.3.8.5 LIBOR Model 201 III.3.8.6 Case Study: Application of PCA to LIBOR Model Calibration 203 III.3.9 Pricing Exotic Options 207 III.3.9.1 Pay-offs to Exotic Options 208 III.3.9.2 Exchange Options and Best/Worst of Two Asset Options 209 III.3.9.3 Spread Options 211 III.3.9.4 Currency Protected Options 213 III.3.9.5 Power Options 214 III.3.9.6 Chooser Options and Contingent Options 214 III.3.9.7 Compound Options 216 III.3.9.8 Capped Options and Ladder Options 216 III.3.3.9 Look-Back and Look-Forward Options 218 III.3.9.10 Barrier Options 219 III.3.9.11 Asian Options 221 III.3.10 Summary and Conclusions 224 III. 4 Volatility 227 III.4. 1 Introduction 227 III.4. 2 Implied Volatility 231 III.4.2.1 ‘Backing Out’ Implied Volatility from a Market Price 231 III.4.2.2 Equity Index Volatility Skew 233 III.4.2.3 Smiles and Skews in Other Markets 236 III.4.2.4 Term Structures of Implied Volatilities 238 III.4.2.5 Implied Volatility Surfaces 239 III.4.2.6 Cap and Caplet Volatilities 240 III.4.2.7 Swaption Volatilities 242 III.4.3 Local Volatility 243 III.4.3.1 Forward Volatility 244 III.4.3.2 Dupire’s Equation 245 III.4.3.3 Parametric Models of Local Volatility 248 III.4.3.4 Lognormal Mixture Diffusion 249 III.4.4 Modelling the Dynamics of Implied Volatility 255 III.4.4.1 Sticky Models 255 III.4.4.2 Case Study I: Principal Component Analysis of Implied Volatilities 257 III.4.4.3 Case Study II: Modelling the ATM Volatility–Index Relationship 261 III 4.4.4 Case Study III: Modelling the Skew Sensitivities 264 III.4.4.5 Applications of Implied Volatility Dynamics to Hedging Options 265 III.4. 5 Stochastic Volatility Models 268 III.4.5. 1 Stochastic Volatility PDE 269 III.4.5. 2 Properties of Stochastic Volatility 271 III.4.5. 3 Model Implied Volatility Surface 275 III.4.5. 4 Model Local Volatility Surface 277 III.4.5. 5 Heston Model 278 III.4.5. 6 GARCH Diffusions 280 III.4.5. 7 CEV and SABR Models 285 III.4.5. 8 Jumps in Prices and in Stochastic Volatility 287 III.4. 6 Scale Invariance and Hedging 289 III.4.6. 1 Scale Invariance and Change of Numeraire 291 III.4.6. 2 Definition of Scale Invariance 291 III.4.6. 3 Scale Invariance and Homogeneity 292 III.4.6. 4 Model Free Price Hedge Ratios 294 III.4.6. 5 Minimum Variance Hedging 297 III.4.6. 6 Minimum Variance Hedge Ratios in Specific Models 299 III.4.6. 7 Empirical Results 300 III.4. 7 Trading Volatility 303 III.4.7. 1 Variance Swaps and Volatility Swaps 304 III.4.7. 2 Trading Forward Volatility 306 III.4.7. 3 Variance Risk Premium 307 III.4.7. 4 Construction of a Volatility Index 308 III.4.7. 5 Effect of the Skew 309 III.4.7. 6 Term Structures of Volatility Indices 309 III.4.7. 7 Vix and Other Volatility Indices 311 III.4.7. 8 Volatility Index Futures 312 III.4.7. 9 Options on Volatility Indices 314 III.4.7.10 Using Realized Volatility Forecasts to Trade Volatility 315 III.4. 8 Summary and Conclusion 316 III. 5 Portfolio Mapping 321 III.5. 1 Introduction 321 III.5. 2 Risk Factors and Risk Factor Sensitivities 323 III.5.2. 1 Interest Rate Sensitive Portfolios 323 III.5.2. 2 Equity Portfolios 324 III.5.2. 3 International Exposures 327 III.5.2. 4 Commodity Portfolios 328 III.5.2. 5 Option Portfolios 328 III.5.2. 6 Orthogonalization of Risk Factors 330 III.5.2. 7 Nominal versus Percentage Risk Factors and Sensitivities 330 III.5. 3 Cash Flow Mapping 332 III.5.3. 1 Present Value Invariant and Duration Invariant Maps 332 III.5.3. 2 PV01 Invariant Cash Flow Maps 333 III.5.3. 3 Volatility Invariant Maps 334 III.5.3. 4 Complex Cash Flow Maps 336 III.5. 4 Applications of Cash Flow Mapping to Market Risk Management 337 III.5.4. 1 Risk Management of Interest Rate Sensitive Portfolios 337 III.5.4. 2 Mapping Portfolios of Commodity Futures 338 III.5. 5 Mapping an Option Portfolio to Price Risk Factors 340 III.5.5. 1 Taylor Expansions 341 III.5.5. 2 Value Delta and Value Gamma 342 III.5.5. 3 Delta–Gamma Approximation: Single Underlying 344 III.5.5. 4 Effect of Gamma on Portfolio Risk 346 III 5 Price Beta Mapping 347 III.5.5. 6 Delta–Gamma Approximation: Several Underlyings 349 III.5.5. 7 Including Time and Interest Rates Sensitivities 351 III.5. 6 Mapping Implied Volatility 353 III.5.6. 1 Vega Risk in Option Portfolios 353 III.5.6. 2 Second Order Approximations: Vanna and Volga 354 III.5.6. 3 Vega Bucketing 355 III.5.6. 4 Volatility Beta Mapping 356 III.5. 7 Case Study: Volatility Risk in FTSE 100 Options 357 III.5.7. 1 Estimating the Volatility Betas 357 III.5.7. 2 Model Risk of Volatility Mapping 360 III.5.7. 3 Mapping to Term Structures of Volatility Indices 361 III.5.7. 4 Using PCA with Volatility Betas 361 III.5. 8 Summary and Conclusions 364 References 367 Index 377

Written by leading market risk academic, Professor Carol Alexander, Pricing, Hedging and Trading Financial Instruments forms part three of the Market Risk Analysis four volume set. This book is an in-depth, practical and accessible guide to the models that are used for pricing and the strategies that are used for hedging financial instruments, and to the markets in which they trade. It provides a comprehensive, rigorous and accessible introduction to bonds, swaps, futures and forwards and options, including variance swaps, volatility indices and their futures and options, to stochastic volatility models and to modelling the implied and local volatility surfaces. All together, the MARKET RISK ANALYSIS four volume set illustrates virtually every concept or formula with a practical, numerical example or a longer, empirical case study. Across all four volumes there are approximately 300 numerical and empirical examples, 400 graphs and figures 30 case studies many of which are contained in interactive Excel spreadsheets available from the accompanying CD-ROM. In this volume alone there are over 200 spreadsheets in 25 workbooks. Here are just some of he illustrative empirical examples and case studies in this volume: Duration-Convexity approximation to bond portfolios, and portfolio immunization;Pricing floaters and vanilla, basis and variance swaps;Coupon stripping and yield curve fitting;Proxy hedging, and hedging international securities and energy futures portfolios;Pricing models for European exotics, including barriers, Asians, look-backs, choosers, capped, contingent, power, quanto, compo, exchange, ‘best-of’ and spread options;Libor model calibration;Dynamic models for implied volatility based on principal component analysis;Calibration of stochastic volatility models (Matlab code);Simulations from stochastic volatility and jump models;Duration, PV01 and volatility invariant cash flow mappings;Delta-gamma-theta-vega mappings for options portfolios;Volatility beta mapping to volatility indices.