Market Risk Analysis, Pricing, Hedging and Trading Financial Instruments

Carol Alexander is a Professor of Risk Management at the ICMA Centre, University of Reading, and Chair of the Academic Advisory Council of the Professional Risk Manager's International Association (PRMIA). She is the author of Market Models: A Guide to Financial Data Analysis(John Wiley & Sons Ltd, 2001) and has been editor and contributor of a very large number of books in finance and mathematics, including the multi-volume Professional Risk Manager's Handbook(McGraw-Hill, 2008 and PRMIA Publications). Carol has published nearly 100 academic journal articles, book chapters and books, the majority of which focus on financial risk management and mathematical finance. Professor Alexander is one of the world's leading authorities on market risk analysis. For further details, see www.icmacentre.rdg.ac.uk/alexander

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

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