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 and 30 case studies many of which are contained in interactive Excel spreadsheets available from the the accompanying CD-ROM . Empirical examples and case studies specific to this volume include:

• 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.

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