Preface ix
Notation and Acronyms xiii
1 The FX Market 1
1.1 FX rates and spot contracts 1
1.2 Outright and FX swap contracts 4
1.3 FX option contracts 10
1.3.1 Exercise 11
1.3.2 Expiry date and settlement date 11
1.3.3 Premium 13
1.3.4 Market standard practices for quoting options 14
1.4 Main traded FX option structures 16
2 Pricing Models for FX Options 21
2.1 Principles of option pricing theory 21
2.1.1 The BlackScholes economy 21
2.1.2 Stochastic volatility economy 26
2.1.3 Change of numeraire 27
2.2 The blackscholes model 29
2.2.1 The forward price to use in the formula 30
2.2.2 BS greeks 31
2.2.3 Retrieving implied volatility and strike 35
2.2.4 Some relationships of the BS formula 38
2.3 The Heston Model 41
2.3.1 Time-dependent parameters in the Heston model 42
2.4 The SABR model 44
2.5 The mixture approach 45
2.5.1 The LMLV model 45
2.5.2 The LMUV model 48
2.5.3 Features of the LMLV and LMUV models and a comparison between them 50
2.5.4 Extension of the LMUV model 51
2.6 Some considerations about the choice of model 53
3 Dynamic Hedging and Volatility Trading 57
3.1 Preliminary considerations 57
3.2 A general framework 59
3.3 Hedging with a constant implied volatility 61
3.4 Hedging with an updating implied volatility 63
3.4.1 A market model for the implied volatility 66
3.5 Hedging Vega 68
3.6 Hedging Delta, Vega, Vanna and Volga 70
3.6.1 VannaVolga hedging with one implied volatility 71
3.6.2 VannaVolga hedging with different implied volatilities 71
3.7 The volatility smile and its phenomenology 75
3.8 Local exposures to the volatility smile 79
3.8.1 Retrieving the strikes of the main structures 79
3.8.2 ATM straddle exposures 81
3.8.3 Risk reversal exposures 81
3.8.4 Vega-weighted butterfly exposures 83
3.9 Scenario hedging and its relationship with VannaVolga hedging 84
3.9.1 Scenario hedging with constant Delta options 86
4 The Volatility Surface 91
4.1 General definitions 91
4.1.1 Arbitrage opportunities under the three different rules 92
4.2 Criteria for an efficient and convenient representation of the volatility surface 94
4.3 Commonly adopted approaches to building a volatility surface 96
4.4 Smile interpolation among strikes: the VannaVolga approach 97
4.4.1 The VannaVolga approach: general setting 97
4.4.2 Computing the VannaVolga weights and option prices 99
4.4.3 Limit and no-arbitrage conditions 102
4.4.4 Approximating implied volatilities 102
4.5 Some features of the VannaVolga approach 104
4.5.1 Hedging error for longer expiries 105
4.5.2 The implied risk-neutral density and smile asymptotics 106
4.5.3 Two consistency results 108
4.6 An alternative characterization of the VannaVolga approach 110
4.7 Smile interpolation among expiries: implied volatility term structure 112
4.8 Admissible volatility surfaces 115
4.9 Taking into account the market butterfly 116
4.10 Building the volatility matrix in practice 120
5 Plain Vanilla Options 131
5.1 Pricing of plain vanilla options 131
5.1.1 Delayed settlement date 131
5.1.2 Cash settlement 133
5.2 Market-making tools 134
5.2.1 Inferring the implied volatility for a given strike 134
5.2.2 Inferring the implied volatility for a given Delta 135
5.2.3 Quoting the Vega-weighted butterfly and the risk reversal 136
5.3 Bid/ask spreads for plain vanilla options 139
5.4 Cutoff times and spreads 141
5.5 Digital options 142
5.5.1 Digital options pricing: the static replica approach 143
5.5.2 Digital options pricing in specific model settings 148
5.5.3 Delayed cash settlement date 150
5.5.4 Bid/ask spreads 150
5.5.5 Quotation conventions 152
5.6 American plain vanilla options 152
5.6.1 Valuation of American plain vanilla options in a BS setting 152
5.6.2 Pricing of American plain vanilla options with the volatility smile 153
6 Barrier Options 155
6.1 A taxonomy of barrier options 155
6.2 Some relationships of barrier option prices 156
6.3 Pricing for barrier options in a BS economy 157
6.3.1 The diffusion equation under single absorbing boundaries 158
6.3.2 Dealing with a constant barrier 159
6.4 Pricing formulae for barrier options 160
6.5 One-touch (rebate) and no-touch options 162
6.6 Double-barrier options 164
6.6.1 Two absorbing states 164
6.6.2 Pricing formula for double-barrier options 165
6.7 Double-no-touch and double-touch options 167
6.8 Probability of hitting a barrier 167
6.9 Greek calculation 168
6.10 Pricing barrier options in other model settings 169
6.11 Pricing barriers with non-standard delivery 170
6.11.1 Delayed settlement date 170
6.11.2 Cash settlement 170
6.12 Market approach to pricing barrier options 171
6.12.1 Inclusion of the smile: the VannaVolga approach for barrier options 171
6.12.2 The VannaVolga approach for barrier options: variations on the theme 177
6.12.3 Slippage at the barrier level 181
6.12.4 Delta-hedging near the barrier level 183
6.12.5 Implicit one-touch and gearing 184
6.12.6 Vega-hedge rebalancing 186
6.13 Bid/ask spreads 188
6.14 Monitoring frequency 191
7 Other Exotic Options 195
7.1 Introduction 195
7.2 At-expiry barrier options 195
7.3 Window barrier options 197
7.4 Firstthen and knock-inknock-out barrier options 199
7.5 Auto-quanto options 202
7.6 Forward start options 204
7.6.1 Including the volatility smile in the pricing 207
7.6.2 Forward implied volatility smiles 210
7.6.3 Forward start barrier and bet options 210
7.6.4 Dealing with notional amounts expressed in numeraire currency 211
7.7 Variance swaps 212
7.8 Compound, Asian and lookback options 215
8 Risk Management Tools and Analysis 217
8.1 Introduction 217
8.2 Implementation of the LMUV model 217
8.2.1 The forward volatility surfaces 221
8.2.2 Calculating the sensitivity to the movements of the volatility surface 223
8.3 Risk monitoring tools 227
8.3.1 FX spot rate-related Greeks 227
8.3.2 Cash-settled options 229
8.3.3 Volatility-related Greeks and sensitivities 229
8.3.4 Barrier implicit one-touch, bets and digitals 231
8.3.5 Interest rate-related Greeks 234
8.4 Risk analysis of plain vanilla options 236
8.4.1 ATM straddle 236
8.4.2 Risk reversal 239
8.4.3 Vega-weighted butterfly 241
8.5 Risk analysis of digital options 244
8.6 Risk analysis of exotic options 249
8.6.1 Barrier options 249
8.6.2 Double barrier options 258
8.6.3 Bet options 262
9 Correlation and FX Options 269
9.1 Preliminary considerations 269
9.2 Correlation in the BS setting 269
9.3 Contracts depending on several FX spot rates 275
9.4 Dealing with correlation and volatility smile 278
9.4.1 VannaVolga extension 278
9.5 Linking volatility smiles 283
References 287
Index 291
