The FX options market represents one of the most liquid and strongly competitive markets in the world, and features many technical subtleties that can seriously harm the uninformed and unaware trader. This book is a unique guide to running an FX options book from the market maker perspective. Striking a balance between mathematical rigour and market practice and written by experienced practitioner Antonio Castagna, the book shows readers how to correctly build an entire volatility surface from the market prices of the main structures.
Starting with the basic conventions related to the main FX deals and the basic traded structures of FX options, the book gradually introduces the main tools to cope with the FX volatility risk. It then goes on to review the main concepts of option pricing theory and their application within a Black-Scholes economy and a stochastic volatility environment. The book also introduces models that can be implemented to price and manage FX options before examining the effects of volatility on the profits and losses arising from the hedging activity.
Coverage includes:
- how the Black-Scholes model is used in professional trading activity
- the most suitable stochastic volatility models
- sources of profit and loss from the Delta and volatility hedging activity
- fundamental concepts of smile hedging
- major market approaches and variations of the Vanna-Volga method
- volatility-related Greeks in the Black-Scholes model
- pricing of plain vanilla options, digital options, barrier options and the less well known exotic options
- tools for monitoring the main risks of an FX options’ book
The book is accompanied by a CD Rom featuring models in VBA, demonstrating many of the approaches described in the book.
Table of Contents:
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 Black–Scholes economy 21
2.1.2 Stochastic volatility economy 26
2.1.3 Change of numeraire 27
2.2 The black–scholes 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 Vanna–Volga hedging with one implied volatility 71
3.6.2 Vanna–Volga 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 Vanna–Volga 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 Vanna–Volga approach 97
4.4.1 The Vanna–Volga approach: general setting 97
4.4.2 Computing the Vanna–Volga 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 Vanna–Volga 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 Vanna–Volga 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 Vanna–Volga approach for barrier options 171
6.12.2 The Vanna–Volga 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 First–then and knock-in–knock-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 Vanna–Volga extension 278
9.5 Linking volatility smiles 283
References 287
Index 291
About the Author :
Antonio Castagna is currently partner and co-founder of the consulting company Iason ltd, providing support to financial institutions for the design of models to price complex derivatives and to measure a wide range of risks, including credit and liquidity. Antonio graduated in Finance from LUISS University, Rome, in 1995 with a thesis on American options and the numerical procedures for their valuation. He began his career in investment banking in IMI Bank, Luxemborug, as a financial analyst in the Risk Control Department before moving to Banca IMI, Milan, first as a market maker of cap/floors and swaptions, before setting up the FX options desk and running the book of plain vanilla and exotic options on the major currencies, whilst also being responsible for the entire FX volatility trading.
Antonio has written a number of papers on credit derivatives, managing of exotic options risks and volatility smiles. He is often invited to academic and post-graduate courses.
