FX Options and Smile Risk_. Antonio Castagna. )WILEY A John Wiley and Sons, Ltd., Publication

Transcription

1 FX Options and Smile Risk_ Antonio Castagna )WILEY A John Wiley and Sons, Ltd., Publication

2 Preface Notation and Acronyms IX xiii 1 The FX Market 1.1 FX rates and spot contracts 1.2 Outright and FX swap contracts 1.3 FX option contracts Exercise Expiry date and settlement date Premium Market standard practices for quoting options 1.4 Main traded FX option structures Pricing Models for FX Options 2.1 Principles of option pricing theory The Black-Scholes economy Stochastic volatility economy Change of numeraire 2.2 The black-scholes model The forward price to use in the formula BS greeks Retrieving implied volatility and strike Some relationships of the BS formula 2.3 ' The Heston Model Time-dependent parameters in the Heston model 2.4 The SABR model 2.5 The mixture approach The LMLV model The LMUV model

3 vi Contents Features of the LMLV and LMUV models and a comparison between them Extension of the LMUV model Some considerations about the choice of model 53 Dynamic Hedging and Volatility Trading Preliminary considerations A general framework Hedging with a constant implied volatility Hedging with an updating implied volatility A market model for the implied volatility Hedging Vega Hedging Delta, Vega, Vanna and Volga Vanna-Volga hedging with one implied volatility Vanna-Volga hedging with different implied volatilities The volatility smile and its phenomenology Local exposures to the volatility smile Retrieving the strikes of the main structures ATM straddle exposures Risk reversal exposures Vega-weighted butterfly exposures Scenario hedging and its relationship with Vanna-Volga hedging Scenario hedging with constant Delta options 86 The Volatility Surface General definitions Arbitrage opportunities under the three different rules Criteria for an efficient and convenient representation of the volatility surface Commonly adopted approaches to building a volatility surface Smile interpolation among strikes: the Vanna-Volga approach The Vanna-Volga approach: general setting Computing the Vanna-Volga weights and option prices Limit and no-arbitrage conditions Approximating implied volatilities Some features of the Vanna-Volga approach Hedging error for longer expiries The implied risk-neutral density and smile asymptotics Two consistency results An alternative characterization of the Vanna-Volga approach Smile interpolation among expiries: implied volatility term structure Admissible volatility surfaces Taking into account the market butterfly Building the volatility matrix in practice 120

4 vii 5 Plain Vanilla Options Pricing of plain vanilla options Delayed settlement date Cash settlement Market-making tools Inferring the implied volatility for a given strike Inferring the implied volatility for a given Delta Quoting the Vega-weighted butterfly and the risk reversal Bid/ask spreads for plain vanilla options Cutoff times and spreads Digital options Digital options pricing: the static replica approach Digital options pricing in specific model settings Delayed cash settlement date Bid/ask spreads Quotation conventions American plain vanilla options Valuation of American plain vanilla options in a BS setting Pricing of American plain vanilla options with the volatility smile Barrier Options A taxonomy of barrier options Some relationships of barrier option prices Pricing for barrier options in a BS economy The diffusion equation under single absorbing boundaries Dealing with a constant barrier Pricing formulae for barrier options One-touch (rebate) and no-touch options Double-barrier options Two absorbing states Pricing formula for double-barrier options Double-no-touch and double-touch options Probability of hitting a barrier Greek calculation Pricing barrier options in other model settings Pricing barriers with non-standard delivery Delayed settlement date Cash settlement Market approach to pricing barrier options Inclusion of the smile: the Vanna-Volga approach for barrier options The Vanna-Volga approach for barrier options: variations on the theme Slippage at the barrier level Delta-hedging near the barrier level Implicit one-touch and gearing Vega-hedge rebalancing 186

5 6.13 Bid/ask spreads Monitoring frequency Other Exotic Options Introduction At-expiry barrier options Window barrier options First-then and knock-in-knock-out barrier options Auto-quanto options Forward start options Including the volatility smile in the pricing Forward implied volatility smiles Forward start barrier and bet options Dealing with notional amounts expressed in numeraire currency Variance swaps Compound, Asian and lookback options 5 8 Risk Management Tools and Analysis Introduction Implementation of the LMUV model The forward volatility surfaces Calculating the sensitivity to the movements of the volatility surface Risk monitoring tools FX spot rate-related Greeks Cash-settled options Volatility-related Greeks and sensitivities Barrier implicit one-touch, bets and digitals Interest rate-related Greeks Risk analysis of plain vanilla options ATM straddle Risk reversal Vega-weighted butterfly Risk analysis of digital options Risk analysis of exotic options Barrier options Double barrier options Bet options Correlation and FX Options Preliminary considerations Correlation in the BS setting Contracts depending on several FX spot rates Dealing with correlation and volatility smile 278? Vanna-Volga extension Linking volatility smiles 283 References 287 Index 291