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.

• Preface ixNotation and Acronyms xiii1 The FX Market 11.1 FX rates and spot contracts 11.2 Outright and FX swap contracts 41.3 FX option contracts 101.3.1 Exercise 111.3.2 Expiry date and settlement date 111.3.3 Premium 131.3.4 Market standard practices for quoting options 141.4 Main traded FX option structures 162 Pricing Models for FX Options 212.1 Principles of option pricing theory 212.1.1 The Black–Scholes economy 212.1.2 Stochastic volatility economy 262.1.3 Change of numeraire 272.2 The black–scholes model 292.2.1 The forward price to use in the formula 302.2.2 BS greeks 312.2.3 Retrieving implied volatility and strike 352.2.4 Some relationships of the BS formula 382.3 The Heston Model 412.3.1 Time-dependent parameters in the Heston model 422.4 The SABR model 442.5 The mixture approach 452.5.1 The LMLV model 452.5.2 The LMUV model 482.5.3 Features of the LMLV and LMUV models and a comparison between them 502.5.4 Extension of the LMUV model 512.6 Some considerations about the choice of model 533 Dynamic Hedging and Volatility Trading 573.1 Preliminary considerations 573.2 A general framework 593.3 Hedging with a constant implied volatility 613.4 Hedging with an updating implied volatility 633.4.1 A market model for the implied volatility 663.5 Hedging Vega 683.6 Hedging Delta, Vega, Vanna and Volga 703.6.1 Vanna–Volga hedging with one implied volatility 713.6.2 Vanna–Volga hedging with different implied volatilities 713.7 The volatility smile and its phenomenology 753.8 Local exposures to the volatility smile 793.8.1 Retrieving the strikes of the main structures 793.8.2 ATM straddle exposures 813.8.3 Risk reversal exposures 813.8.4 Vega-weighted butterfly exposures 833.9 Scenario hedging and its relationship with Vanna–Volga hedging 843.9.1 Scenario hedging with constant Delta options 864 The Volatility Surface 914.1 General definitions 914.1.1 Arbitrage opportunities under the three different rules 924.2 Criteria for an efficient and convenient representation of the volatility surface 944.3 Commonly adopted approaches to building a volatility surface 964.4 Smile interpolation among strikes: the Vanna–Volga approach 974.4.1 The Vanna–Volga approach: general setting 974.4.2 Computing the Vanna–Volga weights and option prices 994.4.3 Limit and no-arbitrage conditions 1024.4.4 Approximating implied volatilities 1024.5 Some features of the Vanna–Volga approach 1044.5.1 Hedging error for longer expiries 1054.5.2 The implied risk-neutral density and smile asymptotics 1064.5.3 Two consistency results 1084.6 An alternative characterization of the Vanna–Volga approach 1104.7 Smile interpolation among expiries: implied volatility term structure 1124.8 Admissible volatility surfaces 1154.9 Taking into account the market butterfly 1164.10 Building the volatility matrix in practice 1205 Plain Vanilla Options 1315.1 Pricing of plain vanilla options 1315.1.1 Delayed settlement date 1315.1.2 Cash settlement 1335.2 Market-making tools 1345.2.1 Inferring the implied volatility for a given strike 1345.2.2 Inferring the implied volatility for a given Delta 1355.2.3 Quoting the Vega-weighted butterfly and the risk reversal 1365.3 Bid/ask spreads for plain vanilla options 1395.4 Cutoff times and spreads 1415.5 Digital options 1425.5.1 Digital options pricing: the static replica approach 1435.5.2 Digital options pricing in specific model settings 1485.5.3 Delayed cash settlement date 1505.5.4 Bid/ask spreads 1505.5.5 Quotation conventions 1525.6 American plain vanilla options 1525.6.1 Valuation of American plain vanilla options in a BS setting 1525.6.2 Pricing of American plain vanilla options with the volatility smile 1536 Barrier Options 1556.1 A taxonomy of barrier options 1556.2 Some relationships of barrier option prices 1566.3 Pricing for barrier options in a BS economy 1576.3.1 The diffusion equation under single absorbing boundaries 1586.3.2 Dealing with a constant barrier 1596.4 Pricing formulae for barrier options 1606.5 One-touch (rebate) and no-touch options 1626.6 Double-barrier options 1646.6.1 Two absorbing states 1646.6.2 Pricing formula for double-barrier options 1656.7 Double-no-touch and double-touch options 1676.8 Probability of hitting a barrier 1676.9 Greek calculation 1686.10 Pricing barrier options in other model settings 1696.11 Pricing barriers with non-standard delivery 1706.11.1 Delayed settlement date 1706.11.2 Cash settlement 1706.12 Market approach to pricing barrier options 1716.12.1 Inclusion of the smile: the Vanna–Volga approach for barrier options 1716.12.2 The Vanna–Volga approach for barrier options: variations on the theme 1776.12.3 Slippage at the barrier level 1816.12.4 Delta-hedging near the barrier level 1836.12.5 Implicit one-touch and gearing 1846.12.6 Vega-hedge rebalancing 1866.13 Bid/ask spreads 1886.14 Monitoring frequency 1917 Other Exotic Options 1957.1 Introduction 1957.2 At-expiry barrier options 1957.3 Window barrier options 1977.4 First–then and knock-in–knock-out barrier options 1997.5 Auto-quanto options 2027.6 Forward start options 2047.6.1 Including the volatility smile in the pricing 2077.6.2 Forward implied volatility smiles 2107.6.3 Forward start barrier and bet options 2107.6.4 Dealing with notional amounts expressed in numeraire currency 2117.7 Variance swaps 2127.8 Compound, Asian and lookback options 2158 Risk Management Tools and Analysis 2178.1 Introduction 2178.2 Implementation of the LMUV model 2178.2.1 The forward volatility surfaces 2218.2.2 Calculating the sensitivity to the movements of the volatility surface 2238.3 Risk monitoring tools 2278.3.1 FX spot rate-related Greeks 2278.3.2 Cash-settled options 2298.3.3 Volatility-related Greeks and sensitivities 2298.3.4 Barrier implicit one-touch, bets and digitals 2318.3.5 Interest rate-related Greeks 2348.4 Risk analysis of plain vanilla options 2368.4.1 ATM straddle 2368.4.2 Risk reversal 2398.4.3 Vega-weighted butterfly 2418.5 Risk analysis of digital options 2448.6 Risk analysis of exotic options 2498.6.1 Barrier options 2498.6.2 Double barrier options 2588.6.3 Bet options 2629 Correlation and FX Options 2699.1 Preliminary considerations 2699.2 Correlation in the BS setting 2699.3 Contracts depending on several FX spot rates 2759.4 Dealing with correlation and volatility smile 2789.4.1 Vanna–Volga extension 2789.5 Linking volatility smiles 283References 287Index 291

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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.

• Preface ixNotation and Acronyms xiii1 The FX Market 11.1 FX rates and spot contracts 11.2 Outright and FX swap contracts 41.3 FX option contracts 101.3.1 Exercise 111.3.2 Expiry date and settlement date 111.3.3 Premium 131.3.4 Market standard practices for quoting options 141.4 Main traded FX option structures 162 Pricing Models for FX Options 212.1 Principles of option pricing theory 212.1.1 The Black–Scholes economy 212.1.2 Stochastic volatility economy 262.1.3 Change of numeraire 272.2 The black–scholes model 292.2.1 The forward price to use in the formula 302.2.2 BS greeks 312.2.3 Retrieving implied volatility and strike 352.2.4 Some relationships of the BS formula 382.3 The Heston Model 412.3.1 Time-dependent parameters in the Heston model 422.4 The SABR model 442.5 The mixture approach 452.5.1 The LMLV model 452.5.2 The LMUV model 482.5.3 Features of the LMLV and LMUV models and a comparison between them 502.5.4 Extension of the LMUV model 512.6 Some considerations about the choice of model 533 Dynamic Hedging and Volatility Trading 573.1 Preliminary considerations 573.2 A general framework 593.3 Hedging with a constant implied volatility 613.4 Hedging with an updating implied volatility 633.4.1 A market model for the implied volatility 663.5 Hedging Vega 683.6 Hedging Delta, Vega, Vanna and Volga 703.6.1 Vanna–Volga hedging with one implied volatility 713.6.2 Vanna–Volga hedging with different implied volatilities 713.7 The volatility smile and its phenomenology 753.8 Local exposures to the volatility smile 793.8.1 Retrieving the strikes of the main structures 793.8.2 ATM straddle exposures 813.8.3 Risk reversal exposures 813.8.4 Vega-weighted butterfly exposures 833.9 Scenario hedging and its relationship with Vanna–Volga hedging 843.9.1 Scenario hedging with constant Delta options 864 The Volatility Surface 914.1 General definitions 914.1.1 Arbitrage opportunities under the three different rules 924.2 Criteria for an efficient and convenient representation of the volatility surface 944.3 Commonly adopted approaches to building a volatility surface 964.4 Smile interpolation among strikes: the Vanna–Volga approach 974.4.1 The Vanna–Volga approach: general setting 974.4.2 Computing the Vanna–Volga weights and option prices 994.4.3 Limit and no-arbitrage conditions 1024.4.4 Approximating implied volatilities 1024.5 Some features of the Vanna–Volga approach 1044.5.1 Hedging error for longer expiries 1054.5.2 The implied risk-neutral density and smile asymptotics 1064.5.3 Two consistency results 1084.6 An alternative characterization of the Vanna–Volga approach 1104.7 Smile interpolation among expiries: implied volatility term structure 1124.8 Admissible volatility surfaces 1154.9 Taking into account the market butterfly 1164.10 Building the volatility matrix in practice 1205 Plain Vanilla Options 1315.1 Pricing of plain vanilla options 1315.1.1 Delayed settlement date 1315.1.2 Cash settlement 1335.2 Market-making tools 1345.2.1 Inferring the implied volatility for a given strike 1345.2.2 Inferring the implied volatility for a given Delta 1355.2.3 Quoting the Vega-weighted butterfly and the risk reversal 1365.3 Bid/ask spreads for plain vanilla options 1395.4 Cutoff times and spreads 1415.5 Digital options 1425.5.1 Digital options pricing: the static replica approach 1435.5.2 Digital options pricing in specific model settings 1485.5.3 Delayed cash settlement date 1505.5.4 Bid/ask spreads 1505.5.5 Quotation conventions 1525.6 American plain vanilla options 1525.6.1 Valuation of American plain vanilla options in a BS setting 1525.6.2 Pricing of American plain vanilla options with the volatility smile 1536 Barrier Options 1556.1 A taxonomy of barrier options 1556.2 Some relationships of barrier option prices 1566.3 Pricing for barrier options in a BS economy 1576.3.1 The diffusion equation under single absorbing boundaries 1586.3.2 Dealing with a constant barrier 1596.4 Pricing formulae for barrier options 1606.5 One-touch (rebate) and no-touch options 1626.6 Double-barrier options 1646.6.1 Two absorbing states 1646.6.2 Pricing formula for double-barrier options 1656.7 Double-no-touch and double-touch options 1676.8 Probability of hitting a barrier 1676.9 Greek calculation 1686.10 Pricing barrier options in other model settings 1696.11 Pricing barriers with non-standard delivery 1706.11.1 Delayed settlement date 1706.11.2 Cash settlement 1706.12 Market approach to pricing barrier options 1716.12.1 Inclusion of the smile: the Vanna–Volga approach for barrier options 1716.12.2 The Vanna–Volga approach for barrier options: variations on the theme 1776.12.3 Slippage at the barrier level 1816.12.4 Delta-hedging near the barrier level 1836.12.5 Implicit one-touch and gearing 1846.12.6 Vega-hedge rebalancing 1866.13 Bid/ask spreads 1886.14 Monitoring frequency 1917 Other Exotic Options 1957.1 Introduction 1957.2 At-expiry barrier options 1957.3 Window barrier options 1977.4 First–then and knock-in–knock-out barrier options 1997.5 Auto-quanto options 2027.6 Forward start options 2047.6.1 Including the volatility smile in the pricing 2077.6.2 Forward implied volatility smiles 2107.6.3 Forward start barrier and bet options 2107.6.4 Dealing with notional amounts expressed in numeraire currency 2117.7 Variance swaps 2127.8 Compound, Asian and lookback options 2158 Risk Management Tools and Analysis 2178.1 Introduction 2178.2 Implementation of the LMUV model 2178.2.1 The forward volatility surfaces 2218.2.2 Calculating the sensitivity to the movements of the volatility surface 2238.3 Risk monitoring tools 2278.3.1 FX spot rate-related Greeks 2278.3.2 Cash-settled options 2298.3.3 Volatility-related Greeks and sensitivities 2298.3.4 Barrier implicit one-touch, bets and digitals 2318.3.5 Interest rate-related Greeks 2348.4 Risk analysis of plain vanilla options 2368.4.1 ATM straddle 2368.4.2 Risk reversal 2398.4.3 Vega-weighted butterfly 2418.5 Risk analysis of digital options 2448.6 Risk analysis of exotic options 2498.6.1 Barrier options 2498.6.2 Double barrier options 2588.6.3 Bet options 2629 Correlation and FX Options 2699.1 Preliminary considerations 2699.2 Correlation in the BS setting 2699.3 Contracts depending on several FX spot rates 2759.4 Dealing with correlation and volatility smile 2789.4.1 Vanna–Volga extension 2789.5 Linking volatility smiles 283References 287Index 291

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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.

• Preface ixNotation and Acronyms xiii1 The FX Market 11.1 FX rates and spot contracts 11.2 Outright and FX swap contracts 41.3 FX option contracts 101.3.1 Exercise 111.3.2 Expiry date and settlement date 111.3.3 Premium 131.3.4 Market standard practices for quoting options 141.4 Main traded FX option structures 162 Pricing Models for FX Options 212.1 Principles of option pricing theory 212.1.1 The Black–Scholes economy 212.1.2 Stochastic volatility economy 262.1.3 Change of numeraire 272.2 The black–scholes model 292.2.1 The forward price to use in the formula 302.2.2 BS greeks 312.2.3 Retrieving implied volatility and strike 352.2.4 Some relationships of the BS formula 382.3 The Heston Model 412.3.1 Time-dependent parameters in the Heston model 422.4 The SABR model 442.5 The mixture approach 452.5.1 The LMLV model 452.5.2 The LMUV model 482.5.3 Features of the LMLV and LMUV models and a comparison between them 502.5.4 Extension of the LMUV model 512.6 Some considerations about the choice of model 533 Dynamic Hedging and Volatility Trading 573.1 Preliminary considerations 573.2 A general framework 593.3 Hedging with a constant implied volatility 613.4 Hedging with an updating implied volatility 633.4.1 A market model for the implied volatility 663.5 Hedging Vega 683.6 Hedging Delta, Vega, Vanna and Volga 703.6.1 Vanna–Volga hedging with one implied volatility 713.6.2 Vanna–Volga hedging with different implied volatilities 713.7 The volatility smile and its phenomenology 753.8 Local exposures to the volatility smile 793.8.1 Retrieving the strikes of the main structures 793.8.2 ATM straddle exposures 813.8.3 Risk reversal exposures 813.8.4 Vega-weighted butterfly exposures 833.9 Scenario hedging and its relationship with Vanna–Volga hedging 843.9.1 Scenario hedging with constant Delta options 864 The Volatility Surface 914.1 General definitions 914.1.1 Arbitrage opportunities under the three different rules 924.2 Criteria for an efficient and convenient representation of the volatility surface 944.3 Commonly adopted approaches to building a volatility surface 964.4 Smile interpolation among strikes: the Vanna–Volga approach 974.4.1 The Vanna–Volga approach: general setting 974.4.2 Computing the Vanna–Volga weights and option prices 994.4.3 Limit and no-arbitrage conditions 1024.4.4 Approximating implied volatilities 1024.5 Some features of the Vanna–Volga approach 1044.5.1 Hedging error for longer expiries 1054.5.2 The implied risk-neutral density and smile asymptotics 1064.5.3 Two consistency results 1084.6 An alternative characterization of the Vanna–Volga approach 1104.7 Smile interpolation among expiries: implied volatility term structure 1124.8 Admissible volatility surfaces 1154.9 Taking into account the market butterfly 1164.10 Building the volatility matrix in practice 1205 Plain Vanilla Options 1315.1 Pricing of plain vanilla options 1315.1.1 Delayed settlement date 1315.1.2 Cash settlement 1335.2 Market-making tools 1345.2.1 Inferring the implied volatility for a given strike 1345.2.2 Inferring the implied volatility for a given Delta 1355.2.3 Quoting the Vega-weighted butterfly and the risk reversal 1365.3 Bid/ask spreads for plain vanilla options 1395.4 Cutoff times and spreads 1415.5 Digital options 1425.5.1 Digital options pricing: the static replica approach 1435.5.2 Digital options pricing in specific model settings 1485.5.3 Delayed cash settlement date 1505.5.4 Bid/ask spreads 1505.5.5 Quotation conventions 1525.6 American plain vanilla options 1525.6.1 Valuation of American plain vanilla options in a BS setting 1525.6.2 Pricing of American plain vanilla options with the volatility smile 1536 Barrier Options 1556.1 A taxonomy of barrier options 1556.2 Some relationships of barrier option prices 1566.3 Pricing for barrier options in a BS economy 1576.3.1 The diffusion equation under single absorbing boundaries 1586.3.2 Dealing with a constant barrier 1596.4 Pricing formulae for barrier options 1606.5 One-touch (rebate) and no-touch options 1626.6 Double-barrier options 1646.6.1 Two absorbing states 1646.6.2 Pricing formula for double-barrier options 1656.7 Double-no-touch and double-touch options 1676.8 Probability of hitting a barrier 1676.9 Greek calculation 1686.10 Pricing barrier options in other model settings 1696.11 Pricing barriers with non-standard delivery 1706.11.1 Delayed settlement date 1706.11.2 Cash settlement 1706.12 Market approach to pricing barrier options 1716.12.1 Inclusion of the smile: the Vanna–Volga approach for barrier options 1716.12.2 The Vanna–Volga approach for barrier options: variations on the theme 1776.12.3 Slippage at the barrier level 1816.12.4 Delta-hedging near the barrier level 1836.12.5 Implicit one-touch and gearing 1846.12.6 Vega-hedge rebalancing 1866.13 Bid/ask spreads 1886.14 Monitoring frequency 1917 Other Exotic Options 1957.1 Introduction 1957.2 At-expiry barrier options 1957.3 Window barrier options 1977.4 First–then and knock-in–knock-out barrier options 1997.5 Auto-quanto options 2027.6 Forward start options 2047.6.1 Including the volatility smile in the pricing 2077.6.2 Forward implied volatility smiles 2107.6.3 Forward start barrier and bet options 2107.6.4 Dealing with notional amounts expressed in numeraire currency 2117.7 Variance swaps 2127.8 Compound, Asian and lookback options 2158 Risk Management Tools and Analysis 2178.1 Introduction 2178.2 Implementation of the LMUV model 2178.2.1 The forward volatility surfaces 2218.2.2 Calculating the sensitivity to the movements of the volatility surface 2238.3 Risk monitoring tools 2278.3.1 FX spot rate-related Greeks 2278.3.2 Cash-settled options 2298.3.3 Volatility-related Greeks and sensitivities 2298.3.4 Barrier implicit one-touch, bets and digitals 2318.3.5 Interest rate-related Greeks 2348.4 Risk analysis of plain vanilla options 2368.4.1 ATM straddle 2368.4.2 Risk reversal 2398.4.3 Vega-weighted butterfly 2418.5 Risk analysis of digital options 2448.6 Risk analysis of exotic options 2498.6.1 Barrier options 2498.6.2 Double barrier options 2588.6.3 Bet options 2629 Correlation and FX Options 2699.1 Preliminary considerations 2699.2 Correlation in the BS setting 2699.3 Contracts depending on several FX spot rates 2759.4 Dealing with correlation and volatility smile 2789.4.1 Vanna–Volga extension 2789.5 Linking volatility smiles 283References 287Index 291

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