FX Barrien Options A Comprehensive Guide for Industry Quants Zareer Dadachanji Director, Model Quant Solutions, Bremen, Germany
Contents List of Figures List of Tables Preface Acknowledgements Foreword Glossary of Mathematical Notation Contract Types 1 Meet the Products 1 1.1 Spot 1 1.1.1 Dollars per euro or euros per dollar? 3 1.1.2 Big figures and small figures 4 1.1.3 The value of Foreign 4 1.1.4 Converting between Domestic and Foreign 6 1.2 Forwards 6 1.2.1 The FX forward market 7 1.2.2 A formula for the forward rate 8 1.2.3 Payoff of a forward contract 10 1.2.4 Valuation of a forward contract 12 1.3 Vanilla options 12 1.3.1 Put-call parity 15 1.4 European digitals 16 1.5 Barrier-contingent vanilla options 16 1.6 Barrier-contingent payments 23 1.7 Rebates 25 1.8 Knock-in-knock-out (KIKO) options 25 1.9 Types of barriers 26 1.10 Structured products 27 1.11 Specifying the contract 28 1.12 Quantitative truisms 29 1.12.1 Foreign exchange symmetry and Inversion 29 1.12.2 Knock-out plus knock-in equals no-barrier contract 29 1.12.3 Put-call parity 30 1.13 Jargon-buster 30 xii xix xx xxiv xxv xxvii xxviii vii
viii [ Contents 2 Living in a Black-Scholes World 33 2.1 The Black-Scholes model equation for spotprice 33 2.2 The process for In S 35 2.3 The Black-Scholes equation for option pricing 38 2.3.1 The lagless approach 38 2.3.2 Derivation of the Black-Scholes PDE 39 2.3.3 Black-Scholes model: hedging assumptions 42 2.3.4 Interpretation of the Black-Scholes PDE 43 2.3.4.1 Term 1: theta term 43 2.3.4.2 Term 2: carry term 43 2.3.4.3 Term 3: gamma term 44 2.3.4.4 Term 4: cash account term 45 2.4 Solving the Black-Scholes PDE 45 2.5 Payments 45 2.6 Forwards 47 2.7 Vanilla options 47 2.7.1 Transformation of the Black-Scholes PDE 48 2.7.1.1 Transformation 1: Time direction 48 2.7.1.2 Transformation 2: Discounting 49 2.7.1.3 Transformation 3: From spot to forward 49 2.7.1.4 Transformation 4: Log-space 51 2.7.2 Solution of the diffusion equation for vanilla options 52 2.7.3 The vanilla option pricing formulae 57 2.7.3.1 Respecting the spot lags 57 2.7.3.2 Expression in terms of forward and discount factors. 57 2.7.3.3 Intrinsic value 58 2.7.3.4 Moneyness 58 2.7.4 Price quotation styles 59 2.7.5 Valuation behaviour of vanilla options 60 2.8 Black-Scholes pricing ofbarrier-contingent vanilla options 64 2.8.1 Knock-outs 65 2.8.2 Knock-ins 69 2.8.3 Quotation methods 70 2.8.4 Valuation behaviour ofbarrier-contingent vanilla options... 70 2.9 Black-Scholes pricing ofbarrier-contingent payments 73 2.9.1 Payment in Domestic 74 2.9.2 Payment in Foreign 76 2.9.3 Quotation methods 76 2.9.4 Valuation behaviour ofbarrier-contingent payments 77 2.10 Discrete barrier options 80
Contents [ ix 2.11 Window barrier options 80 2.12 Black-Scholes numerical valuation methods 81 3 Black-Scholes Risk Management 82 3.1 Spot risk 83 3.1.1 Local spot risk analysis 83 3.1.2 Delta 84 3.1.2.1 Premium-adjusted Delta 84 3.1.2.2 Delta quotation styles 85 3.1.3 Gamma 85 3.1.4 Results for spot Greeks 86 3.1.5 Non-local spot risk analysis 97 3.2 Volatility risk 97 3.2.1 Local volatility risk analysis 98 3.2.1.1 Results for vega and volgamma 100 3.2.1.2 Vanna 110 3.2.2 Non-local volatility risk 112 3.3 Interest rate risk 113 3.4 Theta 115 3.5 Barrier over-hedging 117 3.6 Co-Greeks 120 4 Smile Pricing 121 4.1 The shortcomings of the Black-Scholes model 121 4.2 Black-Scholes with term structure (BSTS) 123 4.3 The implied volatility surface 125 4.4 The FX vanilla option market 126 4.4.1 At-the-money volatility 129 4.4.2 Risk reversal 131 4.4.3 Butterfly 132 4.4.4 The role of the Black-Scholes model in the FX vanilla options market 133 4.5 Theoretical Value (TV) 133 4.5.1 Conventions for extracting market data for TV calculations.. 134 4.5.2 Example broker quote request 135 4.6 Modelling market implied volatilities 136 4.7 The probability density function 137 4.8 Three things we want from a model 141 4.9 The local volatility (LV) model 141 4.9.1 It's the smile dynamics, stupid 155 4.10 Five things we want from a model 156 4.11 Stochastic volatility (SV) modeis 157 4.11.1 SABR model 157
x Contents 4.11.2 Hestonmodel 158 4.11.2.1 Mean-reversion vs volatility 159 4.11.2.2 Calibrating the Heston model 160 4.12 Mixed local/stochastic volatility (LSV) models 162 4.12.1 Term structure of volatility of volatility 170 4.13 Other models and methods 171 4.13.1 Uncertain volatility (UV) models 171 4.13.2 Jump-diffusion models 172 4.13.3 Vanna-volga methods 173 5 Smile Risk Management 175 5.1 Black-Scholes with term structure 175 5.2 Local volatility model 179 5.3 Spot risk under smile models 180 5.4 Theta risk under smile models 182 5.5 Mixed local/stochastic volatility models 182 5.6 Static hedging 183 5.7 Managing risk across businesses 184 6 Numerical Methods 186 6.1 Finite-difference (FD) methods 186 6.1.1 Gridgeometry 187 6.1.2 Finite-difference schemes 189 6.2 Monte Carlo (MC) methods 193 6.2.1 Monte Carlo schedules 194 6.2.2 Monte Carlo algorithms 195 6.2.3 Variance reduction 197 6.2.3.1 Antithetic variables 197 6.2.3.2 Control variates 198 6.2.4 The Brownian Bridge 199 6.2.5 Early termination 200 6.3 Calculating Greeks 200 6.3.1 Bumped Greeks 200 6.3.1.1 Bumping spot near a barrier 201 6.3.1.2 Arbitrage in bucketed vega reports 201 6.3.2 Greeks from finite-difference calculations 202 6.3.3 Greeks from Monte Carlo 203 7 Further Topics 205 7.1 Managed currencies 205 7.2 Stochastic interest rates (SIR) 206 7.3 Real-world pricing 210 7.3.1 Bid-offer spreads 210 7.3.2 Rules-based pricing methods 212
Contents J xi 7.4 Regulation and market abuse 213 Appendix A: Derivation of the Black-Scholes Pricing Equations for Vanilla Options 215 Appendix B: Normal and Lognormal Probability Distribution«220 B.l Normal distribution 220 B.2 Lognormal distribution 220 Appendix C: Derivation of the Local Volatility Function 221 C.l Derivation in terms of call prices 221 C.2 Local volatility from implied volatility 225 C.3 Working in moneyness space 227 C.4 Working in log space 228 C.5 Specialization to BSTS 229 Appendix D: Calibration of Mixed Local/Stochastic Volatility (LSV) Models 230 Appendix E: Derivation of Fokker-Planck Equation for the Local Volatility Model 232 Bibliography 234 Index 237