FX Barrien Options. A Comprehensive Guide for Industry Quants. Zareer Dadachanji Director, Model Quant Solutions, Bremen, Germany

Transcription

1 FX Barrien Options A Comprehensive Guide for Industry Quants Zareer Dadachanji Director, Model Quant Solutions, Bremen, Germany

2 Contents List of Figures List of Tables Preface Acknowledgements Foreword Glossary of Mathematical Notation Contract Types 1 Meet the Products Spot Dollars per euro or euros per dollar? Big figures and small figures The value of Foreign Converting between Domestic and Foreign Forwards The FX forward market A formula for the forward rate Payoff of a forward contract Valuation of a forward contract Vanilla options Put-call parity European digitals Barrier-contingent vanilla options Barrier-contingent payments Rebates Knock-in-knock-out (KIKO) options Types of barriers Structured products Specifying the contract Quantitative truisms Foreign exchange symmetry and Inversion Knock-out plus knock-in equals no-barrier contract Put-call parity Jargon-buster 30 xii xix xx xxiv xxv xxvii xxviii vii

3 viii [ Contents 2 Living in a Black-Scholes World The Black-Scholes model equation for spotprice The process for In S The Black-Scholes equation for option pricing The lagless approach Derivation of the Black-Scholes PDE Black-Scholes model: hedging assumptions Interpretation of the Black-Scholes PDE Term 1: theta term Term 2: carry term Term 3: gamma term Term 4: cash account term Solving the Black-Scholes PDE Payments Forwards Vanilla options Transformation of the Black-Scholes PDE Transformation 1: Time direction Transformation 2: Discounting Transformation 3: From spot to forward Transformation 4: Log-space Solution of the diffusion equation for vanilla options The vanilla option pricing formulae Respecting the spot lags Expression in terms of forward and discount factors Intrinsic value Moneyness Price quotation styles Valuation behaviour of vanilla options Black-Scholes pricing ofbarrier-contingent vanilla options Knock-outs Knock-ins Quotation methods Valuation behaviour ofbarrier-contingent vanilla options Black-Scholes pricing ofbarrier-contingent payments Payment in Domestic Payment in Foreign Quotation methods Valuation behaviour ofbarrier-contingent payments Discrete barrier options 80

4 Contents [ ix 2.11 Window barrier options Black-Scholes numerical valuation methods 81 3 Black-Scholes Risk Management Spot risk Local spot risk analysis Delta Premium-adjusted Delta Delta quotation styles Gamma Results for spot Greeks Non-local spot risk analysis Volatility risk Local volatility risk analysis Results for vega and volgamma Vanna Non-local volatility risk Interest rate risk Theta Barrier over-hedging Co-Greeks Smile Pricing The shortcomings of the Black-Scholes model Black-Scholes with term structure (BSTS) The implied volatility surface The FX vanilla option market At-the-money volatility Risk reversal Butterfly The role of the Black-Scholes model in the FX vanilla options market Theoretical Value (TV) Conventions for extracting market data for TV calculations Example broker quote request Modelling market implied volatilities The probability density function Three things we want from a model The local volatility (LV) model It's the smile dynamics, stupid Five things we want from a model Stochastic volatility (SV) modeis SABR model 157

5 x Contents Hestonmodel Mean-reversion vs volatility Calibrating the Heston model Mixed local/stochastic volatility (LSV) models Term structure of volatility of volatility Other models and methods Uncertain volatility (UV) models Jump-diffusion models Vanna-volga methods Smile Risk Management Black-Scholes with term structure Local volatility model Spot risk under smile models Theta risk under smile models Mixed local/stochastic volatility models Static hedging Managing risk across businesses Numerical Methods Finite-difference (FD) methods Gridgeometry Finite-difference schemes Monte Carlo (MC) methods Monte Carlo schedules Monte Carlo algorithms Variance reduction Antithetic variables Control variates The Brownian Bridge Early termination Calculating Greeks Bumped Greeks Bumping spot near a barrier Arbitrage in bucketed vega reports Greeks from finite-difference calculations Greeks from Monte Carlo Further Topics Managed currencies Stochastic interest rates (SIR) Real-world pricing Bid-offer spreads Rules-based pricing methods 212

6 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