Foreign Exchange Option Pricing: A Practitioner's Guide

Acknowledgements xiii

List of Tables xv

List of Figures xvii

1 Introduction 1

1.1 A Gentle Introduction to FX Markets 1

1.2 Quotation Styles 2

1.3 Risk Considerations 5

1.4 Spot Settlement Rules 5

1.5 Expiry and Delivery Rules 8

1.5.1 Expiry and delivery rules – days or weeks 8

1.5.2 Expiry and delivery rules – months or years 9

1.6 Cutoff Times 10

2 Mathematical Preliminaries 13

2.1 The Black–Scholes Model 13

2.1.1 Assumptions of the Black–Scholes model 13

2.2 Risk Neutrality 13

2.3 Derivation of the Black–Scholes equation 14

2.4 Integrating the SDE for ST 17

2.5 Black–Scholes PDEs Expressed in Logspot 18

2.6 Feynman–Kac and Risk-Neutral Expectation 18

2.7 Risk Neutrality and the Presumption of Drift 20

2.8 Valuation of European Options 23

2.8.1 Forward 26

2.9 The Law of One Price 27

2.10 The Black–Scholes Term Structure Model 28

2.11 Breeden–Litzenberger Analysis 30

2.12 European Digitals 31

2.13 Settlement Adjustments 32

2.14 Delayed Delivery Adjustments 33

2.15 Pricing using Fourier Methods 35

2.15.1 European option pricing involving one numerical integral 37

2.16 Leptokurtosis – More than Fat Tails 38

3 Deltas and Market Conventions 41

3.1 Quote Style Conversions 41

3.2 The Law of Many Deltas 43

3.3 FX Delta Conventions 47

3.4 Market Volatility Surfaces 49

3.5 At-the-Money 50

3.6 Market Strangle 53

3.6.1 Example – EURUSD 1Y 55

3.7 Smile Strangle and Risk Reversal 55

3.8 Visualisation of Strangles 57

3.9 Smile Interpolation – Polynomial in Delta 59

3.10 Smile Interpolation – SABR 60

3.11 Concluding Remarks 62

4 Volatility Surface Construction 63

4.1 Volatility Backbone – Flat Forward Interpolation 65

4.2 Volatility Surface Temporal Interpolation 67

4.3 Volatility Surface Temporal Interpolation – Holidays and Weekends 70

4.4 Volatility Surface Temporal Interpolation – Intraday Effects 73

5 Local Volatility and Implied Volatility 77

5.1 Introduction 77

5.2 The Fokker–Planck Equation 78

5.3 Dupire’s Construction of Local Volatility 83

5.4 Implied Volatility and Relationship to Local Volatility 86

5.5 Local Volatility as Conditional Expectation 87

5.6 Local Volatility for FX Markets 88

5.7 Diffusion and PDE for Local Volatility 89

5.8 The CEV Model 90

5.8.1 Asymptotic expansion 91

6 Stochastic Volatility 95

6.1 Introduction 95

6.2 Uncertain Volatility 95

6.3 Stochastic Volatility Models 96

6.4 Uncorrelated Stochastic Volatility 107

6.5 Stochastic Volatility Correlated with Spot 108

6.6 The Fokker–Planck PDE Approach 111

6.7 The Feynman–Kac PDE Approach 113

6.8 Local Stochastic Volatility (LSV) Models 117

7 Numerical Methods for Pricing and Calibration 129

7.1 One-Dimensional Root Finding – Implied Volatility Calculation 129

7.2 Nonlinear Least Squares Minimisation 130

7.3 Monte Carlo Simulation 131

7.4 Convection–Diffusion PDEs in Finance 147

7.5 Numerical Methods for PDEs 153

7.6 Explicit Finite Difference Scheme 155

7.7 Explicit Finite Difference on Nonuniform Meshes 163

7.8 Implicit Finite Difference Scheme 165

7.9 The Crank–Nicolson Scheme 167

7.10 Numerical Schemes for Multidimensional PDEs 168

7.11 Practical Nonuniform Grid Generation Schemes 173

7.12 Further Reading 176

8 First Generation Exotics – Binary and Barrier Options 177

8.1 The Reflection Principle 179

8.2 European Barriers and Binaries 180

8.3 Continuously Monitored Binaries and Barriers 183

8.4 Double Barrier Products 194

8.5 Sensitivity to Local and Stochastic Volatility 195

8.6 Barrier Bending 197

8.7 Value Monitoring 202

9 Second Generation Exotics 205

9.1 Chooser Options 206

9.2 Range Accrual Options 206

9.3 Forward Start Options 207

9.4 Lookback Options 209

9.5 Asian Options 212

9.6 Target Redemption Notes 214

9.7 Volatility and Variance Swaps 214

10 Multicurrency Options 225

10.1 Correlations, Triangulation and Absence of Arbitrage 226

10.2 Exchange Options 229

10.3 Quantos 229

10.4 Best-ofs and Worst-ofs 233

10.5 Basket Options 239

10.6 Numerical Methods 241

10.7 A Note on Multicurrency Greeks 242

10.8 Quantoing Untradeable Factors 243

10.9 Further Reading 244

11 Longdated FX 245

11.1 Currency Swaps 245

11.2 Basis Risk 247

11.3 Forward Measure 249

11.4 LIBOR in Arrears 250

11.5 Typical Longdated FX Products 253

11.6 The Three-Factor Model 255

11.7 Interest Rate Calibration of the Three-Factor Model 257

11.8 Spot FX Calibration of the Three-Factor Model 259

11.9 Conclusion 264

References 265

Further Reading 271

Index 273