Iain Clark – Foreign Exchange Option Pricing. A Practitioner’s Guide

Description

Description

This guide covers overseas alternate choices from the perspective of the finance practitioner. It incorporates every little thing a quant or dealer working in a financial institution or hedge fund would wish to know in regards to the arithmetic of overseas alternate—not simply the theoretical arithmetic coated in different books but in addition complete protection of implementation, pricing and calibration.

With content material developed with enter from merchants and with examples utilizing real-world knowledge, this guide introduces most of the extra generally requested merchandise from FX choices buying and selling desks, along with the fashions that seize the chance traits essential to cost these merchandise precisely. Crucially, this guide describes the numerical strategies required for calibration of those fashions – an space usually uncared for within the literature, which is however of paramount significance in follow. Thorough remedy is given in a single unified textual content to the next options:

• Correct market conventions for FX volatility floor building
• Adjustment for settlement and delayed supply of choices
• Pricing of vanillas and barrier choices beneath the volatility smile
• Barrier bending for limiting barrier discontinuity danger close to expiry
• Industry power partial differential equations in a single and several other spatial variables utilizing finite variations on nonuniform grids
• Fourier remodel strategies for pricing European choices utilizing attribute features
• Stochastic and native volatility fashions, and a combined stochastic/native volatility mannequin
• Three-factor long-dated FX mannequin
• Numerical calibration strategies for all of the fashions on this work
• The augmented state variable method for pricing strongly path-dependent choices utilizing both partial differential equations or Monte Carlo simulation

Connecting mathematically rigorous idea with follow, that is the important information to overseas alternate choices within the context of the true monetary market.

 

Table of Contents

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 supply guidelines – days or perhaps weeks 8

1.5.2 Expiry and supply guidelines – 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 mannequin 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 utilizing Fourier Methods 35

2.15.1 European possibility 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.Three 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 growth 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.Four 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

 

Author Information

Dr Iain J. Clark, (London, UK), is Head of Foreign Exchange Quantitative Analysis at Dresdner Kleinwort in London, the place he arrange and runs the workforce liable for creating pricing libraries for the entrance workplace. Previously, he was Director of the Quantitative Research Group in Lehman Brothers, Fixed Income Quantitative Analyst at BNP Paribas and has additionally labored in FX Commodities Derivatives analysis at JP Morgan. He holds an MSc in Mathematics from the University of Edinburgh, and a PhD in Applied Mathematics from the University of Queensland, Australia. Dr Clark is an everyday speaker at key finance occasions, and has offered at London Imperial College, The Bachelier Society Annual Conference, London Imperial College, world enterprise Strategies annual Conference, Risk occasions, Marcus Evans occasions and plenty of extra.