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Foreign Exchange Option Pricing: A Practitioners Guide


This book covers foreign exchange options from the point of view of the finance practitioner. It contains everything a quant or trader working in a bank or hedge fund would need to know about the mathematics of foreign exchange—not just the theoretical mathematics covered in other books but also comprehensive coverage of implementation, pricing and calibration.  

With content developed with input from traders and with examples using real-world data, this book introduces many of the more commonly requested products from FX options trading desks, together with the models that capture the risk characteristics necessary to price these products accurately. Crucially, this book describes the numerical methods required for calibration of these models – an area often neglected in the literature, which is nevertheless of paramount importance in practice. Thorough treatment is given in one unified text to the following features:

  • Correct market conventions for FX volatility surface construction
  • Adjustment for settlement and delayed delivery of options
  • Pricing of vanillas and barrier options under the volatility smile
  • Barrier bending for limiting barrier discontinuity risk near expiry
  • Industry strength partial differential equations in one and several spatial variables using finite differences on nonuniform grids
  • Fourier transform methods for pricing European options using characteristic functions
  • Stochastic and local volatility models, and a mixed stochastic/local volatility model
  • Three-factor long-dated FX model
  • Numerical calibration techniques for all the models in this work
  • The augmented state variable approach for pricing strongly path-dependent options using either partial differential equations or Monte Carlo simulation

Connecting mathematically rigorous theory with practice, this is the essential guide to foreign exchange options in the context of the real financial marketplace.

Table of Contents

Mathematical Preliminaries 

Deltas and Market Conventions

Volatility Surface Construction

Local Volatility and Implied Volatility

Stochastic Volatility

Numerical Methods for Pricing and Calibration

First Generation Exotics – Binary and Barrier Options

Second Generation Exotics

Multicurrency Options

Long-dated FX Options


List of Tables.

List of Figures.

1 Introduction.

1.1 A Gentle Introduction to FX Markets.

1.2 Quotation Styles.

1.3 Risk Considerations.

1.4 Spot Settlement Rules.

1.5 Expiry and Delivery Rules.

1.6 Cutoff Times.

2 Mathematical Preliminaries.

2.1 The Black–Scholes Model.

2.2 Risk Neutrality.

2.3 Derivation of the Black–Scholes equation.

2.4 Integrating the SDE for ST.

2.5 Black–Scholes PDEs Expressed in Logspot.

2.6 Feynman–Kac and Risk-Neutral Expectation.

2.7 Risk Neutrality and the Presumption of Drift.

2.8 Valuation of European Options.

2.9 The Law of One Price.

2.10 The Black–Scholes Term Structure Model.

2.11 Breeden–Litzenberger Analysis.

2.12 European Digitals.

2.13 Settlement Adjustments.

2.14 Delayed Delivery Adjustments.

2.15 Pricing using Fourier Methods.

2.16 Leptokurtosis – More than Fat Tails.

3 Deltas and Market Conventions.

3.1 Quote Style Conversions.

3.2 The Law of Many Deltas.

3.3 FX Delta Conventions.

3.4 Market Volatility Surfaces.

3.5 At-the-Money.

3.6 Market Strangle.

3.7 Smile Strangle and Risk Reversal.

3.8 Visualisation of Strangles.

3.9 Smile Interpolation – Polynomial in Delta.

3.10 Smile Interpolation – SABR.

3.11 Concluding Remarks.

4 Volatility Surface Construction.

4.1 Volatility Backbone – Flat Forward Interpolation.

4.2 Volatility Surface Temporal Interpolation.

4.3 Volatility Surface Temporal Interpolation – Holidays and Weekends.

4.4 Volatility Surface Temporal Interpolation – Intraday Effects.

5 Local Volatility and Implied Volatility.

5.1 Introduction.

5.2 The Fokker–Planck Equation.

5.3 Dupire's Construction of Local Volatility.

5.4 Implied Volatility and Relationship to Local Volatility.

5.5 Local Volatility as Conditional Expectation.

5.6 Local Volatility for FX Markets.

5.7 Diffusion and PDE for Local Volatility.

5.8 The CEV Model.

6 Stochastic Volatility.

6.1 Introduction.

6.2 Uncertain Volatility.

6.3 Stochastic Volatility Models.

6.4 Uncorrelated Stochastic Volatility.

6.5 Stochastic Volatility Correlated with Spot.

6.6 The Fokker–Planck PDE Approach.

6.7 The Feynman–Kac PDE Approach.

6.8 Local Stochastic Volatility (LSV) Models.

7 Numerical Methods for Pricing and Calibration.

7.1 One-Dimensional Root Finding – Implied Volatility Calculation.

7.2 Nonlinear Least Squares Minimisation.

7.3 Monte Carlo Simulation.

7.4 Convection–Diffusion PDEs in Finance.

7.5 Numerical Methods for PDEs.

7.6 Explicit Finite Difference Scheme.

7.7 Explicit Finite Difference on Nonuniform Meshes.

7.8 Implicit Finite Difference Scheme.

7.9 The Crank–Nicolson Scheme.

7.10 Numerical Schemes for Multidimensional PDEs.

7.11 Practical Nonuniform Grid Generation Schemes.

7.12 Further Reading.

8 First Generation Exotics – Binary and Barrier Options.

8.1 The Reflection Principle.

8.2 European Barriers and Binaries.

8.3 Continuously Monitored Binaries and Barriers.

8.4 Double Barrier Products.

8.5 Sensitivity to Local and Stochastic Volatility.

8.6 Barrier Bending.

8.7 Value Monitoring.

9 Second Generation Exotics.

9.1 Chooser Options.

9.2 Range Accrual Options.

9.3 Forward Start Options.

9.4 Lookback Options.

9.5 Asian Options.

9.6 Target Redemption Notes.

9.7 Volatility and Variance Swaps.

10 Multicurrency Options.

10.1 Correlations, Triangulation and Absence of Arbitrage.

10.2 Exchange Options.

10.3 Quantos.

10.4 Best-ofs and Worst-ofs.

10.5 Basket Options.

10.6 Numerical Methods.

10.7 A Note on Multicurrency Greeks.

10.8 Quantoing Untradeable Factors.

10.9 Further Reading.

11 Longdated FX.

11.1 Currency Swaps.

11.2 Basis Risk.

11.3 Forward Measure.

11.4 LIBOR in Arrears.

11.5 Typical Longdated FX Products.

11.6 The Three-Factor Model.

11.7 Interest Rate Calibration of the Three-Factor Model.

11.8 Spot FX Calibration of the Three-Factor Model.

11.9 Conclusion.


Further Reading.