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Foreign Exchange Option Pricing: A Practitioner's Guide

Book Description

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

  1. Cover
  2. Half Title page
  3. Title page
  4. Copyright page
  5. Dedication
  6. Acknowledgements
  7. List of Tables
  8. List of Figures
  9. Chapter 1: Introduction
    1. 1.1 A Gentle Introduction to FX Markets
    2. 1.2 Quotation Styles
    3. 1.3 Risk Considerations
    4. 1.4 Spot Settlement Rules
    5. 1.5 Expiry and Delivery Rules
    6. 1.6 Cutoff Times
  10. Chapter 2: Mathematical Preliminaries
    1. 2.1 The Black–Scholes Model
    2. 2.2 Risk Neutrality
    3. 2.3 Derivation of the Black–Scholes Equation
    4. 2.4 Integrating the SDE for ST
    5. 2.5 Black–Scholes PDEs Expressed in Logspot
    6. 2.6 Feynman–Kac and Risk-Neutral Expectation
    7. 2.7 Risk Neutrality and the Presumption of Drift
    8. 2.8 Valuation of European Options
    9. 2.9 The Law of One Price
    10. 2.10 The Black–Scholes Term Structure Model
    11. 2.11 Breeden–Litzenberger Analysis
    12. 2.12 European Digitals
    13. 2.13 Settlement Adjustments
    14. 2.14 Delayed Delivery Adjustments
    15. 2.15 Pricing Using Fourier Methods
    16. 2.16 Leptokurtosis – More Than Fat Tails
  11. Chapter 3: Deltas and Market Conventions
    1. 3.1 Quote Style Conversions
    2. 3.2 The Law of Many Deltas
    3. 3.3 FX Delta Conventions
    4. 3.4 Market Volatility Surfaces
    5. 3.5 At-The-Money
    6. 3.6 Market Strangle
    7. 3.7 Smile Strangle and Risk Reversal
    8. 3.8 Visualisation of Strangles
    9. 3.9 Smile Interpolation – Polynomial in Delta
    10. 3.10 Smile Interpolation – SABR
    11. 3.11 Concluding Remarks
  12. Chapter 4: Volatility Surface Construction
    1. 4.1 Volatility Backbone – Flat Forward Interpolation
    2. 4.2 Volatility Surface Temporal Interpolation
    3. 4.3 Volatility Surface Temporal Interpolation – Holidays and Weekends
    4. 4.4 Volatility Surface Temporal Interpolation – Intraday Effects
  13. Chapter 5: Local Volatility and Implied Volatility
    1. 5.1 Introduction
    2. 5.2 The Fokker–Planck Equation
    3. 5.3 Dupire’s Construction of Local Volatility
    4. 5.4 Implied Volatility and Relationship to Local Volatility
    5. 5.5 Local Volatility as Conditional Expectation
    6. 5.6 Local Volatility for Fx Markets
    7. 5.7 Diffusion and PDE for Local Volatility
    8. 5.8 The CEV Model
  14. Chapter 6: Stochastic Volatility
    1. 6.1 Introduction
    2. 6.2 Uncertain Volatility
    3. 6.3 Stochastic Volatility Models
    4. 6.4 Uncorrelated Stochastic Volatility
    5. 6.5 Stochastic Volatility Correlated with Spot
    6. 6.6 The Fokker–Planck PDE Approach
    7. 6.7 The Feynman–KAC PDE Approach
    8. 6.8 Local Stochastic Volatility (LSV) Models
  15. Chapter 7: Numerical Methods for Pricing and Calibration
    1. 7.1 One-Dimensional Root Finding – Implied Volatility Calculation
    2. 7.2 Nonlinear Least Squares Minimisation
    3. 7.3 Monte Carlo Simulation
    4. 7.4 Convection–Diffusion Pdes in Finance
    5. 7.5 Numerical Methods for PDEs
    6. 7.6 Explicit Finite Difference Scheme
    7. 7.7 Explicit Finite Difference on Nonuniform Meshes
    8. 7.8 Implicit Finite Difference Scheme
    9. 7.9 The Crank–Nicolson Scheme
    10. 7.10 Numerical Schemes for Multidimensional PDES
    11. 7.11 Practical Nonuniform Grid Generation Schemes
    12. 7.12 Further Reading
  16. Chapter 8: First Generation Exotics – Binary and Barrier Options
    1. 8.1 The Reflection Principle
    2. 8.2 European Barriers and Binaries
    3. 8.3 Continuously Monitored Binaries and Barriers
    4. 8.4 Double Barrier Products
    5. 8.5 Sensitivity to Local And Stochastic Volatility
    6. 8.6 Barrier Bending
    7. 8.7 Value Monitoring
  17. Chapter 9: Second Generation Exotics
    1. 9.1 Chooser Options
    2. 9.2 Range Accrual Options
    3. 9.3 Forward Start Options
    4. 9.4 Lookback Options
    5. 9.5 Asian Options
    6. 9.6 Target Redemption Notes
    7. 9.7 Volatility and Variance SWAPS
  18. Chapter 10: Multicurrency Options
    1. 10.1 Correlations, Triangulation and Absence of Arbitrage
    2. 10.2 Exchange Options
    3. 10.3 Quantos
    4. 10.4 Best-OFS and Worst-OFS
    5. 10.5 Basket Options
    6. 10.6 Numerical Methods
    7. 10.7 A Note on Multicurrency Greeks
    8. 10.8 Quantoing Untradeable Factors
    9. 10.9 Further Reading
  19. Chapter 11: Longdated FX
    1. 11.1 Currency SWAPS
    2. 11.2 Basis Risk
    3. 11.3 Forward Measure
    4. 11.4 Libor in Arrears
    5. 11.5 Typical Longdated FX Products
    6. 11.6 The Three-Factor Model
    7. 11.7 Interest Rate Calibration of the Three-Factor Model
    8. 11.8 Spot FX Calibration of the Three-Factor Model
    9. 11.9 Conclusion
  20. References
  21. Further Reading
  22. Index