Foreign Exchange Option Pricing E-Book

Iain J. Clark

• Herausgeber: John Wiley & Sons
• Kategorie: Fachliteratur
• Serie: Wiley Finance Series
• Sprache: Englisch

Seitenzahl: 431

Veröffentlichungsjahr: 2011

Beschreibung

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.

Leseprobe

Contents

Cover

Half Title page

Title page

Copyright page

Dedication

Acknowledgements

List of Tables

List of Figures

Chapter 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

Chapter 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

Chapter 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

Chapter 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

Chapter 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

Chapter 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

Chapter 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

Chapter 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

Chapter 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

Chapter 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

Chapter 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

References

Further Reading

Index

Foreign Exchange Option Pricing

For other titles in the Wiley Finance seriesplease see www.wiley.com/finance

This edition first published 2011© 2011 Iain J. Clark

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Library of Congress Cataloging-in-Publication Data

Clark, Iain J.Foreign exchange option pricing : a practitioner’s guide / Iain J. Clark.p. cm.ISBN 978-0-470-68368-21. Options (Finance)–Prices. 2. Stock options. 3. Foreign exchange rates. I. Title. HG6024.A3C563 2011332.4′5–dc22

2010030438

A catalogue record for this book is available from the British Library.

ISBN 978-0-470-68368-2

For Isabel

Acknowledgements

I would like to thank everyone at Standard Bank, particularly Peter Glancey and Marcelo Labre, for their patience during the execution of this work. This is an industry book and it would not have happened without the help and encouragement of everyone I’ve worked with, most recently at Standard Bank and in previous years at JP Morgan, BNP Paribas, Lehman Brothers, Dresdner Kleinwort and Commerzbank. As such, special thanks are due to David Kitson, Jérôme Lebuchoux, Marek Musiela, Nicolas Jackson, Robert Campbell, Dominic O’Kane, Ronan Dowling, Tim Sharp, Ian Robertson, Alex Langnau and John Juer.

A special debt of gratitude to Messaoud Chibane, outstanding quant and very good friend, who has encouraged me every step of the way over the years.

For parts of Chapter 11, I am indebted to the excellent work on longdated modelling of my former team members at Dresdner Kleinwort: Andrey Gal, Chia Tan, Olivier Taghi and Lars Schouw.

I must also thank Pete Baker, Aimee Dibbens, Karen Weller and Lori Boulton at Wiley Finance for their help and patience with me during the completion and production of this work, as well as all the rest of the Wiley team who have done such an excellent job to bring this book to publication. While I take sole responsibility for any errors that remain in this work, I am very grateful to Pat Bateson and Rachael Wilkie for their thoroughness in checking the manuscript. Thanks are due to my literary agent Isabel White for seeing the potential for me to write a book on this topic.

I would also like to thank my amazing wife, to whom I owe more than I can possibly say. I am as always grateful to my parents John and Joan for their love, support and tolerance of my difficult questions and interest in science and mathematics – I’m glad to say some things haven’t changed so much! Also to my extended family, whom I don’t get to see as often as I would like, thanks for keeping us in your thoughts and all your messages of encouragement. They mean a lot to an author.

Finally, to my young nieces and nephews in Canada – Andrew, Bradley, Isabel, Mackenzie and William–who asked me if my mathematics book for grown-ups was going to have ‘very hard sums’ like 1 000010 – 1 000 000 012, I have a very hard sum just for you:

This book is for you and for all students, young and old, of the mathematical arts. I wish you all the very best with your studies and your work.

Web page for this book

www.fxoptionpricing.com

List of Tables

Table 1.1 Currency pair quotation conventions and market terminology

Table 1.2 Currency pair exceptions to T + 2 settlement

Table 1.3 Cutoff times

Table 2.1 European digitals

Table 3.1 Delta conventions for common currency pairs

Table 3.2 Premium currency for major currency pairs

Table 3.3 Sample market volatility surface for EURUSD

Table 3.4 Sample market volatility surface for USDJPY

Table 3.5 1Y EURUSD smile with polynomial delta parameterisation

Table 3.6 1Y EURUSD Black–Scholes prices under delta polynomial

Table 3.7 1Y EURUSD smile with SABR parameterisation

Table 3.8 1Y EURUSD Black–Scholes prices under SABR

Table 4.1 Flat forward volatility interpolation by smile strike

Table 4.2 EURUSD market strangle and risk reversal at 1Y and 2Y

Table 4.3 EURUSD smile at 1Y and 2Y

Table 4.4 EURUSD smile at 1Y and 2Y with consistent market conventions

Table 4.5 Interpolated 18M EURUSD smile

Table 4.6 Typical term structure of volatility on a Friday

Table 4.7 Sample shortdated EURUSD day weights

Table 4.8 Sample intraday EURUSD weights

Table 5.1 A trivial upward sloping two-period term structure of implied volatility

Table 5.2 Forward volatility consistent with upward sloping implied volatility

Table 5.3 Implied and forward volatilities for a typical ATM volatility structure

Table 5.4 Example of implied volatility surface with convexity only beyond 1Y

Table 5.5 Example of local volatility surface with convexity only beyond 1Y

Table 6.1 Common stochastic volatility models in the literature

Table 6.2 Parameters of the Heston model

Table 6.3 Violation of Heston Feller condition in typical FX markets

Table 6.4 Parameters of the Stein and Stein model

Table 6.5 Violation of Heston Feller condition even after 65% mixing weight

Table 7.1 Standard errors halved by quadrupling Nsims

Table 7.2 Standard error multiplied by

Table 7.3 European call: standard error multiplied by for increasing Nsims

Table 7.4 European call: standard error multiplied by for increasing Nsims and Ntimes

Table 7.5 Importance sampling example

Table 8.1 American digitals/binaries

Table 8.2 Barrier options

Table 10.1 Arbitrage trade for inconsistent currency triangle

Table 11.1 Typical basis swap spreads as of September 2007 (Source: Lehman Brothers)