Smile Pricing Explained

Transcription

1 Smile Pricing Explained

2 Financial Engineering Explained About the series Financial Engineering Explained is a series of concise, practical guides to modern finance, focusing on key, technical areas of risk management and asset pricing. Written for practitioners, researchers and students, the series discusses a range of topics in a non-mathematical but highly intuitive way. Each self-contained volume is dedicated to a specific topic and offers a thorough introduction with all the necessary depth, but without too much technical ballast. Where applicable, theory is illustrated with real world examples, with special attention to the numerical implementation. Series Editor: Wim Schoutens, Department of Mathematics, Catholic University of Leuven. Series Advisory Board: Peter Carr, Executive Director, NYU Mathematical Finance; Global Head of Market Modeling, Morgan Stanley. Ernst Eberlein, Department of Mathematical Stochastics, University of Freiburg. Matthias Scherer, Chair of Mathematical Finance, Technische Universität München. Titles in the series: Equity Derivatives Explained, Mohamed Bouzoubaa The Greeks and Hedging Explained, Peter Leoni Forthcoming titles: Smile Pricing Explained, Peter Austing Interest Rates Explained Volume 1, Jörg Kienitz Interest Rates Explained Volume 2, Jörg Kienitz Dependence Modeling Explained, Matthias Scherer and Jan-Frederik Mai Submissions: Wim Schoutens wim@schoutens.be Financial Engineering Explained series Series Standing Order ISBN: You can receive future titles in this series as they are published by placing a standing order. Please contact your bookseller or, in case of difficulty, write to us at the address below with your name and address, the title of the series and the ISBN quoted above. Customer Services Department, Macmillan Distribution Ltd, Houndmills, Basingstoke, Hampshire RG21 6XS, England

3 Smile Pricing Explained Peter Austing Imperial College, London

4 Peter Austing 2014 All rights reserved. No reproduction, copy or transmission of this publication may be made without written permission. No portion of this publication may be reproduced, copied or transmitted save with written permission or in accordance with the provisions of the Copyright, Designs and Patents Act 1988, or under the terms of any licence permitting limited copying issued by the Copyright Licensing Agency, Saffron House, 6 10 Kirby Street, London EC1N 8TS. Any person who does any unauthorised act in relation to this publication may be liable to criminal prosecution and civil claims for damages. The author has asserted his right to be identified as the author of this work in accordance with the Copyright, Designs and Patents Act First published 2014 by PALGRAVE MACMILLAN Palgrave Macmillan in the UK is an imprint of Macmillan Publishers Limited, registered in England, company number , of Houndmills, Basingstoke, Hampshire RG21 6XS. Palgrave Macmillan in the US is a division of St Martin s Press LLC, 175 Fifth Avenue, New York, NY Palgrave Macmillan is the global academic imprint of the above companies and has companies and representatives throughout the world. Palgrave and Macmillan are registered trademarks in the United States, the United Kingdom, Europe and other countries. ISBN ISBN (ebook) DOI / This book is printed on paper suitable for recycling and made from fully managed and sustained forest sources. Logging, pulping and manufacturing processes are expected to conform to the environmental regulations of the country of origin. A catalogue record for this book is available from the British Library. A catalog record for this book is available from the Library of Congress.

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6 vii Contents List of Symbols Acknowledgements Preface xi xiii xiv 1 Introduction to Derivatives Hedging with Forward Contracts Speculation with Forward Contracts Arbitrage Vanilla Options Interest Rates Valuing a Forward Contract Key Points Further Reading 9 2 Stochastic Calculus Brownian Motion Stochastic Model for Stock Price Evolution Ito s Lemma The Product Rule Log-Normal Stock Price Evolution The Markov Property Term Structure Ito s Lemma in More than One Dimension Key Points Further Reading 20 3 Martingale Pricing Setting the Scene Tradeable Assets Zero Coupon Bond Rolling Money Market Account Choosing a Numeraire Changing the Measure Girsanov s Theorem Martingales Continuous Martingales Black Scholes Formula for a Call Option 28

7 viii Contents 3.11 At-the-Money Options The Black Scholes Equation An Elegant Derivation of the Black Scholes Formula Key Points Further Reading 39 4 Dynamic Hedging and Replication Dynamic Hedging in the Absence of Interest Rates Dynamic Hedging with Interest Rates Delta Hedging The Greeks Gamma, Vega and Time Decay Vega and Volatility Trading Key Points Further Reading 46 5 Exotic Options in Black Scholes European Options Asian Options Continuous Barrier Options The Reflection Principle The Reflection Principle with Log-Normal Dynamic Valuing Barrier Options in Black Scholes Discretely Monitored Barrier Options Key Points Further Reading 57 6 Smile Models The Volatility Smile Smile Implied Probability Distribution The Forward Kolmogorov Equation Local Volatility Key Points Further Reading 70 7 Stochastic Volatility Properties of Stochastic Volatility Models The Heston Model What Makes the Heston Model Special Solving for Vanilla Prices The Feller Boundary Condition The SABR Model The Ornstein Uhlenbeck Process Mixture Models Regime Switching Model Calibrating Stochastic Volatility Models Key Points Further Reading 95

8 Contents ix 8 Numerical Techniques Monte Carlo Monte Carlo in One Dimension Monte Carlo in More than One Dimension Variance Reduction in Monte Carlo Limitations of Monte Carlo The PDE Approach Stable and Unstable Schemes Choice of Scheme Other Ways of Improving Accuracy More Complex Contracts in PDE Solving Higher Dimension PDEs Key Points Further Reading Local Stochastic Volatility The Fundamental Theorem of On-smile Pricing Arbitrage in Implied Volatility Surfaces Two Extremes of Smile Dynamic Sticky Strike Dynamic Sticky Delta Dynamic Local Stochastic Volatility Simplifying Models Spot Volatility Correlation Term Structure Vega for a Barrier Option Simplifying Stochastic Volatility Parameters Risk Managing with Local Stochastic Volatility Models Practical Calibration Impact of Mixing on Contract Values Key Points Further Reading Volatility Products Overview Variance Swaps The Variance Swap Contract Idealised Variance Swap Trade Valuing the Idealised Trade Beauty in Variance Swaps Delta and Gamma of a Variance Swap Practical Considerations Volatility Swaps Volatility Swap in Stochastic Volatility Models and LSV Volatility Swap Versus Variance Swap Valuing a Volatility Swap Stochastic versus Local Volatility 163

9 x Contents 10.4 Forward Volatility Agreements Practicalities Key Points Further Reading Multi-Asset Overview Local Volatility with Constant Correlation Copulas Correlation Smile Marking Correlation Smile Common Correlation Products The Triangle Rule Modelling Local Correlation Practicalities Local Stochastic Correlation Valuing European Contracts Special Properties of Best-of Options Valuing a Best-of Option in Black Scholes Construction of a Joint PDF Using the Density Function for Pricing Numeraire Symmetry Baskets as Correlation Instruments Summary Key Points Further Reading 197 Afterword 198 Appendix: Measure Theory and Girsanov s Theorem 200 References 207 Further Reading 213 Index 216

10 xi List of Symbols Symbol Description Page A t Value of tradeable asset at time t 22 B Shorthand for bond price B t at time t, 33 Barrier level 143 B(S) Bump function 65 B t Value of a bond at time t, 22 Notation for Brownian motion in alternative measure 25 C({S t }) Contract payout given spot path {S t } 21 C Cholesky decomposition of correlation matrix 101 Sensitivity of a contract to change in spot level 43 δ(x) Dirac delta function 64 d 1, d 2 Standard parameters used in Black Scholes formula 31 dp Infinitesimal probability measure 25 dq dp Radon Nikodým derivative 25 dw Shorthand for Brownian increment dw t 33 dw t Brownian increment at time t 10 η Vol-of-vol or vol-of-var 74 F Shorthand for the forward level at valuation time to an expiry time T 30 F i Forward level at discrete time t i, 50 Forward levels for multiple assets distinguished using integer indices F 1,F 2, 188 F i (S) Smile implied cumulative probability distribution for asset i 174 F t Shorthand for the forward to expiry time T as measured at time t 83 F t Filtration at time t 17, 204 Ɣ Second order sensitivity of contract price to spot level 44 K Strike 2 L Barrier level, 53 Matrix discretisation of differential operator 109 L Differential operator 106 λ Mean reversion rate 74 m T M T Minimum value taken by a Brownian motion in time interval [0, T] 51 Maximum value taken by a Brownian motion in time interval [0, T] 144

11 xii List of Symbols μ Drift of a stochastic process, 14 Mean return 149 N Notional amount, 2 Number in a sequence, e.g. Monte Carlo paths 99 N(0,1) Standard normal distribution 11 N(x) Cumulative normal function 31 N 2 (x 1,x 2 ;ρ) Bivariate cumulative normal function with correlation ρ 174 Set of all possible outcomes of a random process 200 P Probability measure 25 PV Present value of a contract 30 ρ Sensitivity of contract price to interest rate, 44 Correlation, 19 Correlation matrix 101 r Continuously compounding interest rate 6 r(t) Continuously compounding interest rate applying instantaneously at time t 6 r dom Continuously compounding interest rate of the natural pricing currency (domestic currency) for an asset 7, 31 r yield Continuously compounding yield rate of an asset 7, 31 σ Constant volatility, 14 Terminal volatility, 18 Shorthand for stochastic instantaneous volatility σ t applying at time t 72 σ ATM At-the-money volatility 93 σ imp (K,T) Implied volatility at strike K and expiry time T 58 σ local (S,t) Local volatility at spot S and time t 66 σ realised Volatility realised in a time interval [0,T] 72 σ t Instantaneous volatility applying at time t 18 σ (t) Alternative representation of instantaneous volatility σ t 31 S Shorthand for spot price applying at time t 33 S i Spot level at discrete time t i, 50 Spot levels for multiple assets distinguished using integer indices S 1,S 2, 172 S t Spot level at time t 2 S T Spot level at contract expiry time T 2 Sensitivity of contract price to passage of time 44 t Time, measured in years from now 2 T Time to contract expiry in years from now 2 V Realised variance 149 vega Sensitivity of contract price to volatility 44 W Shorthand for Brownian motion W t at time t 33 W t Brownian motion at time t 10 W i Multiple Brownian motions distinguished using integer indices W 1,W 2, 19 Z Standard normal random variable 17 Z T Radon Nikodým derivative applying in time interval [0,T] 25, 205

12 xiii Acknowledgements I am indebted to the many friends and colleagues from whom I have learned this trade, and who have generously offered their insight and support including Quentin Adam, Mariam Aitichou, Jennifer Austing, Richard Austing, Guillaume Bascoul, Marko Bastianic, Marouane Benchekroun, Oleg Butkovsky,IainClark, Jeremy Cohen, John Darlington, Houman Falakshahi, Gareth Farnan, Markus Fritz, Ian Hamilton, Johnson Han, Duncan Harrison, Robert Hayes, Peter Jäckel, Amy Kam, Piotr Karasinski, Vladislav Krasin, Mark Lenssen, Minying Lin, Alex Lipton, Vladimir Lucic, Arthur Mountain, Jean-Pierre O Brien, Neil Oliver, Vladimir Piterbarg, Juliette Pubellier, Tino Senge, David Shelton, Peter Spoida, Richard Summerbell, Lin Sun, Neil Waldie, Zoe Wang, Claudia Yastremiz and Mathieu Zaradzki.

13 xiv Preface In modern derivatives pricing, Black Scholes theory is only a starting point. Asset volatilities are not constant, but change with market conditions. Large price moves tend to be associated with periods of high market turbulence and this leads to a smile shaped curve of the volatility implied from vanilla option prices. Smile pricing is a core area of practice and research in modern quantitative finance. There are a number of models that seek to explain the volatility smile. Two famous examples are Dupire s local volatility model, and the Heston stochastic volatility model. While they agree on vanilla option prices, their asset dynamics are very different, leading to large disagreement in exotic option prices. This book aims to provide a clear but thorough explanation of the concepts of smile modelling that are at the forefront of modern derivatives pricing theory. The key models used in practice are covered, together with numerical techniques and calibration. I have kept the needs of students and time-pressed practitioners very much in mind while writing. Topics are presented succinctly, with unnecessary complexity carefully avoided. Intuition is provided before mathematics so that readers may enjoy the book without having to follow every mathematical detail. Extended calculations are rarely necessary, but where they are (as in the solution of the Heston model for example) guidance is provided for those who wish to understand the result without ploughing through the maths. Smile Pricing Explained is a self-contained textbook and desktop reference. In addition it tells a story, of which each chapter is an integral part. We start, naturally enough, right at the beginning, by using the principle of no arbitrage to value simple forward contracts. Then models are built up, starting with Black Scholes and adding complexity and numerical techniques until we can create a full local stochastic volatility model. It is only having developed all this technology that we are able to step back and understand just what it is that makes a derivative pricing model good. Peter Austing Imperial College