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Customer Services Department, Macmillan Distribution Ltd, Houndmills, Basingstoke, Hampshire RG21 6XS, England An Option Greeks Primer – Building Intuition with Delta Hedging and Monte Carlo Simulation Using Excel Jawwad Ahmed Farid Fellow Society of Actuaries © Jawwad Farid 2015 Softcover reprint of the hardcover 1st edition 2015 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 unauthorized 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 1988. First published 2015 by PALGRAVE MACMILLAN Palgrave Macmillan in the UK is an imprint of Macmillan Publishers Limited, registered in England, company number 785998, 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 10010. 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: 978–1–137–37166–9 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 catalogue record for this book is available from the Library of Congress. ISBN 47572-8 ISBN 978-1-137-37167-6 (eBook) DOI 10.1057/9781137371676 For Amin, Salwa and Taha. As promised. May you be blessed with teachers who broaden your minds and change your lives. This page intentionally left blank Contents List of Figures xi Preface xx Acknowledgements xxiv About the Author xxvi Glossary xxvii Part I Refresher Introduction: Context 3 1 Options 3 2 Option price drivers 3 3 Greeks 4 4 Hedging and squaring 4 5 Empirical and implied volatility 4 6 In and out of money options 6 7 Deep out of money options and lottery tickets 7 8 Monte Carlo simulations in Excel 9 9 Model and methodology basics 9 10 The Black–Scholes–Merton pricing model 11 11 N ’(d1 ) and N’(d2 ) 12 12 Understanding Black–Scholes: an intuitive derivation of N(d1 ) and N(d2 ) 12 i Underlying assumptions 13 ii Estimating the price move 14 iii Plugging in the values 15 iv Receipt of stock and N(d1 ) 16 v Difference between N(d1 ) and N(d2 ) 17 1 Delta and Gamma 19 1 The five Greeks 19 2 Introducing Delta 21 i Let’s talk about Delta 21 ii Using Delta: creating the replicating portfolio 26 iii Dissecting Delta 29 3 Introducing Gamma 32 i Against strike 33 vii viii Contents ii Against time 34 iii Against volatility 42 Appendix 1 – Calculation Examples for At or Near Money Options 44 Appendix 2 – Using Greeks 50 Part II Delta Hedging 2 A Simulation Model for Delta Hedging – European Call Options 61 1 Setting the groundwork 62 2 Assumptions 62 3 Simulating the stock price 64 4 Calculating Delta 65 5 Calculating total borrowing 66 6 Putting it all together 69 7 Next steps, and questions 70 3 Delta Hedging European Put Options 71 1 Tweaking the original Monte Carlo simulation model 72 2 Assumptions 73 3 Simulating the underlying 74 4 Calculating the amount lent 76 5 Putting the rest of the sheet together 77 4 Calculating Cash P&L for a Call Option 79 1 Dissecting the P&L model 80 i Interest paid and principal borrowed for the hedge 82 ii Calculating the trading loss on account of selling low 84 iii Putting it all together 85 2 The vexing question of trading gain (loss) 88 3 Next steps, and questions 89 5 Calculating Cash P&L for a Put Option 91 1 Dissecting the P&L model 91 i Interest paid and principal borrowed for the hedge 92 ii Calculating the trading gain/loss 93 iii Putting it all together 95 Part III Building Surfaces in Excel 6 Understanding Volatility 101 1 The many flavours of volatility 101 2 Enter volatility surface 103 3 The difference between implied and local volatilities 105 i Implied volatilities 105 ii Local volatilities 106 Contents ix 7 Building Volatility Surfaces 109 1 Creating the implied volatility dataset 109 2 Building local volatility surfaces in Excel 113 3 Next steps, and questions 124 8 Forward Implied Volatilities 125 1 Comparing local, implied and forward volatilities 128 Part IV Hedging Higher-Order Greeks 9 Vega, Volga and Vanna 133 1 Vega 133 2 Vanna 134 3 Volga 135 4 Plotting Vega and Gamma 136 i Against strike 136 ii Against time 137 5 Shadow Gamma – including the impact of volatility changes 138 6 Vega, Gamma, Vanna and Volga surfaces 142 7 Next steps, and questions 144 10 Hedging Higher-Order Greeks 146 1 Hedging Gamma and Vega – framework 148 i Hedging higher-order Greeks for a single short position 149 11 Reviewing the Solver Solution 158 1 Hedging Gamma and Vega for a book of options 162 2 Hedging portfolio Vega and Gamma using Solver 166 i Constraints review 168 ii Solver solution – first pass 169 3 Minimizing hedge portfolio cost 171 4 Optimizing Delta 175 5 Next Steps, and questions 178 Part V Applications 12 Rebalancing, Implied Vol and Rho 181 1 Assumptions and securities 181 2 Rebalancing frequency and efficiency of the hedge: implications for profitability? 182 i The Gamma correction 185 3 Volatility and profitability: the question of implied volatility 186 i Implied volatility and P&L – four scenarios to set things right 188 x Contents ii Relative P&L comparison 192 iii More vexing questions 195 4 Dissecting Rho 195 5 Risk-free rates and profitability: the question of Rho 197 13 Understanding Theta 202 1 Theta against Gamma and Rho 202 i Theta for a call option 204 ii Theta for a put option 204 iii Rho for a call option 204 iv Rho for a put option 205 2 Theta plots for at the money call and put options 205 14 Option Prices and Time to Expiry 210 1 Theta relative to underlying asset value S 210 2 Theta and option time premium relative to volatility 213 3 Theta relative to time to maturity 214 4 Theta and option time premium relative to risk-free rate 215 Appendix 1 – Implied Volatility 216 Appendix 2 – Formula and Plot Reference 219 i Greeks formula summarized reference – continuous edition 221 ii Greeks formula reference – discrete edition 225 iii Greeks suspects gallery 226 iv Greeks suspects gallery – Vega and Rho edition 229 Appendix 3 – Drift, Diffusion and Volatility Drag Using MC Simulation 234 Notes 240 Further Reading 242 Index 245 List of Figures 1 Changes in gold, silver, WTI and US dollar prices 5 2 WTI prices and trading volatility 6 3 Example OIL ETF – call and put prices at a historically low implied volatility 7 4 Example