The Volatility Smile. Park Curry David

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      Emanuel Derman

      The Volatility Smile

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      The Volatility Smile

      EMANUEL DERMAN

      MICHAEL B. MILLER

      with contributions by David Park

      Cover image: Under the Wave off Kanagawa by Hokusai © Fine Art Premium / Corbis Images

      Cover design: Wiley

      Copyright © 2016 by Emanuel Derman and Michael B. Miller. All rights reserved.

      Published by John Wiley & Sons, Inc., Hoboken, New Jersey.

      Published simultaneously in Canada.

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       Library of Congress Cataloging-in-Publication Data:

      Names: Derman, Emanuel, author. | Miller, Michael B. (Michael Bernard), 1973-author.

      Title: The volatility smile / Emanuel Derman, Michael B. Miller.

      Description: Hoboken, New Jersey: Wiley, 2016. | Series: The Wiley finance series | Includes index.

      Identifiers: LCCN 2016012191 (print) | LCCN 2016019398 (ebook) | ISBN 9781118959169 (hardback) | ISBN 9781118959176 (pdf) | ISBN 9781118959183 (epub)

      Subjects: LCSH: Finance–Mathematical models. | Securities–Valuation. | BISAC: BUSINESS & ECONOMICS / Finance.

      Classification: LCC HG106 .D48 2016 (print) | LCC HG106 (ebook) | DDC 332.63/228301–dc23

      LC record available at https://lccn.loc.gov/2016012191

      My job, I believe, is to persuade others that my conclusions are sound. I will use an array of devices to do this: theory, stylized facts, time-series data, surveys, appeals to introspection, and so on.

– Fischer Black

      Preface

      Academic books and papers on finance have become regrettably formal over the past 30 years, filled with postulates, theorems, and lemmas. This axiomatic approach is suitable for presenting pure mathematics, but, in our view, is inappropriate for the field of finance. In finance, ideas should come first; mathematics is simply the language that we use to express ideas and elaborate their consequences.

      We feel that the best way to learn and teach financial theory is to walk a middle line between the traditionally math-inclined academic and the stereotypically math-skeptical trader. This book tries to present a treatment of the volatility smile that combines the insight that comes from models with the practicality of the trading desk.

      The first two chapters of this book provide a close look at the theory of modeling and the principles of valuation, themes that we return to again and again throughout the book. Chapters 3 through 13 explore the Black-Scholes-Merton option pricing model. At the heart of this model is a clash with the actual behavior of markets, the contradiction of the volatility smile. We show how, despite this flaw, there are productive ways to use not only the model itself, but the principles underlying it. Finally, in Chapters 14 through 24, we explore more advanced option models consistent with the smile. These models can be grouped into three families: local volatility, stochastic volatility, and jump-diffusion. While these newer models address many of the shortcomings of the Black-Scholes-Merton model, they are themselves imperfect. As markets evolve and traders gain experience, old models inevitably fail and need modification, or are replaced by newer models. Our hope is that the principles in this book will provide readers with the ability to develop and use their own models.

      Acknowledgments

      Emanuel Derman: Over the years I have benefited from enlightening conversations with, among many others, Iraj Kani, Mike Kamal, Joe Zou, the late Fischer Black, Peter Carr, Paul Wilmott, Nassim Taleb, Elie Ayache, Jim Gatheral, and Bruno Dupire. In particular, the influence of the work of Peter Carr and Paul Wilmott will be obvious in many chapters.

      We thank Sebastien Bossu, Jesse Cole, and Tim Leung for helpful comments on the manuscript.

      About the Authors

      Emanuel Derman is a professor at Columbia University, where he directs the program in financial engineering. He was born in South Africa but has lived most of his professional life in Manhattan. He started out as a theoretical physicist, doing research on unified theories of elementary particle interactions. At AT&T Bell Laboratories in the 1980s he developed programming languages for business modeling. From 1985 to 2002 he worked on Wall Street, where he codeveloped the Black-Derman-Toy interest rate model and the local volatility model. His previous books, My Life as a Quant and Models.Behaving.Badly, were both among BusinessWeek's top 10 annual books.

      Michael B. Miller is the founder and CEO of Northstar Risk Corp. Before starting Northstar, he was the chief risk officer for Tremblant Capital and before that the head of quantitative risk management at Fortress Investment Group. He is the author of Mathematics and Statistics for Financial Risk Management, now in