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FORECASTING VOLATILITYby Stephen Figlewski

FINAL DRAFT April 24, 2004

New York University Stern School of Business 44 West 4th Street, Suite 9-160 New York, NY 10012-1126 212-998-0712 Voice 212-995-4220 FAX sfiglews@stern.nyu.edu

PREFACEThis monograph puts together results from several lines of research that I have pursued over a period of years, on the general topic of volatility forecasting for option pricing applications. It is not meant to be a complete survey of the extensive literature on the subject, nor is it a definitive set of prescriptions on how to get the best volatility forecast. While at the outset, I had hoped to find the Best Method to obtain a volatility input for use in pricing options, as the reader will quickly determine, it seems that I have been more successful in uncovering the flaws and difficulties in the methods that are widely used than I have been in determining a single optimal strategy myself. Since I am not revealing the optimal approach to volatility forecasting, the major value of this work, if any, is more to share with the reader a variety of observations and thoughts about volatility prediction, that I have arrived at after investigating the problem from a number of different angles. Two major themes emerge, both having to do with the connection, or perhaps more correctly, the possibility of a disconnection between theory and practice in dealing with volatility prediction and its role in option valuation. Two general classes of theories are involved. First, there is the statistical theory involved in modeling price behavior in financial markets. In Chapter I we bring out the distinction between a physical process and an economic process in terms of the stability of their internal structure and the prospects for making accurate predictions about them. We argue that simply applying the theoretical estimation methodology appropriate for physical processes to the economic process of price behavior in a financial market can lead one to build models that are too complex and hold inappropriately high expectations about the potential accuracy of volatility forecasts from those models. The second area where conflict between theory and practice arises is in the use of implied volatility from option market prices. The conflict comes from the disparity between the trading strategies arbitrage-based derivatives valuation models assume investors follow and what actual market participants do. In theory, the implied volatility is the market=s well-informed prediction of future volatility. In practice, however, the arbitrage trading that is supposed to force option prices into conformance with the market=s volatility expectations may be very hard to execute. It will also be less profitable and entail more risk than simple market making that maximizes order flow and earns profits from the bid-ask spread. The latter, however, does little to enforce theoretical pricing in the face of the forces of supply and demand in the market. In both cases, I try to point out important implications for estimating volatility that tend to be overlooked by those following the more traditional lines of thought. I hope the reader will find some of these insights to be of value. In the long course of this research, there have been many people who helped in many ways.

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First, I would like to thank a long series of able and patient research assistants who are responsible for much of the empirical work discussed below. They include Linda Canina, N.K. Chidambaran, Amar Gande, Clifton Green, Jeffrey Heisler, Edith Hotchkiss, and Sundar Polavaram. Other benefactors have helped by providing data. Thanks to Ajay Dravid, Arthur Ferri, Mark Flannery, Richard Levich, and Bill May. Over the years, helpful comments and suggestions have come from so many sources and at so many different presentations of portions of this work, that it is no longer feasible to mention even all of the seminars and conferences at which they were received. I therefore issue a generic thank you to all who have offered their helpful thoughts on this research over the years. You know who you are. One person who gave me valuable comments on the final manuscript, however, needs to be mentioned. Special thanks to Rob Engle whose gentle but firm defense of GARCH methods led me to reconsider whether I had given them a fair enough examination in the first version, and to do the additional research that persuaded me I had not. Finally, I am grateful to the Bank and Financial Analysts Association for the original funding that got this project started.

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Chapter I. INTRODUCTIONVolatility has become a topic of enormous importance to almost anyone who is involved in the financial markets, even as a spectator. To many among the general public, the term is simply synonymous with risk: high volatility is thought of as a symptom of market disruption. To them, volatility means that securities are not being priced fairly and the capital market is not functioning as well as it should. But for those who deal with derivative securities, understanding volatility, forecasting it accurately, and managing the exposure of their investment portfolios to its effects are crucial. Modern option pricing theory, beginning with Black and Scholes [1973], accords volatility a central role in determining the fair value for an option, or any derivative instrument with option features. While the returns volatility of the underlying asset is only one of five parameters in the basic Black-Scholes (BS) option pricing formula, its importance is magnified by the fact that it is the only one that is not directly observable. Stock price, strike price, time to option expiration, and the interest rate are all known or can be easily obtained from the market, but volatility must be forecasted. Although the realized volatility over recent periods can easily be computed from historical data, an option's theoretical value today depends on the volatility that will be experienced in the future, over the option=s entire remaining lifetime. Simply projecting observed past volatility into the future is a common way to make a forecast, but it is only one of several common methods, and need not be the most accurate. Moreover, there are numerous variations in exactly how historical price data are used in predicting volatility. Volatility forecasting is vital for derivatives trading, but it remains very much an art rather than a science, particularly among derivatives traders.

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From the beginning, volatility prediction has posed significant problems for those interested in applying derivatives valuation models, but the difficulty has become greater in recent years as the maturities of available instruments have lengthened dramatically. In the 1970s, most options trading was in equity options with maturities of only a few months. While it was recognized that a security's return volatility could be expected to change over time, as long as this only occurs gradually, it should be possible to get a reasonably good short term forecast by simply assuming that volatility over the near future will remain about the same as what was realized in the recent past. That assumption becomes less tenable the longer the maturity of the option that is being priced. Today there is active trading in derivatives of all kinds with maturities that may be 10 years or more. How should one go about calculating the appropriate volatility parameter to value a 10 year cap contract on the Deutschemark / dollar exchange rate? However one decides to make such a forecast, it is bound to be subject to considerable error. How much uncertainty is there around the best possible prediction for a time span like that? These are some of the issues we will focus on in this monograph. In the next chapters we will discuss and evaluate the major procedures for forecasting volatility, always with an eye toward prediction rather than modeling and explaining volatility behavior. Moreover, we will be most concerned with forecast accuracy, not with theoretical or econometric elegance, since elegance often comes at the expense of robustness in out-of-sample forecasting. The remainder of this introduction will consider the fundamental question of what volatility actually is and why people need to forecast it. One of the major difficulties in resolving the arguments about whether derivatives trading increases the volatility of the market is that the term is understood in different ways by different people. Restricting our attention to professional derivatives traders and securities firms who use mathematical option pricing models, and to the

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academics who build them, one might expect fairly close agreement about how to define volatility, at least as far as how it is used in the models. It turns out, however, that even among those whose object is to obtain a parameter estimate to put into a standard theoretical valuation model, there are wide disparities in what they do, and in what they ought to do, for their particular purposes. These disparities arise from differences in how volatility affects their trading strategies, and in how they understand the fundamental mechanism of security valuation in a financial market. Many of the issues we will discuss are not particularly well recognized even by the professionals involved, who for the most part think they are all doing basically the same thing.

I.1. What is Volatility?Empirical and theoretical research on security prices since the 1950s has largely supported the "efficient markets" or "random walk" model. Actually, the term random walk has a precise mathematical meaning that is not a fully accurate description of how security prices should and do move over time. But the expression was used in some of the earliest research on the topic, and being more colorful than the more precise "martingale" or "supermartin

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