Option Returns and the Cross-Sectional Predictability of Implied - PDF document

Option Returns and the Cross-Sectional Predictability of Implied Volatility ∗ Amit Goyal Alessio Saretto Goizueta Business School The Krannert School Emory University † Purdue University ‡ August 2006 Abstract We study the cross-section of realized stock option returns and find an economically important source of predictability in the cross-sectional distribution of implied volatility. A zero-cost trading strategy that is long (short) in straddles with a large positive (negative) forecast of the change in implied volatility forecast produces an economically important and statistically significant average monthly return. The results are robust to different market conditions, to firm risk-characteristics, to various industry groupings, to options liquidity characteristics, and are not explained by linear factor models. Compared to the market prediction, the implied volatility estimate obtained from the cross-sectional forecasting model is a more precise and efficient estimate of future realized volatility. JEL Classifications: C21, G13, G14 ∗ We thank Tarun Chordia, Laura Frieder, Robert Geske, Raffaella Giacomini, Mark Grinblatt, Richard Roll, Pedro Santa-Clara, Jay Shanken, Walter Torous and the seminar participants at Emory University, Purdue University, and UCLA for valuable suggestions. † Atlanta, GA 30322, phone: (404) 727-4825, e-mail: amit goyal@bus.emory.edu. ‡ West Lafayette, IN 47907, phone: (765) 496-7591, e-mail: asaretto@purdue.edu.

1 Introduction Volatility is central to the pricing of options as contracts on more volatile stocks are more expensive than those on less volatile stocks, ceteris paribus . An accurate predic- tion of future volatility, therefore, delivers important economic information to traders. It is not surprising that there is an extensive literature on predicting volatility. The fu- ture volatility of stocks is usually predicted using weighted averages of historical actual realized volatility, variants of GARCH models, range-based approaches, and options im- plied volatility (IV). 1 Granger and Poon (2003) survey the extant literature and conclude that the market forecast embedded in IV is the best forecast of future realized volatility. However, most studies focus on predicting the volatility of a single asset (frequently the S&P 500 index) using time-series methods. Instead, we directly examine the behavior of the cross-section of implied volatilities of all U.S. equity options. We show that there is important information in the cross-section of stock implied volatilities that leads to better predictions of future volatility than those provided by the market’s IV itself. To the best of our knowledge, this is the first paper to investigate the predictability of the cross-section of individual equity option implied volatilities. We obtain IV estimates from one month to maturity, at-the-money options since these are the most liquid contracts. Doing so also ensures that, across stocks, our sample is homogenous with respect to the contract characteristics. Since IV of at-the- money options is directly related to the underlying volatility, it carries similar statistical properties as it is very persistent. We use a system of Fama and MacBeth (1973) cross- sectional regressions to estimate a mean-reversion cross-sectional model of IV augmented with variables that improve the forecasting power. We find that a stock with an IV below the cross-sectional average and below its own twelve-month moving average has a higher IV in the next month. Similarly, a stock with IV above the cross-sectional average and above its own twelve-month moving average has a lower IV in the next month. Thus, cross-sectional regressions indicate a high degree of mean-reversion in implied volatilities. We then study the economic implications of these forecasts through options port- folio strategies. We use the out-of-sample volatility predictions produced by the cross- sectional forecasting model to sort stocks into deciles. We calculate equally-weighted portfolio monthly returns on calls and puts on stocks in each decile. To minimize the impact of microstructure effects, we eliminate stale quotes and skip a day between the 1 The literature is too voluminous to cite here. For an incomplete list, see Alizadeh, Brandt, and Diebold (2002), Andersen, Bollerslev, Diebold, and Ebens (2001), Bollerslev, Chou, and Kroner (1992), Christensen and Prabhala (1998), French, Schwert, and Stambaugh (1989), and Schwert (1989). 1

estimation of the forecasts and portfolio formation. We find that all of these portfolios are quite profitable. Calls and puts portfolios have high average returns which, however, are marked by high volatility that leads to low Sharpe ratios and negative certainty equivalents. A zero-cost trading strategy, involving a long position in a portfolio of op- tions with a large predicted increase in volatility and a short position in a portfolio of options with a large predicted decrease in volatility, is very attractive. For instance, the monthly Sharpe ratio of the calls (puts) long-short portfolio is 0.435 (0.134). We also calculate equally-weighted monthly returns on straddles. Since at-the-money straddles have very low deltas, they are postulated to benefit from the volatility forecast more directly than calls or puts. We find this to be the case as straddles portfolios still have high returns but considerably lower volatility than portfolios of calls or puts. A long- short portfolio of straddles has a monthly Sharpe ratio of 0.626. The returns to straddles portfolios are comparable to those in Coval and Shumway (2001), who report absolute returns of around 3% per week for zero-beta straddles on the S&P 500. We conduct several tests to understand the nature of these profits. The long-short straddles portfolio has higher average returns when aggregate volatility (proxied by volatility of S&P 500 index, VIX) is increasing than when it is decreasing. We also find that average returns are higher for high beta, small market capitalization, and past loser stocks. However, the profits due to IV predictability persist in any beta, size, book-to-market, and momentum portfolios indicating that the “volatility effect” is not subsumed by other effects typical of the cross-section of stock returns. Moreover, the long-short volatility strategy is quite profitable in each industry and not concentrated in any particular industry (for instance, technology, etc.). We also examine whether returns to the long-short strategy are related to aggregate risk. We use linear factor models comprising the Fama and French (1993) factors, the Carhart (1997) momentum factor, and an aggregate volatility factor proxied by changes in VIX. 2 We find that the return on the long-short straddles portfolio is negatively related to movements in the three stock market factors. This suggests that the long- short strategy is attractive because it hedges the sources of aggregate risk that are priced in the stock market. The return on the straddles portfolio is also positively related to the changes in VIX. Since a volatility premium explanation would predict a negative loading on ∆VIX, a positive loading implies that the long-short straddles portfolio is 2 Option payoffs are non-linearly related to payoffs of stocks. Therefore, a linear factor model is unlikely to characterize the cross-section of option returns. We use a linear model merely to illustrate that the option returns described in this paper are not related to aggregate sources of risk in an obvious way. 2