Implied Volatility Smiles: Evidence From Options on Individual Equities1

We investigate the relative importance of various factors in explaining the volatility smile observed in the prices of options on individual stocks traded on the Chicago Board Options Exchange. First, we verify that, on average, the slope of the volatility smile on stock options is slightly negative, but not as steep as the smile for S&P 500 index options. Second, we find that stocks that have larger betas tend to have steeper smiles, and small stocks tend to have steeper smiles. This suggests that deviations from Black-Scholes may be related to market risk. Third, volatility smiles tend to be more negatively sloped for more actively-traded stocks. Fourth, the steepness of the S&P500 smile is significantly related to the put/call volume ratio, but not other measures of market sentiment. Fifth, the put-call ratio may have some role in explaining volatility smiles on individual stocks, but the estimated sign of the effect is not robust to sample specification. Last, contrary to the predictions of leverage-based models, firms with more leverage seem to have flatter volatility smiles. It has long been known that observed option prices tend to violate the cross-sectional restrictions imposed by the constant variance Black-Scholes and binomial models. For example, MacBeth and Merville (1979) and Rubinstein (1985) have documented that the volatility parameters implied by the market prices of different options on the same underlying stock tend to differ. This systematic difference between observed prices and theoretical values based on the constant volatility model has come to be known as the “volatility smile” or the “volatility skew.” In this paper, we attempt to empirically distinguish between alternative explanations for the observed violations of the constant volatility model by examining a large cross section of exchange traded stock options. There has been little academic research detailing the empirical characteristics of volatility smiles on individual stock options. This lack of attention may be attributed, in part, to computational difficulties or to logistical difficulties in managing the raw data. Possibly, the tremendous growth in the use of derivatives linked to fixed income instruments, currencies and stock indexes may have diverted research attention away from individual stock options. Using individual stock options on a cross section of heterogeneous firms allows us to distinguish between the effect of firm-specific factors and market-wide factors on the shape of the volatility smile. A better understanding of the factors driving the deviations from the constant volatility model will help indicate what theoretical aspects of option pricing are most important. The Black-Scholes and binomial models assume that the volatility of the underlying stock is constant and that there are no transactions costs. To explain the systematic violation of the constant volatility model, while still ruling out arbitrage, one may examine three possibilities. The first possibility is that the stock may follow a constant variance process but trading costs make it impossible to follow the dynamic replicating strategy. In this case the deviations from BlackScholes pricing may be related to supply/demand imbalances or market risk. The second possibility is that the value of the underlying firm’s assets may follow a constant variance process, but due to leverage, the stock price does not. The last possibility is that the underlying process for the value