Volatility Smirk Research Articles

Volatility Dynamics for the S&P500: Evidence from Realized Volatility, Daily Returns and Option Prices

Most recent empirical option valuation studies build on the affine square root (SQR) stochastic volatility model. The SQR model is a convenient choice, because it yields closed-form solutions for option prices. However, relatively little is known about the resulting biases. We investigate alternatives to the SQR model, by comparing its empirical performance with that of five different but equally parsimonious stochastic volatility models. We provide empirical evidence from three different sources: realized volatilities, S&P500 returns, and an extensive panel of option data. The three sources of data we employ all point to the same conclusion: the SQR model is misspecified. The best of the alternative volatility specifications is a model with linear rather than square root diffusion for variance, which we refer to as the VAR model. This model captures the stylized facts in realized volatilities, it performs well in fitting various samples of index returns, and it has the lowest option implied volatility mean squared error in- and out-of-sample. It fits the option data better than the SQR model in several dimensions: it improves the fit of at-the-money options, and it provides a more realistic volatility term structure and implied volatility smirk.

Crash–O–Phobia

From a large options data set on major equity indexes across the world, the authors find that worldwide, implied volatilities of options on equity indexes exhibit strikingly similar behaviors. When plotted against moneyness, implied volatilities show a heavily skewed smirk pattern, implying that out–of–the–money put options are more expensive than the corresponding out–of–the–money call options and that the risk–neutral return distribution for these indexes is heavily negatively skewed. Furthermore, as the option maturity increases from one month to five years, the implied volatility smirk does not flatten out but steepens, indicating that the risk–neutral distribution of equity index returns becomes even more negatively skewed at longer horizons. The average term structure of the implied volatility level is relatively flat, and the standard deviation of the implied volatility declines as maturity increases. Although fairly persistent, the implied volatility series are stationary. Finally, principal component analysis reveals that equity index volatility movements across the world share one global component.

Crash-O-Phobia: A Domestic Fear or A Worldwide Concern?

From a large options data set on major equity indexes across the world, we find that worldwide, implied volatilities of options on equity indexes exhibit strikingly similar behaviors. When plotted against moneyness, implied volatilities show a heavily skewed smirk pattern, implying that out-of-the-money put options are more expensive than the corresponding out-of-the-money call options, and that the risk-neutral return distribution for these indexes is heavily negatively skewed. Furthermore, as the option maturity increases from one month to five years, the implied volatility smirk does not flatten out, but steepens, indicating that the risk-neutral distribution of equity index returns becomes even more negatively skewed at longer horizons. The average term structure of the implied volatility level is relatively flat, and the standard deviation of the implied volatility declines as maturity increases. Although fairly persistent, the implied volatility series are stationary. Finally, principal component analysis reveals that equity index volatility movements across the world share one global component.

The Finite Moment Log Stable Process and Option Pricing

ABSTRACTWe document a surprising pattern in S&P 500 option prices. When implied volatilities are graphed against a standard measure of moneyness, the implied volatility smirk does not flatten out as maturity increases up to the observable horizon of two years. This behavior contrasts sharply with the implications of many pricing models and with the asymptotic behavior implied by the central limit theorem (CLT). We develop a parsimonious model which deliberately violates the CLT assumptions and thus captures the observed behavior of the volatility smirk over the maturity horizon. Calibration exercises demonstrate its superior performance against several widely used alternatives.