The Skew Risk Premium in the Equity Index Market

Abstract

This is the accepted version of the paper.This version of the publication may differ from the final published version. Permanent AbstractWe measure the skew risk premium in the equity index market through the skew swap. We argue that just as variance swaps can be used to explore the relationship between the implied variance in option prices and realized variance, so too can skew swaps be used to explore the relationship between the skew in implied volatility and realized skew. Like the variance swap, the skew swap corresponds to a trading strategy, necessary to assess risk premia in a model-free way. We find that almost half of the implied volatility skew can be explained by the skew risk premium. We provide evidence that skew and variance premia are manifestations of the same underlying risk factor in the sense that strategies designed to exploit one of the risk premia but to hedge out the other make zero excess returns. * We are thankful to

Variance and skew risk premiums for the volatility market: The VIX evidence

“…It is of interest to compare the results obtained here for the volatility index market, through the use of VIX options, and those obtained for the equity index market, through the use of S&P500 options, analyzed in Kozhan et al (2013) as these are two different asset classes (i.e., volatility vs. equity). It would allow us to assess whether there are structural differences between these two markets.…”

“…, , the log return of a position on the forward contract. The availability of these derivative products allows us to compute the variance and skew risk premiums in a model-free way as shown in the literature with the important contribution provided by Kozhan et al (2013) that we will closely follow (see also Neuberger, 2012).…”

“…To the best of our knowledge, we are only aware of the work of Huang and Shaliastovich (2014) that exploits VIX options to extract volatility higher moments (in their case the second moment, that is to say, the volatility of the VIX or the volatility of volatility) and performs a joint analysis with the second moment extracted from S&P500 options (i.e., the square of the VIX) along with high frequency quantities such as realized volatility and bipower variation (these two latter quantities allow the authors to isolate the role of jumps). As our work follows Kozhan et al (2013), it differs from Huang and Shaliastovich (2014) by focusing also on the skewness of the VIX distribution and this aspect is important as it controls the shape of the VIX option smile and is also related to the "inverse" leverage effect, or positive skew, for the volatility market. 1 What is more, it is really at the skewness level that the volatility index option market departs from the equity index option market.…”

“…and as explained in Kozhan et al (2013) it depends on the skewness of the underlying asset distribution (i.e., the skewness of the log of VIX distribution) and is implied from the options. If we define…”

“…It would be more accurate to term Equations 5 and 8 as risk-neutral and realized third moments but nevertheless we will use the word skew. Following Kozhan et al (2013), most if not all of the regressions will involve the skew swap excess return given by Equation (9) and whether the normalization by Var ( ) t T , 3/2 is performed or not is irrelevant for that quantity with the consequence that our results can be compared to those of Kozhan et al (2013). Lastly, in Kozhan et al (2013), Table 3 in that paper mentions the "normalized" implied skew and realized skew while the corresponding section (3.1 Time variation in risk premiums) refers only to third implied and realized moments (see Equations 31 and 32 in that paper).…”