Nonparametic risk-neutral return and volatility distributions

Using a no-arbitrage condition we develop a nonparametric technique to extract the risk-neutral distribution of both asset returns and instantaneous volatilities from plain vanilla option prices. Our technique extends existing approaches that lead to risk-neutral return distributions only. In order to estimate the risk-neutral volatility distribution, we do not need to that derivatives on volatility are traded. More generally, as our method yields a nonparametric estimate of the joint risk-neutral return/volatilitydistribution, we can also estimate conditional distributions of returns given future volatility levels. This opens the possibility to answer several important questions on risk-neutral volatility distributions and, thus, volatility risk premiums. Using S&P-500 data, we confirm negative volatility risk premiums and a right-shift in the future volatility distribution for higher initial volatility levels, but find additionally positive risk-neutral volatility skewness. Moreover, volatility skewness is more pronounced inlow volatility periods. This is consistent with a large aversion towards unexpected positive volatility shocks. With respect to the risk-neutral return distribution, we confirm overall negative skewness, but find that conditionally on decreasing volatility levels, the negative return skewness disappears. Concerning the risk-neutral dependence between return and volatility, we confirm that this dependence is negative. Compared to parametric models, we find that risk-neutral volatility of volatility is much smaller thanpredicted by the popular Heston (1993) model. This indicates the necessity of a jump component in the risk-neutral return process. Furthermore, the risk-neutral volatility of volatility cannot be described by a single di®usion risk-neutral volatility process.

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