Nonparametric Risk-Neutral Return and Volatility Distributions

Abstract

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 riskneutral 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/volatility distribution, 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&P500 data, we conflrm negative volatility risk premiums and a right-shift in the future volatility distribution for higher initial volatility levels, but flnd additionally positive risk-neutral volatility skewness. Moreover, volatility skewness is more pronounced in low volatility periods. This is consistent with a large aversion towards unexpected positive volatility shocks. With respect to the risk-neutral return distribution, we conflrm overall negative skewness, but flnd that conditionally on decreasing volatility levels, the negative return skewness disappears. Concerning the risk-neutral dependence between return and volatility, we conflrm that this dependence is negative. Compared to parametric models, we flnd that risk-neutral volatility of volatility is much smaller than predicted 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 difiusion risk-neutral volatility process.