The Risk-Neutral Distribution of Option Returns

Abstract

Options provide a payoff structure that is very attractive to lottery investors. Despite the growing importance of options in investors' portfolios, little is known about the actual distribution of option returns. We derive solutions for the risk-neutral variance, skewness, and kurtosis of call and put option returns and document several properties of these ex-ante moments. We find that the volatility, skewness, and kurtosis of both call and put returns are higher (lower) for options that are further out-of-the-money (in-the-money). The risk-neutral volatility of call returns is higher for options with shorter times to expiration, while the risk-neutral skewness and kurtosis are higher for calls with longer times to expiration. For put options, all three moments are higher for short-maturity options. The risk-neutral moments of call returns are increasing in the volatility of the underlying security, while the opposite is true for put returns. Risk-neutral call return moments exhibit strong positive time-series correlation, as do put return moments. Call return moments have strong negative time-series correlation with put return moments. Comparing the risk-neutral return moments to physical return moments, we find that the magnitudes of the risk-neutral and physical moments differ substantially, indicating risk premia associated with option return moments.