The relation between physical and risk-neutral cumulants

Abstract

Variance swaps are natural instruments for investors taking directional bets on volatility and are often used for portfolio protection. But the crucial observation suggests that derivative professionals may desire to hedge beyond volatility risk and there exists the need to hedge higher-moment market risks, such as skewness and kurtosis risks. We propose new derivative contracts: skewness swap and kurtosis swap, which trade the forward realized third and fourth cumulants. Using S&P 500 index options data from 1996 to 2005, we document the returns of these swap contracts, i.e., skewness risk premium and kurtosis risk premium. We find that the skewness risk premium is significantly negative and kurtosis risk premium for 90 day maturity is significantly positive.