Variance Risk Premiums

Abstract

We propose a direct and robust method for quantifying the variance risk premium on financial assets. We show that the risk-neutral expected value of return variance, also known as the variance swap rate, is well approximated by the value of a particular portfolio of options. We propose to use the difference between the realized variance and this synthetic variance swap rate to quantify the variance risk premium. Using a large options data set, we synthesize variance swap rates and investigate the historical behavior of variance risk premiums on five stock indexes and 35 individual stocks. (JEL G10, G12, G13) It has been well documented that return variance is stochastic. When investing in a security, an investor faces at least two sources of uncertainty, namely the uncertainty about the return as captured by the return variance, and the uncertainty about the return variance itself. It is important to know how investors deal with the uncertainty in return variance to effectively manage risk and allocate assets, to accurately price and hedge derivative securities, and to understand the behavior of financial asset prices in general. We develop a direct and robust method for quantifying the return variance risk premium on an asset using the market prices of options written on this asset. Our method uses the notion of a variance swap, which is an over-thecounter contract that pays the difference between a standard estimate of the realized variance and the fixed variance swap rate. Since variance swaps cost zero to enter, the variance swap rate represents the risk-neutral expected value of the realized variance. We show that the variance swap rate can be synthesized accurately by a particular linear combination of option prices. We propose to