Estimating a Density Ratio Model for Stock Market Risk and Option Demand

Option-implied risk-neutral densities are widely used for constructing forward-looking risk measures. Meanwhile, investor risk aversion introduces a multiplicative pricing kernel between the risk-neutral and true conditional densities of the underlying asset’s return. This paper proposes a simple local estimator of the pricing kernel based on inverse density weighting, and characterizes its asymptotic bias and variance. The estimator can be used to correct biased density forecasts, and performs well in a simulation study. A local exponential linear variant of the estimator is proposed to include conditioning variables. In an application, we estimate a demand-based model for S&P 500 index options using net positions data, and attribute the U-shaped pricing kernel to heterogeneous beliefs about conditional volatility.