Uniform Confidence Bands for Pricing Kernels

Pricing kernels implicit in option prices play a key role in assessing the risk aversion over equity returns. We deal with nonparametric estimation of the pricing kernel (PK) given by the ratio of the risk-neutral density estimator and the historical density (HD). The former density can be represented as the second derivative w.r.t. the European call option price function, which we estimate by nonparametric regression. HD is estimated nonparametrically too. In this framework, we develop the asymptotic distribution theory of the Empirical Pricing Kernel (EPK) in the L-infinity sense. Particularly, to evaluate the overall variation of the pricing kernel, we develop a uniform confidence band of the EPK. Furthermore, as an alternative to the asymptotic approach, we propose a bootstrap confidence band. The developed theory is helpful for testing parametric specifications of pricing kernels and has a direct extension to estimating risk aversion patterns. The established results are assessed and compared in a Monte-Carlo study. As a real application, we test risk aversion over time induced by the EPK.