A Bayesian Estimate of the Pricing Kernel

Abstract

The article presents a Bayesian nonparametric approach to model the Pricing Kernel (PK), defined as the present value of the ratio between the risk neutral density, q, and a modified physical density, p*. The risk neutral density is estimated from option data and the modified physical density is defined as the sum with Poisson-Dirichlet weights of the risk neutral density rescaled for the equity premium and the traditional physical density, estimated from the time series of the underlying log-returns. The nonparametric approach does not impose any a priori restriction on the shape of the PK and the Bayesian component allows one to include in the physical density estimation some forward-looking information coming from the option market. As a result, the heterogeneity between the physical and the risk neutral densities, identified as a major driver of the PK puzzle, disappears and both densities are conditional on a comparable information set. The PK estimates gain accuracy in the tail estimation and display a monotonically decreasing shape over a large support of returns and across multiple time to maturities, consistently with the classical theory. The monotonic decreasing nature of the relation between PK and market returns is tested and validated by the results of the nonparametric Monotonic Relation (MR) test.

Keywords: Pricing Kernel, pricing kernel puzzle, Poisson-Dirichlet Process

JEL Classification: G13, G19