# Nonparametric Estimation of Risk-Neutral Densities

Maria Grith, Wolfgang K. Härdle, Melanie Schienle
### Abstract

This chapter deals with nonparametric estimation of the risk neutral density. We present three different approaches which do not require parametric functional assumptions on the underlying asset price dynamics nor on the distributional form of the risk neutral density. The first estimator is a kernel smoother of the second derivative of call prices, while the second procedure applies kernel type smoothing in the implied volatility domain. In the conceptually different third approach we assume the existence of a stochastic discount factor (pricing kernel) which establishes the risk neutral density conditional on the physical measure of the underlying asset. Via direct series type estimation of the pricing kernel we can derive an estimate of the risk neutral density by solving a constrained optimization problem. The methods are compared using European call option prices. The focus of the presentation is on practical aspects such as appropriate choice of smoothing parameters in order to facilitate the application of the techniques.

Publication

In *Handbook of Computational Finance*, Jin-Chuan Duan, James E. Gentle, and Wolfgang Härdle (eds). Springer Verlag, 2011, 277-305