# Parametric Estimation of Risk Neutral Density Functions

Maria Grith, Volker Krätschmer
### Abstract

This chapter deals with the estimation of risk neutral distributions for pricing index options resulting from the hypothesis of the risk neutral valuation principle. After justifying this hypothesis, we shall focus on parametric estimation methods for the risk neutral density functions determining the risk neutral distributions. We we shall differentiate between the direct and the indirect way. Following the direct way, parameter vectors are estimated which characterize the distributions from selected statistical families to model the risk neutral distributions. The idea of the indirect approach is to calibrate characteristic parameter vectors for stochastic models of the asset price processes, and then to extract the risk neutral density function via Fourier methods. For every of the reviewed methods the calculation of option prices under hypothetically true risk neutral distributions is a building block. We shall give explicit formula for call and put prices w.r.t. reviewed parametric statistical families used for direct estimation. Additionally, we shall introduce the Fast Fourier Transform method of call option pricing developed in Carr and Madan [J. Comput. Finance 2(4):61–73, 1999]. It is intended to compare the reviewed estimation methods empirically.

Publication

In *Handbook of Computational Finance*, Jin-Chuan Duan, James E. Gentle, and Wolfgang Härdle (eds), Springer Verlag, 2011, pp 253-275