Nonparametric Estimation of Risk-Neutral Densities

Abstract

Summary. 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.

Application of Predictive Methods to Financial Data Sets

“…There are many kernels like cosine, Gaussian, Epanechnikov and Triangular kernels. The general formulae for kernel density estimation are Grith et al (2010) derived the risk neutral density using the kernel density estimation. neutral measure the discounted expectation of a finan-The main idea comes from this fact that under the risk neutral measure the discounted expectation of a financial derivative is its price.…”

Conditional risk-neutral density from option prices by local polynomial kernel smoothing with no-arbitrage constraints

“…In the simulation, we have a very controlled environment with equally-spaced strikes, an equal number of observations for each strike; however, prices are highly perturbed. As it is also demonstrated in the literature associated with RND estimation, obtaining a good fit for the mean pricing function does not constitute a great challenge, in opposition to its second derivative estimation (Aït-Sahalia and Duarte, 2003;Yatchew and Härdle, 2006;Birke and Pilz, 2008;Grith et al, 2012).…”

Imposing no‐arbitrage conditions in implied volatilities using constrained smoothing splines

“…A related literature is linked to non‐parametric estimation of the risk‐neutral density, as in 8, 10, 23, 24. In this literature, the objective is to recover the risk‐neutral density as the non‐parametric estimate of the second derivative of the pricing function.…”