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.

Non-structural approach to implied moments extraction

“…Throughout the recent literature, there are papers using semiparametric and nonparametric approaches for estimating implied probability distribution (Breeden & Litzenberger, 2014;Datta, Londono, & Ross, 2017;Malz, 2014;Smith, 2012;Tavin, 2011), some of them are comparing parametric and nonparametric approaches (Aparicio & Hodges, 1998;Mizrach, 2010;Xiao & Zhou, 2017), while several are dealing with parametric (Arneri c et al, 2015;Cheng, 2010;Gemmill & Saflekos, 2000;Grith & Kr€ atschmer, 2011;Khrapov, 2014;S€ oderlind, 2000;V€ ah€ amaa, 2005) or nonparametric approaches only (Andersen & Wagener, 2002;Bahaludin & Abdullah, 2017;Figlewski, 2009;Grith, H€ ardle, & Schienle, 2011;Jackwerth & Rubinstein, 1996). There are only few papers that compare various non-structural models for implied probability distribution estimation (Bliss & Panigirtzoglou, 2002;Coutant et al, 2001;Gemmill & Saflekos, 2000;Jackwerth, 1999;Jondeau et al, 2007;Jondeau & Rockinger, 2000;Santos & Guerra, 2015).…”

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.…”

Nonparametric estimates of pricing functionals

“…One of the first works applying non-parametric regression à la Nadaraya-Watson in pricing problems is [1], where the authors, among other things, report impressive empirical results on the precision of semi-parametric estimators applied to pricing European call options on the S&P500. To date, a large literature deals with this approach to pricing -see, e.g., [7] and references therein, as well as [6,14] for related ideas.…”