A New Nonparametric Estimate of the Risk-Neutral Density with Application to Variance Swap

Abstract

In this paper, we develop a new nonparametric approach for estimating the risk-neutral density of asset price and reformulate its estimation into a double-constrained optimization problem. We implement our approach in R and evaluate it using the S&P 500 market option prices from 1996 to 2015. A comprehensive cross-validation study shows that our approach outperforms the existing nonparametric quartic B-spline and cubic spline methods, as well as the parametric method based on the Normal Inverse Gaussian distribution. More specifically, our approach is capable of recovering option prices much better over a broad spectrum of strikes and expirations. While the other methods essentially fail for long-term options (1 year or 2 years to maturity), our approach still works reasonably well. As an application, we use the proposed density estimator to price long-term variance swaps, and our prices match reasonably well with those of the variance future downloaded from the Chicago Board Options Exchange website.