State price density estimation via nonparametric mixtures

TL;DR

This paper introduces a nonparametric mixture approach to estimate the state price density directly from option prices, providing an efficient and convergent method despite the infinite-dimensional problem.

Contribution

It proposes a novel nonparametric mixture model for state price density estimation from option prices, with a finite-dimensional solution and proven convergence rates.

Findings

Finite-dimensional representation of the estimator

Efficient computation of the estimator

Convergence to the true density at a nearly parametric rate

Abstract

We consider nonparametric estimation of the state price density encapsulated in option prices. Unlike usual density estimation problems, we only observe option prices and their corresponding strike prices rather than samples from the state price density. We propose to model the state price density directly with a nonparametric mixture and estimate it using least squares. We show that although the minimization is taken over an infinitely dimensional function space, the minimizer always admits a finite dimensional representation and can be computed efficiently. We also prove that the proposed estimate of the state price density function converges to the truth at a ``nearly parametric'' rate.