Nonparametric estimates of option prices via Hermite basis functions

TL;DR

This paper introduces a nonparametric method for approximating European option prices using Hermite polynomial expansions of the underlying's return density, extending Black-Scholes models and evaluating empirical performance.

Contribution

It develops a novel nonparametric pricing approach based on Hermite polynomial density approximations and assesses its empirical effectiveness in option pricing.

Findings

Models perform well for options near observed strikes

Empirical accuracy depends on polynomial order and calibration method

Outperforms simple interpolation for certain strike ranges

Abstract

We consider approximate pricing formulas for European options based on approximating the logarithmic return's density of the underlying by a linear combination of rescaled Hermite polynomials. The resulting models, that can be seen as perturbations of the classical Black-Scholes one, are nonpararametric in the sense that the distribution of logarithmic returns at fixed times to maturity is only assumed to have a square-integrable density. We extensively investigate the empirical performance, defined in terms of out-of-sample relative pricing error, of this class of approximating models, depending on their order (that is, roughly speaking, the degree of the polynomial expansion) as well as on several ways to calibrate them to observed data. Empirical results suggest that such approximate pricing formulas, when compared with simple nonparametric estimates based on interpolation and extrapolation on the implied volatility curve, perform reasonably well only for options with strike price not too far apart from the strike prices of the observed sample.