  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.
