Time changes and option pricing: non parametric estimation of the stochastic clock

The representation property of any semi-martingale as a Brownian motion subordinated by a possibly correlated stochastic clock is used to model the risk-neutral measure of asset prices. A procedure for the non-parametric calibration of the density of the stochastic clock using cross section of market prices is proposed. The approach maps the set of market option prices contained in bid/ask intervals to a set of compatible probability measures. In contrast to other estimation approaches that derive only one density, the proposed method derives a set containing all the measures that are compatible with the observed option prices.