Log-normal mixtures in option pricing: non parametric calibration

Time subordination represents a simple but powerful tool for modeling asset dynamics. This is theoretically founded on the property of local semi-martingales which can be represented as a time changed Brownian motion as shown by Monroe (1978). Given the non- negativeness of asset prices, here, time-changed geometric Brownian motions are considered. In particular, this formulation allows to represent option prices as averages of Black and Scholes prices. The problem of a non-parametric calibration of the subordinator distribution is focused. In particular, it is shown that a cross section of market option prices imposes a set of linear inequalities that the distribution of the subordinator has to satisfy. This results in an infinite dimensional optimization problem or in a set of semi-infinite dimensional programming models. The advantages with respect to a parametric specification consist of the fact that the problem remains linear in the unknowns. Furthermore, differently from a fully unrestricted non-parametric approach, the return distribution is always continuous. Empirical results are reported by using market option prices on the FTSE index.