On certain representations of pricing functionals (Q6536771)

From the abstract: ``We revisit two classical problems: the determination of the law of the underlying with respect to a risk-neutral measure on the basis of option prices, and the pricing of options with convex payoffs in terms of prices of call options with the same maturity (all options are European).''\N\NThe pricing functional \(\pi\) is defined in the paper as the price at time zero of a European option with maturity \(T\) and payoff profile \(g : R_+ \to R \) calculated with respect to a risk-neutral probability measure in the standard way.\N\NThe author considers problem of reconstructing the law of the underlying from option prices, topics considered since the 1970s (see, for example, [\textit{D. T. Breeden} and \textit{R. H. Litzenberger}, ``Prices of state-contingent claims implicit in option prices'', J Bus 51, No. 4, 621--651 (1978)]).\N\NAlso considered is the problem of how, knowing the values of the \(\pi\) on the set of payoff functions \(G\), compute \(\pi(f)\) for certain functions \(f\) that do not belong to \(G\). It is shown that prices of call options for all strikes determine the prices of options with payoff functions that can be written as the difference of two convex functions.\N\NA related problem on the reconstruction of put option prices in an approximation scheme is also discussed.\N\NIt is worth to note that some results do not rely on any special assumptions on the law of underlying, and thus extend existing results in the literature.