On certain representations of pricing functionals

TL;DR

This paper revisits classical problems in financial mathematics related to determining the underlying asset's law from option prices and pricing convex payoffs using call options, offering new proofs that are broadly applicable.

Contribution

It provides novel, assumption-light proofs for inverse problems in option pricing, extending existing results without relying on specific underlying distributions.

Findings

Generalized solutions for inverse option pricing problems

Proofs that do not depend on specific underlying laws

Potential extension of classical results in the literature

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). The formulation of both problems is expressed in a language loosely inspired by the theory of inverse problems, and several proofs of the corresponding solutions are provided that do not rely on any special assumptions on the law of the underlying and that may, in some cases, extend results currently available in the literature.