Adapted Wasserstein Distances and Stability in Mathematical Finance

Abstract

Assume that an agent models a financial asset through a measure Q with the goal to price / hedge some derivative or optimize some expected utility. Even if the model Q is chosen in the most skilful and sophisticated way, she is left with the possibility that Q does not provide an "exact" description of reality. This leads us to the following question: will the hedge still be somewhat meaningful for models in the proximity of Q? If we measure proximity with the usual Wasserstein distance (say), the answer is NO. Models which are similar w.r.t. Wasserstein distance may provide dramatically different information on which to base a hedging strategy. Remarkably, this can be overcome by considering a suitable "adapted" version of the Wasserstein distance which takes the temporal structure of pricing models into account. This adapted Wasserstein distance is most closely related to the nested distance as pioneered by Pflug and Pichler \cite{Pf09,PfPi12,PfPi14}. It allows us to establish Lipschitz properties of hedging strategies for semimartingale models in discrete and continuous time. Notably, these abstract results are sharp already for Brownian motion and European call options.