Peacocks nearby: approximating sequences of measures

TL;DR

This paper investigates how well sequences of probability measures can be approximated by peacocks, which are families of measures increasing in convex order, using various metrics including Wasserstein distance, with applications in financial market consistency checks.

Contribution

It provides explicit conditions for the existence of a peacock close to a given measure sequence under different distances, extending classical results and enabling practical financial applications.

Findings

Established conditions for peacock approximation within Wasserstein distance

Extended approximation results to stop-loss, Lévy, and Prokhorov distances

Applied findings to verify consistency of option market quotes

Abstract

A peacock is a family of probability measures with finite mean that increases in convex order. It is a classical result, in the discrete time case due to Strassen, that any peacock is the family of one-dimensional marginals of a martingale. We study the problem whether a given sequence of probability measures can be approximated by a peacock. In our main results, the approximation quality is measured by the infinity Wasserstein distance. Existence of a peacock within a prescribed distance is reduced to a countable collection of rather explicit conditions. This result has a financial application (developed in a separate paper), as it allows to check European call option quotes for consistency. The distance bound on the peacock than takes the role of a bound on the bid-ask spread of the underlying. We also solve the approximation problem for the stop-loss distance, the L\'evy distance, and the Prokhorov distance.