Robust bounds for the American Put

TL;DR

This paper derives model-free upper bounds for American put options using European put prices and constructs the worst-case model with a corresponding superhedge, based on the left-curtain martingale transport.

Contribution

It introduces a method to find the maximal American put price consistent with European options and constructs the associated worst-case model and superhedge.

Findings

Derived the highest possible American put price from European put prices.

Constructed the worst-case model using left-curtain martingale transport.

Provided a cheapest superhedge for the American put.

Abstract

We consider the problem of finding a model-free upper bound on the price of an American put given the prices of a family of European puts on the same underlying asset. Specifically we assume that the American put must be exercised at either $T_1$ or $T_2$ and that we know the prices of all vanilla European puts with these maturities. In this setting we find a model which is consistent with European put prices and an associated exercise time, for which the price of the American put is maximal. Moreover we derive a cheapest superhedge. The model associated with the highest price of the American put is constructed from the left-curtain martingale transport of Beiglb\"{o}ck and Juillet.