American options in an imperfect market with default

TL;DR

This paper develops a framework for pricing and hedging American options in an imperfect market with default risk, using nonlinear BSDEs and reflected BSDEs to characterize superhedging prices for both sellers and buyers.

Contribution

It introduces a novel approach linking American option superhedging prices to solutions of nonlinear reflected BSDEs in markets with default risk.

Findings

Seller's superhedging price equals the value of a reflected BSDE with a lower barrier.

Existence of superhedging strategies for both sellers and buyers under certain regularity conditions.

Characterization of buyer's superhedging price via a nonlinear reflected BSDE with an upper barrier.

Abstract

We study pricing and (super)hedging for American options in an imperfect market model with default, where the imperfections are taken into account via the nonlinearity of the wealth dynamics. The payoff is given by an RCLL adapted process $(\xi_t)$. We define the {\em seller's superhedging price} of the American option as the minimum of the initial capitals which allow the seller to build up a superhedging portfolio. We prove that this price coincides with the value function of an optimal stopping problem with nonlinear expectations induced by BSDEs with default jump, which corresponds to the solution of a reflected BSDE with lower barrier. Moreover, we show the existence of a superhedging portfolio strategy. We then consider the {\em buyer's superhedging price}, which is defined as the supremum of the initial wealths which allow the buyer to select an exercise time $\tau$ and a portfolio strategy $\varphi$ so that he/she is superhedged. Under the additional assumption of left upper semicontinuity along stopping times of $(\xi_t)$, we show the existence of a superhedge $(\tau, \varphi)$ for the buyer, as well as a characterization of the buyer's superhedging price via the solution of a nonlinear reflected BSDE with upper barrier.