American options in an imperfect complete market with default

¹ Department of Mathematics, King’s College London, Strand, London, WC2R 2LS, United Kingdom, email: roxana.dumitrescu@kcl.ac.uk
² LPMA, Université Paris 7 Denis Diderot, Boite courrier 7012, 75251 Paris Cedex 05, France, email: quenez@math.univ-paris-diderot.fr
³ INRIA Paris, 2 rue Simone Iff, CS 42112, 75589 Paris Cedex 12, France, and Université Paris-Est, email: agnes.sulem@inria.fr

Abstract

We study pricing and 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 (ξ_t). We define the seller's 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 a nonlinear expectation 𝓔^g (induced by a BSDE), which corresponds to the solution of a nonlinear reflected BSDE with obstacle (ξ_t). Moreover, we show the existence of a superhedging portfolio strategy. We then consider the buyer's price of the American option, which is defined as the supremum of the initial prices which allow the buyer to select an exercise time τ and a portfolio strategy φ so that he/she is superhedged. We show that the buyer's price is equal to the value function of an optimal stopping problem with a nonlinear expectation, and that it can be characterized via the solution of a reflected BSDE with obstacle (ξ_t). Under the additional assumption of left upper semicontinuity along stopping times of (ξ_t), we show the existence of a super-hedge (τ, φ) for the buyer.