Abstract:
We study pricing and hedging for American options in a market model with default, where there are imperfections are taken into account via the nonlinearity of the wealth dynamics. In the first part of the talk, we review some recents results obtained in the case of a complete market. In the second part of the talk, we address the pricing and superhedging issues in the case of an incomplete market. We characterise the seller’s price as the value of a mixed optimal control/optimal stopping problem with nonlinear expectations, which is shown to admit a representation as the minimal supersolution of a nonlinear constrained reflected BSDE with lower obstacle. We then show that the buyer’s price corresponds to the value of a stochastic control/optimal stopping game with $E^{g}$-expectations, which is related to the maximal sub solution of a constrained reflected BSDE with lower obstacle. We moreover show that under appropriate conditions the game admits a value. We emphasise the fact that we are able to derive a dynamic representation of the buyer’s price, our approach being based on some recent results obtained in collaboration with Elie, Sabbagh and Zhou.