Game options in an imperfect market with default

Abstract: We study pricing and superhedging strategies for game options in an imperfect market with default. We extend the results obtained by Kifer in \cite{Kifer} in the case of a perfect market model to the case of an imperfect market with default, when the imperfections are taken into account via the nonlinearity of the wealth dynamics. We introduce the {\em seller's price} of the game option as the infimum of the initial wealths which allow the seller to be superhedged. We {prove} that this price coincides with the value function of an associated {\em generalized} Dynkin game, recently introduced in \cite{DQS2}, expressed with a nonlinear expectation induced by a nonlinear BSDE with default jump. We moreover study the existence of superhedging strategies. We then address the case of ambiguity on the model, - for example ambiguity on the default probability - and characterize the robust seller's price of a game option as the value function of a {\em mixed generalized} Dynkin game. We study the existence of a cancellation time and a trading strategy which allow the seller to be super-hedged, whatever the model is.