Abstract : We study the superhedging prices and the associated superhedging strategies for European options in a nonlinear incomplete market model with default. The underlying market model consists of one risk-free asset and one risky asset, whose price may admit a jump at the default time. The portfolio processes follow nonlinear dynamics with a nonlinear driver $f$. By using a dynamic programming approach, we first provide a dual formulation of the seller's (superhedging) price for the European option as the supremum, over a suitable set of equivalent probability measures $Q \in \mathcal{Q}$, of the $f$ - evaluation/expectation under $Q$ of the payoff. We also establish a characterization of the seller's (superhedging) price as the initial value of the minimal supersolution of a constrained backward stochastic differential equation with default. Moreover, we provide some properties of the terminal profit made by the seller, and some results related to replication and no-arbitrage issues. Our results rely on first establishing a nonlinear optional and a nonlinear predictable decomposition for processes which are $\mathcal{E}^f$-strong supermartingales under $Q$ for all $Q \in \mathcal{Q}$. Joint work with M. Grigorova and A. Sulem.

Keywords : incomplete market; superhedging; nonlinear option pricing; constrained BSDE; control problem with 𝑓-expectation; nonlinear optional decomposition; pricing-hedging duality