Hedging of derivatives by local minimization of convex risk criteria.

Summary In this thesis, we are interested in hedging derivatives in incomplete markets. The chosen approach can be seen as an extension of M. Schweizer's work on local minimization of quadratic risk. Indeed, while remaining within the framework of asset modeling by semimartingales, our method consists in replacing the quadratic risk criterion by a more general risk criterion, in the form of a convex functional of the local cost. We first obtain existence, uniqueness and characterization results for optimal strategies in a frictionless market, in discrete and continuous time. Then we explain these strategies in the framework of diffusion models with and without jumps. We also extend our method to the case where liquidity is no longer infinite. Finally, we show through numerical simulations the effects of the choice of the risk functional on the constitution of the optimal portfolio.