Optimal approximate coverage of European options.

Summary This thesis studies the discrete-time hedging of European options. In the first part, we introduce hedging restrictions in the black-scholes model: we assume that the market-maker can only hedge a fixed maximum number of times at random times of his choice. We identify the strategy that minimizes the variance of the hedging error. We show that the minimum variance is a solution of a sequence of optimal stopping problems that lead to variational inequalities (i. V. ). Using the viscosity solution technique, we study the existence and uniqueness of solutions of these i. V. And we show the convergence of the solution of the discretized problem by the finite difference method to the solution of the continuous problem. Finally, we extend these results to other criteria. In the second part, we determine the smallest initial wealth needed to over-cover the option in the black-scholes model in the following real-world setting: the market-maker can only hedge at random times of his choice. When the number of covers is fixed, we show that this price corresponds to the buy-and-hold strategy for a call, or the corresponding strategy for any option with a continuous payoff. In the case where the number can depend on the trajectory of the spot and the delta of the contingent asset black-scholes option is a finite variation process (which excludes all standard options in general), we show that the smallest price is the black-scholes price of the option.