Option hedging in a market with impact and numerical schemes for particle system based EDSRs.

Summary The classical theory of derivatives valuation is based on the absence of transaction costs and infinite liquidity. However, these assumptions are no longer true in the real market, especially when the transaction is large and the assets illiquid. The first part of this thesis focuses on proposing a model that incorporates both the transaction cost and the impact on the price of the underlying asset. We start by deriving the continuous time asset dynamics as the limit of the discrete time dynamics. Under the constraint of a zero position on the asset at the beginning and at maturity, we obtain a quasi-linear equation for the price of the derivative, in the sense of viscosity. We offer the perfect hedging strategy when the equation admits a regular solution. As for the hedging of a covered European option under the gamma constraint, the dynamic program principle used previously is no longer valid. Following the techniques of the stochastic target and the partial differential equation, we show that the price of the over-replication has become a viscosity solution of a nonlinear equation of parabolic type. We also construct the ε-optimal strategy, and propose a numerical scheme.The second part of this thesis is devoted to studies on a new numerical scheme of EDSR, based on the branching process. We first approximate the Lipschitzian generator by a sequence of local polynomials, and then apply the Picard iteration. Each Picard iteration can be represented in terms of a branching process. We demonstrate the convergence of our scheme on the infinite time horizon. A concrete example is discussed at the end in order to illustrate the performance of our algorithm.