Equilibrium, market imperfections and valuation of derivatives.

Summary This thesis deals with option pricing problems in incomplete markets. It is divided into two parts, one dealing with generic incomplete problems in discrete time and the second with special cases of imperfections in continuous time. In the first part, the fundamental assumption is the equilibrium between supply and demand. In a multi-period model based on consumption, an agent seeks to value a financial product by maximizing his marginal utility at each date. In markets where only the process actually followed by the underlying asset is known, the arbitrage conditions do not sufficiently constrain the space of admissible "risk-neutral" probabilities. We show that, in several situations, the equilibrium assumption allows us to reduce this space by using the necessary properties satisfied by the admissible transition probabilities and to obtain a narrower range of prices, without specifying the agent's utility function (VNM). In the second part of the paper, we are interested in incompleteness situations arising from imperfections in the coverage of the derivative to be valued. The problems studied are those of transaction costs and illiquidity on the underlying. We treat the seller's problem by an optimal control procedure on the control variables. In the situation with transaction costs, we show that the use of a correlated asset in the hedge is a simple method to reduce the risk premium and thus be more competitive. In the context of a market with restrictions on the quantity of the underlying asset traded, management is more "static" than in the unconstrained setting. Our results allow us to better represent the management methods used in the markets, while keeping the parameterization simple, since we only add the agent's risk aversion factor.