Utility maximization in incomplete markets in the presence of claims or random endowments

Abstract

This thesis deals with two optimization problems of rational investors, who want to maximize their expected utility in the presence of random endowments or claims in incomplete markets. In Part I we focus on the problem of hedging a European contingent claim under exponential preferences in an incomplete market where prices are modeled as stochastic exponentials of additive processes. We use the martingale optimality principle and the theory of BSDEs to write the value function. First we consider the problem of the existence of an optimal strategy when the classical hypothesis of convexity of the constraint set is relaxed. We give a representation of a candidate optimal strategy as the limit inferior of strategies which are optimal for a sequence of approximating compact problems. Then, we consider the same problem in a market where the stock price is modelled as a Lévy-driven pure jump process, and analyze its welldefinedness. This leads to some conditions on the market model. Finally, we discuss the example of a cross-hedging problem and, under some assumptions on the structure of the claim, we give explicit solutions. In Part II we deal with the utility maximization and optimal asset allocation problem in the presence of a stream of stochastic contributions that cannot be fully hedged through trading in the financial market. We rely on the dynamic programming approach and the theory of viscosity solutions to solve the optimization problem in the presence of a random endowment. The properties of the value function, particularly the homogeneity, are used to reduce the HJB equation by one dimension. We can therefore apply finite difference methods to solve numerically the PDE. These results are then used to investigate target date funds. We show that stochastic contributions can play an essential role in the determination of optimal investment strategies, and we find that an age-increasing equity holding can be optimal too.