Portfolio Optimization and Optimal Martingale Measures in Markets with Jumps

Abstract

We discuss optimal portfolio selection with respect to utility functions of type exp(-ax), a>0 (exponential problem) and -|1-ax/p|^p (p-th problem). We consider N risky assets and a risk-free bond. Risky assets are modeled by continuous semimartingales or exponential Lévy processes.These dynamic expected utility maximization problems are solved by transforming the model into a constrained static version and applying convex duality. The connection between the static and the dynamic problems is drawn as follows: firstly we construct explicit portfolios (dynamic solution) attaining the optimal static values for the p-th problem, secondly we establish uniform convergence in probability of these portfolios and the corresponding wealth processes to the dynamic solution of the exponential problem. Moreover, convergence of the optimal wealth processes in a supremum norm and convergence of the terminal values in L^v, v>=1 follows under the assumption of upward bounded jumps. By construction these results yield an explicit portfolio for the exponential problem. To establish our results, we need to prove several properties on the solutions of the dual problems, i.e. the q-optimal martingale measure and the minimal entropy martingale measure.In fact, in the presence of unbounded jumps the q-optimal martingale measure (the dual solution of the static p-th problem) may fail to be equivalent.Depending on the specific formulation of the portfolio selection problem, i.e. whether or not consumption is allowed, we have to consider the signed or the absolutely continuous version of the q-optimal martingale measure. However, techniques usually applied in order to characterize the equivalent case are not suitable. An explicit form of the q-optimal signed martingale measure is therefore established by a new verification procedure via a hedging argument reversing the above duality. Admitting consumption, a superhedging argument yields an explicit form of the absolutely continuous martingale measure. The convergence of both versions of the q-optimal martingale measures to the minimal entropy martingale measure, when q tends to 1 is proved, implying the convergence of the optimal strategies (portfolio and eventually consumption) of the p-th problem to the exponential one.We close the thesis by a comparison of the achieved results in the continuousand discontinuous case.