Derivative pricing and logarithmic portfolio optimization in incomplete markets
This thesis studies the problem of derivative pricing in incomplete markets and the problem of portfolio optimization for logarithmic utility. In an incomplete market the no arbitrage criterion does not suffice to value contingent claims any more. Each equivalent martingale measure yields a possible price. Therefore additional criteria have to be imposed for derivative pricing. One approach is to consider the martingale measure wich minimizes a certain distance. One result of this thesis is the characterization of such minimal distance martingale measures in general semimartingale models. We do not consider special distances but the whole class of f-divergence distances defined by strictly convex, differentiable functions. Another problem studied in this thesis is the determination of optimal portfolios for logarithmic utility in general semimartingale models. The solution is given explicitly in terms of the semimartingale characteristics of the price process, containing earlier results as special cases. A sufficient condition is given which turns out to be necessary as well. The sufficient part is extended in two respects: Firstly, we allow for random convex constraints, for instance, the case of short sale constraints. Secondly, the consumption clock may be stochastic as well.
derivative pricing, utility maximization, incomplete markets
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