Pricing, Hedging and Measuring Risk in Markets with Transaction Costs

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The objective of this project is to solve problems of hedging and pricing in markets where transaction costs are present. Closely related is the question on how to measure risk in these markets. The difficulty lies in the fact that transaction costs lead in a natural way to set-valued constructions. Set-valued risk measures for random solvency cones are studied. An example is the superhedging price of a multivariate claim under transaction costs. This link makes it possible to study price bounds that are more reasonable from a practical point of view like good deal bounds and indifference prices, which have already proven to be a powerful pricing mechanism in frictionless incomplete markets. Another goal is to solve a hedging problem in markets with transaction costs when the risk of falling short a contingent claim is evaluated by a set-valued risk measure. The above problems are solved by means of a recent convex duality theory, particularly designed for set-valued functions. Especially, dual problems for the main set-valued problem and its subproblems are established and conditions for strong duality will be studied. The chosen model itself, a set-valued optimization problem, as well as the proposed solution methods are new in mathematical finance andbeyond: The approach also extends concepts from optimization theory.Models for markets with transaction costs are more realistic than frictionless models. The finance industry benefits from research in hedging, pricing and risk managing techniques as there is a need to cope with bid-ask-spreads for traded assets caused by transaction costs. In complex market situations it is an advantage to have flexible tools for risk evaluation in terms of more than one currency and corresponding risk managing techniques. Thus, this research project leads to a better understanding of multivariate risks.