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Abstract
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In our analysis, we consider a risk-averse market participant who is confronted with pricing, hedging and optimal investment problems in incomplete markets. Preferences have to be introduced in order to evaluate optimal prices and hedging strategies. We adapt the perspective of a rational market participant who aims to maximize his expected utility according to his level of risk aversion. Based on these preferences, the utility indifference price is that price where the decision maker is indifferent - in terms of maximal expected utility - between holding the contingent claim or not. The hedging strategy for the contingent claim is defined as the adjustment of the optimal portfolio strategy, induced by the additional liability from the claim. Mathematically speaking, we are confronted with a stochastic optimization problem in which concave functionals are maximized on spaces of stochastic integrals. We look for both the maximal value but also the corresponding optimal strategy.
We assume that the preferences of the market participant may be described by the exponential utility function, motivated by its suitable features. In case of the exponential utility function, the utility maximization problem can, via the Legendre transform, be solved by considering its dual problem, namely to find a martingale measure minimizing some functional and whose density process can be written in a particular form. A constructive characterization allows to uniquely describe the optimal martingale measure. We determine a defining equation for the optimal martingale measure, which we develop from a representation result of Grandits and Rheinl\"ander (2002). This equation, analyzed in a L\'evy process setting, provides an inspired guess for the shape of the optimal martingale measure. With this approach, the optimal martingale measure may be determined by the solution of a boundary problem which includes a nonlinear partial integro-differential equation.
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