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We consider an equilibrium model within which to price financial securities in incomplete markets when agents have translation invariant preference functionals. In discrete time, the model can be solved in full generality. We prove that an equilibrium exists if the agents are sensitive to large losses and that the problem of dynamic equilibrium pricing can be reduced a recursive sequence of static one-period equilibrium problems. If the flow of information is generated by independent random walks this implies that the equilibrium dynamics can be described by a coupled system if backward stochastic difference equations which renders our model amenable to a numerical analysis. Using numerical simulations we show how the model helps to explain the skew and smile patterns frequently observed in options markets. For the particular case where agents have exponential utility functions we extend our analysis to continuous time and derive a closed form solution for equilibrium stock and option prices.
The first part of the talk is based on joint work with Patrick Cheridito, Michael Kupper and Traian Pirvu.
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