Pricing Via Utility Maximization and Entropy

In a financial market model with constraints on the portfolios, define the price for a claim C as the smallest real number p such that sup π E[ U ( X T x + p , π − C )]≥ sup π E[ U ( X T x , π )], where U is the negative exponential utility function and X x , π is the wealth associated with portfolio π and initial value x . We give the relations of this price with minimal entropy or fair price in the flavor of Karatzas and Kou (1996) and superreplication. Using dynamical methods, we characterize the price equation, which is a quadratic Backward SDE, and describe the optimal wealth and portfolio. Further use of Backward SDE techniques allows for easy determination of the pricing function properties.