Dynamic Exponential Utility Indifference Valuation

We study the dynamics of the exponential utility indifference value process C(B; α) for a contingent claim B in a semimartingale model with a general continuous filtration. We prove that C(B; α) is (the first component of) the unique solution of a backward stochastic differential equation with a quadratic generator and obtain BMO estimates for the components of this solution. This allows us to prove several new results about Ct(B; α). We obtain continuity in B and local Lipschitz-continuity in the risk aversion α, uniformly in t, and we extend earlier results on the asymptotic behavior as α ց 0 or α ր ∞ to our general setting. Moreover, we also prove convergence of the corresponding hedging strategies. 0. Introduction. One of the important problems in mathematical finance is the valuation of contingent claims in incomplete financial markets. In mathematical terms, this can be formulated as follows. We have a semi-martingale S modeling the discounted prices of the available assets and a random variable B describing the payoff of a financial instrument at a given time T. The gains from a trading strategy ϑ with initial capital x are described by the stochastic integral x + ϑ dS = x + G(ϑ). If B admits a representation as B = x + G T (ϑ) for some pair (x, ϑ), the claim B is called attainable, and its value at any time t ≤ T must equal x + G 0,t (ϑ) due to absence-of-arbitrage considerations. Incompleteness means that there are some nonattainable B, and the question is how to value those. In this paper we use the utility indifference approach to this problem. For a given utility function U and an initial capital x t at time t, we define the value C t (x t , B) implicitly by the requirement that ess sup ϑ E[U (x t +G t,T (ϑ))|F t ] = ess sup ϑ E[U (x t +C t (x t , B)+G t,T (ϑ)−B)|F t ].