Utility-based hedging and pricing with a nontraded asset for jump processes

Optimal strategies for hedging a claim on a nontraded asset X are analyzed. The claim is valued and hedged in an exponential utility maximization frame using a correlated traded asset S. The traded asset is described as a geometric jump process and the nontraded asset as a jump-diffusion process having common jump times with S. The classical dynamic programming approach leads to characterizing the value function as a solution to the Hamilton-Jacobi-Bellman equation. Closed form formulas for the value function, in terms of a new probability measure Q equivalent to the real world probability measure P, and for the optimal investment strategy are given. Admissibility for the optimal strategy is discussed and, via a duality result, an explicit expression for the density of the minimal entropy martingale measure (MEMM) is provided. Closed form formulas are also given for the writer's indifference price in terms of Q and the MEMM.