Hedging and Utility Valuation of a Defaultable Claim Driven by Hawkes Processes

This article studies the problem of hedging a defaultable claim via the maximization of the mean value of exponential utility, over a set of admissible strategies. The dynamics of the underlying asset is assumed to be governed by mutually exciting Hawkes processes, which captures the jumps clustering phenomenon observed in the market.\ The resulting market is incomplete and does not allow perfect replication. Hence, a dynamic programming approach is adopted to characterize the value function as the largest solution to a suitable Backward Stochastic Differential Equation (BSDE) with a non-Lipschitz generator. The value function of the optimal investment problem and of the indifference prices are represented in terms of limits of the sequence of value functions of suitable Lipschitz BSDEs and, further, a result of uniqueness is achieved. Finally, numerical experiments are performed to demonstrate the applicability of the proposed framework and to understand the impact of the jump-clustering on the values of the claims.