One of the central issues of debate among practitioners and academicians after the subprime crisis is “how to price and hedge vulnerable contingent claims, i.e., the contracts that may default before maturity.” The main ingredient to accomplish this task is modeling the default event and the expected losses after default. Two categories of models have been proposed in the literature to model the default, namely structural models and reduced-form models.The structural models in the literature are either based on the Poisson jump-diffusion models or Lévy processes in general. However, empirical observations suggest relinquishing the widely applied Lévy jump-diffusion models and motivating the development of models capable of incorporating the default clustering and contagion effect. In this article, we employ a mutually exciting Hawkes process, capable of addressing jump clustering and contagion phenomena, to model the dynamics of the underlying defaultable asset. Due to the incompleteness of the market, one cannot perform the full replication. Hence, we design the problem of hedging a defaultable claim via maximization of the mean value of exponential utility from the terminal wealth over a set of admissible strategies. We then follow the dynamic programming-based approach and characterize the value function as the largest solution to a suitable BSDE. Further, we prove the uniqueness result by establishing that the value function of an optimal investment is the unique limit of a non-increasing sequence of value functions, each of which is the unique solution of a suitable BSDE with a Lipschitz generator. Finally, the paper presents some numerical experiments to demonstrate the applicability of the proposed modeling framework and how the inclusion of jump-clustering in the model affects the prices of defaultable claims compared to models without jumps clustering.