Pricing equity-linked pure endowments with risky assets that follow Lévy processes

We investigate the pricing problem for pure endowment contracts whose life contingent payment is linked to the performance of a tradable risky asset or index. The heavy tailed nature of asset return distributions is incorporated into the problem by modeling the price process of the risky asset as a finite variation Lévy process. We price the contract through the principle of equivalent utility. Under the assumption of exponential utility, we determine the optimal investment strategy and show that the indifference price solves a non-linear partial-integro-differential equation (PIDE). We solve the PIDE in the limit of zero risk aversion, and obtain the unique risk-neutral equivalent martingale measure dictated by indifference pricing. In addition, through an explicit–implicit finite difference discretization of the PIDE we numerically explore the effects of the jump activity rate, jump sizes and jump skewness on the pricing and the hedging of these contracts.