We study the pricing and hedging of contingent claims that are subject to Event Risk which we define as rare and unpredictable events whose occurrence may be correlated to, but cannot be hedged perfectly with standard marketed instruments. The super and sub-replication costs of such event sensitive contingent claims (ESCC), in general, provide little guidance for the pricing of these claims. Instead, we study utility based prices of ESCC under two scenarios of resolution of uncertainty for event risk: when the event is continuously monitored or when it is revealed only at the payment date. In both cases, we are able transform the incomplete market optimal portfolio choice problem of an agent endowed with an ESCC into a complete market problem with a state and possibly path dependent utility function. For negative exponential utility, we obtain an explicit representation of the utility based prices under both information resolution scenarios and this in turn leads us to a simple characterization of the early resolution premium. For CRRA utility functions we propose a simple numerical scheme to compute both late and early resolution prices and study the impact of size of the position, wealth and expected return on these prices.