When the price process for a long-lived asset is of a mixed jump–diffusion type, pricing of options on that asset by arbitrage is not possible if trading is allowed only in the underlying asset and a riskless bond. Using a general equilibrium framework, we derive and analyze option prices when the underlying asset is the market portfolio with discontinuous returns. The premium for the risk of jumps and the diffusion risk forms a significant part of the prices of the options. In this economy, an attempted replication of call and put options by the Black–Scholes type of trading strategies may require substantial infusion of funds when jumps occur. We study the cost and risk implications of such dynamic hedging plans.