When jumps break the hedging argument for calls

Describe pure-jump and jump-diffusion models for a stock price. Under each, can a call still be priced by a no-arbitrage replication argument using only the stock and a riskless bond? Explain when it works and when it fails.

Only in special cases. If the stock follows a pure jump process with a known jump size (e.g. Cox-Ross), each instant has two possible outcomes, so the stock and bond still span the risk and a call can be replicated. In a jump-diffusion (Merton) there are two risks, the Brownian part and jumps of random size, but only one risky hedge instrument, so the market is incomplete: replication fails unless you add more traded options or assume (as Merton did) that jump risk is diversifiable and unpriced.