In a jump diffusion, explain why the risk-neutral drift of the stock is r - lambda_Q k rather than r, and say precisely which parameters no-arbitrage fails to determine.

In a jump diffusion, explain why the risk-neutral drift of the stock is r - lambda_Q k rather than r, and say precisely which parameters no-arbitrage fails to determine.

Approach: Impose that the discounted stock is a martingale under Q and split its expected return into a diffusion part and a jump part, then count how many free parameters the single martingale condition constrains.

The jump term contributes lambda_Q k dt of expected return by itself, so the diffusion drift must be r - lambda_Q k for the discounted stock to be a Q-martingale, which fixes the risk-neutral drift of the total return at r. Writing S_t = S_0 exp(...) and taking the expected relative change, the compensated jump part (J - 1) dN - lambda_Q k dt has mean zero, and the compensator lambda_Q k dt is exactly what must be subtracted from the drift. The single martingale condition constrains only the sum, so the market is incomplete: the jump intensity lambda_Q and the law of the jump size under Q can each differ from their physical values, and any pair satisfying the one equation defines a valid equivalent martingale measure. A continuum of such measures prices the underlying identically and prices options differently, which is why jump parameters must be calibrated to option quotes rather than estimated from the stock's own history.

Follow-up: Under the Esscher transform, what functional form does the jump size density take under Q, and which single parameter indexes the family?

Key concepts: market incompleteness, equivalent martingale measure, jump compensator, risk-neutral drift.