  Modelling stock prices via jump processes is common in financial markets. In
practice, to hedge a contingent claim one typically uses the so-called
delta-hedging strategy. This strategy stems from the Black--Merton--Scholes
model where it perfectly replicates contingent claims. From the theoretical
viewpoint, there is no reason for this to hold in models with jumps. However in
practice the delta-hedging strategy is widely used and its potential
shortcoming in models with jumps is disregarded since such models are typically
incomplete and hence most contingent claims are non-attainable. In this note we
investigate a complete model with jumps where the delta-hedging strategy is
well-defined for regular payoff functions and is uniquely determined via the
risk-neutral measure. In this setting we give examples of (admissible)
delta-hedging strategies with bounded discounted value processes, which
nevertheless fail to replicate the respective bounded contingent claims. This
demonstrates that the deficiency of the delta-hedging strategy in the presence
of jumps is not due to the incompleteness of the model but is inherent in the
discontinuity of the trajectories.
