A note on delta hedging in markets with jumps
Modeling 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 inpractice 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 boundeddiscounted 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.
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