A Note on Delta Hedging in Markets with Jumps

TL;DR

This paper examines the limitations of delta-hedging strategies in markets with jump processes, showing that such strategies can fail to replicate claims even in complete models due to trajectory discontinuities.

Contribution

It demonstrates that the shortcomings of delta-hedging in jump models are inherent and not solely due to market incompleteness, providing explicit examples in a complete setting.

Findings

Delta-hedging strategies can fail in complete jump models.

Discontinuities in price trajectories cause replication issues.

Hedging deficiencies are intrinsic to jumps, not market incompleteness.

Abstract

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.