Robustness of Delta Hedging in a Jump-Diffusion Model

Abstract

Suppose an investor aims at Delta hedging a European contingent claim $h(S(T))$ in a jump-diffusion model, but incorrectly specifies the stock price's volatility and jump sensitivity, so that any hedging strategy is calculated under a misspecified model. When does the erroneously computed strategy super-replicate the true claim in an appropriate sense? If the misspecified volatility and jump sensitivity dominate the true ones, we show that following the misspecified Delta strategy does super-replicate $h(S(T))$ in expectation among a wide collection of models. We also show that if a robust pricing operator with a whole class of models is used, the corresponding hedge is dominating the contingent claim under each model in expectation. Our results rely on proving stochastic flow properties of the jump-diffusion and the convexity of the value function. In the pure Poisson case, we establish that an overestimation of the jump sensitivity results in an almost sure one-sided hedge. Moreover, in general the misspecified price of the option dominates the true one if the volatility and the jump sensitivity are overestimated.