Robustness of Delta Hedging in a Jump-Diffusion Model

TL;DR

This paper investigates the conditions under which misspecified delta hedging strategies in jump-diffusion models can super-replicate European options, highlighting the effects of overestimating volatility and jump sensitivity.

Contribution

It provides theoretical conditions for super-replication under model misspecification in jump-diffusion settings, including convexity and stochastic flow properties.

Findings

Misspecified strategies dominate the true claim when volatility and jump sensitivity are overestimated.

Overestimation of jump sensitivity leads to almost sure one-sided hedges in pure Poisson cases.

Misspecified option prices are higher than true prices when parameters are overestimated.

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.