Is the difference between deep hedging and delta hedging a statistical arbitrage?

Abstract: The recent work of Horikawa and Nakagawa (2024) claims that under a complete market admitting statistical arbitrage, the difference between the hedging position provided by deep hedging and that of the replicating portfolio is a statistical arbitrage. This raises concerns as it entails that deep hedging can include a speculative component aimed simply at exploiting the structure of the risk measure guiding the hedging optimisation problem. We test whether such finding remains true in a GARCH-based market model, which is an illustrative case departing from complete market dynamics. We observe that the difference between deep hedging and delta hedging is a speculative overlay if the risk measure considered does not put sufficient relative weight on adverse outcomes. Nevertheless, a suitable choice of risk measure can prevent the deep hedging agent from engaging in speculation.