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

TL;DR

This paper investigates whether the difference between deep hedging and delta hedging constitutes a statistical arbitrage, especially in incomplete markets modeled with GARCH, and how risk measure choices influence speculative behavior.

Contribution

It demonstrates that in GARCH-based markets, the difference can be speculative unless the risk measure emphasizes adverse outcomes, highlighting the importance of risk measure selection.

Findings

Difference can be speculative without proper risk measures

Choosing appropriate risk measures prevents speculative behavior

Deep hedging's relation to statistical arbitrage depends on market completeness

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.