# Deep Hedging: Learning to Remove the Drift under Trading Frictions with Minimal Equivalent Near-Martingale Measures

arXiv: https://arxiv.org/abs/2111.07844 · HTML: https://mlsanity.com/paper/2111.07844/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- Builds on 3 of its 17 references that are in the map.
- First posted on arXiv 15 Nov 2021.

## Builds on (most cited first, 3 of 3)

- [Deep Hedging](https://mlsanity.com/paper/1802.03042/) (2018)
- [A Data-driven Market Simulator for Small Data Environments](https://mlsanity.com/paper/2006.14498/) (2020)
- [Deep Hedging: Learning to Simulate Equity Option Markets](https://mlsanity.com/paper/1911.01700/) (2019)

## Built on by (most cited first, 11 of 11)

- [Quantum Deep Hedging](https://mlsanity.com/paper/2303.16585/) (2023)
- [Multi-Asset Spot and Option Market Simulation](https://mlsanity.com/paper/2112.06823/) (2021)
- [Deep Hedging: Continuous Reinforcement Learning for Hedging of General Portfolios across Multiple Risk Aversions](https://mlsanity.com/paper/2207.07467/) (2022)
- [SigFormer: Signature Transformers for Deep Hedging](https://mlsanity.com/paper/2310.13369/) (2023)
- [Risk-Neutral Market Simulation](https://mlsanity.com/paper/2202.13996/) (2022)
- [Application of Deep Reinforcement Learning to At-the-Money S&P 500 Options Hedging](https://mlsanity.com/paper/2510.09247/) (2025)
- [DYSANOS Generative Dynamic Smooth Arbitrage-free Non-parametric Option Surfaces](https://mlsanity.com/paper/2608.12587/) (2026)
- [Sig-Splines: universal approximation and convex calibration of time series generative models](https://mlsanity.com/paper/2307.09767/) (2023)
- [Generating drawdown-realistic financial price paths using path signatures](https://mlsanity.com/paper/2309.04507/) (2023)
- [Deep Hedging of Green PPAs in Electricity Markets](https://mlsanity.com/paper/2503.13056/) (2025)
- [Risk-Averse Reinforcement Learning with Itakura-Saito Loss](https://mlsanity.com/paper/2505.16925/) (2025)

## Abstract (arXiv)

We present a machine learning approach for finding minimal equivalent martingale measures for markets simulators of tradable instruments, e.g. for a spot price and options written on the same underlying. We extend our results to markets with frictions, in which case we find "near-martingale measures" under which the prices of hedging instruments are martingales within their bid/ask spread. By removing the drift, we are then able to learn using Deep Hedging a "clean" hedge for an exotic payoff which is not polluted by the trading strategy trying to make money from statistical arbitrage opportunities. We correspondingly highlight the robustness of this hedge vs estimation error of the original market simulator. We discuss applications to two market simulators.