Deep Hedging

32 Pages Posted: 20 Feb 2018 Last revised: 8 Aug 2022

See all articles by Hans Buehler

Hans Buehler

XTX Markets

Lukas Gonon

Ludwig-Maximilians-Universität München

Josef Teichmann

ETH Zurich; Swiss Finance Institute

Ben Wood

JP Morgan Chase

Date Written: February 8, 2018


We present a framework for hedging a portfolio of derivatives in the presence of market frictions such as transaction costs, market impact, liquidity constraints or risk limits using modern deep reinforcement machine learning methods.

We discuss how standard reinforcement learning methods can be applied to non-linear reward structures, i.e. in our case convex risk measures. As a general contribution to the use of deep learning for stochastic processes, we also show in section 4 that the set of constrained trading strategies used by our algorithm is large enough to ∈-approximate any optimal solution.

Our algorithm can be implemented efficiently even in high-dimensional situations using modern machine learning tools. Its structure does not depend on specific market dynamics, and generalizes across hedging instruments including the use of liquid derivatives. Its computational performance is largely invariant in the size of the portfolio as it depends mainly on the number of hedging instruments available.

We illustrate our approach by showing the effect on hedging under transaction costs in a synthetic market driven by the Heston model, where we outperform the standard “complete market” solution.

This is the "stochastic analysis" version of the paper. A version in machine learning notation is available here

There are now some code examples on GitHub.

Keywords: reinforcement learning, approximate dynamic programming, machine learning, market frictions, transaction costs, hedging, risk management, portfolio optimization

Suggested Citation

Buehler, Hans and Gonon, Lukas and Teichmann, Josef and Wood, Ben, Deep Hedging (February 8, 2018). Available at SSRN: or

Hans Buehler

XTX Markets ( email )

14-18 Handyside Street
London, Greater London N1C 4DN
United Kingdom


Lukas Gonon

Ludwig-Maximilians-Universität München ( email )

Josef Teichmann (Contact Author)

ETH Zurich ( email )

Rämistrasse 101
Zürich, 8092


Swiss Finance Institute ( email )

c/o University of Geneva
40 Bd du Pont-d'Arve
CH-1211 Geneva 4

Ben Wood

JP Morgan Chase ( email )

United Kingdom

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