Deep Hedging: Learning Risk-Neutral Implied Volatility Dynamics

19 Pages Posted: 29 Mar 2021 Last revised: 14 Jul 2021

See all articles by Hans Buehler

Hans Buehler

JP Morgan

Murray Phillip

affiliation not provided to SSRN

Mikko Pakkanen

Imperial College London - Department of Mathematics

Ben Wood

JP Morgan Chase

Date Written: March 20, 2021

Abstract

We present a numerically efficient approach for machine-learning a risk-neutral measure for paths of simulated spot and option prices up to a finite horizon under convex transaction costs and convex trading constraints.

This approach can then be used to implement a stochastic implied volatility model in the following two steps:
1) Train a market simulator for option prices, for example as discussed in our recent work here;

2) Find a risk-neutral density, specifically in our approach the minimal entropy martingale measure.

The resulting model can be used for risk-neutral pricing, or for Deep Hedging in the case of transaction costs or trading constraints.

To motivate the proposed approach, we also show that market dynamics are free from "statistical arbitrage" in the absence of transaction costs if and only if they follow a risk-neutral measure. We additionally provide a more general characterization in the presence of convex transaction costs and trading constraints.

These results can be seen as an analogue of the fundamental theorem of asset pricing for statistical arbitrage under trading frictions and are of independent interest.

Keywords: Stochastic Implied Volatility, Deep Hedging, Minimal Entropy Martingale Measure, Statistical Arbitrage, Machine Learning, Deep Learning, Reinforcement Learning

JEL Classification: C15, C45, C5, C53, C6, C63, G0

Suggested Citation

Buehler, Hans and Phillip, Murray and Pakkanen, Mikko and Wood, Ben, Deep Hedging: Learning Risk-Neutral Implied Volatility Dynamics (March 20, 2021). Available at SSRN: https://ssrn.com/abstract=3808555 or http://dx.doi.org/10.2139/ssrn.3808555

Hans Buehler (Contact Author)

JP Morgan ( email )

4/F, 25 Bank Street
London, E14 5JP
United Kingdom

Murray Phillip

affiliation not provided to SSRN

Mikko Pakkanen

Imperial College London - Department of Mathematics ( email )

South Kensington Campus
Imperial College
LONDON, SW7 2AZ
United Kingdom

Ben Wood

JP Morgan Chase ( email )

London
United Kingdom

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