Deep Hedging: Learning Risk-Neutral Implied Volatility Dynamics

TL;DR

This paper introduces a numerically efficient method for learning risk-neutral measures and implied volatility dynamics, enabling improved option pricing and hedging under transaction costs and trading constraints.

Contribution

It proposes a novel approach to learn risk-neutral measures from simulated paths, incorporating convex transaction costs and constraints, extending fundamental asset pricing theories.

Findings

Market dynamics are free from statistical arbitrage without transaction costs.

The method effectively trains market simulators for option prices.

It characterizes risk-neutral measures under trading frictions.

Abstract

We present a numerically efficient approach for 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, as discussed for example in our recent; 2. Find a risk-neutral density, specifically 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.