Learning Risk-Neutral Implied Volatility Dynamics for Deep Hedging

Summary

The document outlines a numerical method for learning a risk-neutral measure over simulated paths of spot and option prices up to a finite horizon. The setting includes convex transaction costs and convex trading constraints. The learned measure is the minimal entropy martingale measure, obtained after training a market simulator for option prices, and can support stochastic implied-volatility modeling, risk-neutral pricing, or deep hedging.

The paper also characterizes when market dynamics are free of statistical arbitrage: without transaction costs, this holds if and only if the dynamics follow a risk-neutral measure. It presents a broader characterization when convex costs and trading constraints apply, as an analogue of the fundamental theorem of asset pricing under frictions. The excerpt describes the framework but gives no numerical benchmarks or evidence about performance on real market data.

Key ideas

• The method learns a risk-neutral measure for simulated spot and option price paths.
• It uses the minimal entropy martingale measure in a two-stage modeling approach.
• The resulting model is intended for risk-neutral pricing and deep hedging under frictions.
• The paper characterizes statistical arbitrage both without costs and with convex frictions and constraints.
• The excerpt provides no real-market performance benchmarks.

Tags

# Deep Hedging: Learning Risk-Neutral Implied Volatility Dynamics


# Deep Hedging: Learning Risk-Neutral Implied Volatility Dynamics









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.

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