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Deep Hedging: Learning Risk-Neutral Implied Volatility Dynamics

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  • Hans Buehler
  • Phillip Murray
  • Mikko S. Pakkanen
  • Ben Wood

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.

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  • Hans Buehler & Phillip Murray & Mikko S. Pakkanen & Ben Wood, 2021. "Deep Hedging: Learning Risk-Neutral Implied Volatility Dynamics," Papers 2103.11948, arXiv.org, revised Jul 2021.
    • Handle: RePEc:arx:papers:2103.11948
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    References listed on IDEAS

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    1. Rama Cont & Jose da Fonseca, 2002. "Dynamics of implied volatility surfaces," Quantitative Finance, Taylor & Francis Journals, vol. 2(1), pages 45-60.
    2. Sasha Stoikov, 2018. "The micro-price: a high-frequency estimator of future prices," Quantitative Finance, Taylor & Francis Journals, vol. 18(12), pages 1959-1966, December.
    3. Maxime Bergeron & Nicholas Fung & John Hull & Zissis Poulos, 2021. "Variational Autoencoders: A Hands-Off Approach to Volatility," Papers 2102.03945, arXiv.org.
    4. Magnus Wiese & Lianjun Bai & Ben Wood & Hans Buehler, 2019. "Deep Hedging: Learning to Simulate Equity Option Markets," Papers 1911.01700, arXiv.org.
    5. Hans Föllmer & Thomas Knispel, 2013. "Convex risk measures: Basic facts, law-invariance and beyond, asymptotics for large portfolios," World Scientific Book Chapters, in: Leonard C MacLean & William T Ziemba (ed.), HANDBOOK OF THE FUNDAMENTALS OF FINANCIAL DECISION MAKING Part II, chapter 30, pages 507-554, World Scientific Publishing Co. Pte. Ltd..
    6. Thomas Goll & Ludger Rüschendorf, 2001. "Minimax and minimal distance martingale measures and their relationship to portfolio optimization," Finance and Stochastics, Springer, vol. 5(4), pages 557-581.
    7. Marco Frittelli, 2000. "The Minimal Entropy Martingale Measure and the Valuation Problem in Incomplete Markets," Mathematical Finance, Wiley Blackwell, vol. 10(1), pages 39-52, January.
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