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Functional central limit theorems for rough volatility

Author

Listed:
  • Blanka Horvath
  • Antoine Jacquier
  • Aitor Muguruza
  • Andreas Sojmark

Abstract

The non-Markovian nature of rough volatility processes makes Monte Carlo methods challenging and it is in fact a major challenge to develop fast and accurate simulation algorithms. We provide an efficient one for stochastic Volterra processes, based on an extension of Donsker's approximation of Brownian motion to the fractional Brownian case with arbitrary Hurst exponent $H \in (0,1)$. Some of the most relevant consequences of this `rough Donsker (rDonsker) Theorem' are functional weak convergence results in Skorokhod space for discrete approximations of a large class of rough stochastic volatility models. This justifies the validity of simple and easy-to-implement Monte-Carlo methods, for which we provide detailed numerical recipes. We test these against the current benchmark Hybrid scheme~\cite{BLP17} and find remarkable agreement (for a large range of values of~$H$). This rDonsker Theorem further provides a weak convergence proof for the Hybrid scheme itself, and allows to construct binomial trees for rough volatility models, the first available scheme (in the rough volatility context) for early exercise options such as American or Bermudan options.

Suggested Citation

  • Blanka Horvath & Antoine Jacquier & Aitor Muguruza & Andreas Sojmark, 2017. "Functional central limit theorems for rough volatility," Papers 1711.03078, arXiv.org, revised Nov 2023.
    • Handle: RePEc:arx:papers:1711.03078
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    References listed on IDEAS

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    2. Elisa Al`os & David Garc'ia-Lorite & Aitor Muguruza, 2018. "On smile properties of volatility derivatives and exotic products: understanding the VIX skew," Papers 1808.03610, arXiv.org.
    3. Blanka Horvath & Aitor Muguruza & Mehdi Tomas, 2019. "Deep Learning Volatility," Papers 1901.09647, arXiv.org, revised Aug 2019.
    4. Christian Bayer & Benjamin Stemper, 2018. "Deep calibration of rough stochastic volatility models," Papers 1810.03399, arXiv.org.
    5. Philipp Harms, 2019. "Strong convergence rates for Markovian representations of fractional processes," Papers 1902.01471, arXiv.org, revised Aug 2020.
    6. Christian Bayer & Eric Joseph Hall & Ra'ul Tempone, 2020. "Weak error rates for option pricing under linear rough volatility," Papers 2009.01219, arXiv.org, revised Dec 2021.
    7. Blanka Horvath & Antoine Jacquier & Peter Tankov, 2018. "Volatility options in rough volatility models," Papers 1802.01641, arXiv.org, revised Jan 2019.

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