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Multilevel Monte Carlo simulation for VIX options in the rough Bergomi model

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  • Florian Bourgey
  • Stefano De Marco

Abstract

We consider the pricing of VIX options in the rough Bergomi model. In this setting, the VIX random variable is defined by the one-dimensional integral of the exponential of a Gaussian process with correlated increments, hence approximate samples of the VIX can be constructed via discretization of the integral and simulation of a correlated Gaussian vector. A Monte-Carlo estimator of VIX options based on a rectangle discretization scheme and exact Gaussian sampling via the Cholesky method has a computational complexity of order $\mathcal{O}(\varepsilon^{-4})$ when the mean-squared error is set to $\varepsilon^2$. We demonstrate that this cost can be reduced to $\mathcal{O}(\varepsilon^{-2} \log^2(\varepsilon))$ combining the scheme above with the multilevel method, and further reduced to the asymptotically optimal cost $\mathcal{O}(\varepsilon^{-2})$ when using a trapezoidal discretization. We provide numerical experiments highlighting the efficiency of the multilevel approach in the pricing of VIX options in such a rough forward variance setting.

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  • Florian Bourgey & Stefano De Marco, 2021. "Multilevel Monte Carlo simulation for VIX options in the rough Bergomi model," Papers 2105.05356, arXiv.org, revised Jun 2022.
    • Handle: RePEc:arx:papers:2105.05356
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    References listed on IDEAS

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    Cited by:

    1. Florian Bourgey & Stefano De Marco & Emmanuel Gobet, 2022. "Weak approximations and VIX option price expansions in forward variance curve models," Papers 2202.10413, arXiv.org, revised May 2022.
    2. Antoine Jacquier & Aitor Muguruza & Alexandre Pannier, 2021. "Rough multifactor volatility for SPX and VIX options," Papers 2112.14310, arXiv.org, revised Nov 2023.

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