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Long versus short time scales: the rough dilemma and beyond

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Abstract

Using a large dataset on major FX rates, we test the robustness of the rough fractional volatility model over different time scales, by including smoothing and measurement errors into the analysis. Our findings lead to new stylized facts in the log–log plots of the second moments of realized variance increments against lag which exhibit some convexity in addition to the roughness and stationarity of the volatility. The very low perceived Hurst exponents at small scales are consistent with the rough framework, while the higher perceived Hurst exponents for larger scales lead to a nonlinear behaviour of the log–log plot that has not been described by models introduced so far.

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Notes

  1. A fractional Brownian motion (fBm) \(B_H\) of Hurst exponent \(H\in (0,1)\) is a Gaussian process with a non-trivial covariance function, namely a non-Markovian process that allows for long or short memory, according to resp. \(H>0.5\) or \(H<0.5\). The case \(H=0.5\) corresponds to the classic (Markovian) Brownian motion.

  2. http://realized.oxford-man.ox.ac.uk/data/download. The Oxford-Man Institute’s Realized Library contains a selection of daily nonparametric estimates of volatility of financial assets, including realized variance and realized kernel estimates.

  3. See, for example, the papers on the website https://sites.google.com/site/roughvol/home.

  4. In reality, we are using a version of \(M_{k,\tau ,N}(X)\) with overlapping increments. This allows us to slightly increase the convergence of this empirical absolute moment Lo and MacKinlay (1988).

  5. Note that here we assume an additive model for the variance, not for the volatility. We can justify this choice by the additive nature of the variance, which makes it possible to apply the central limit theorem and to get an asymptotic distribution of the measurement error (Rootzen 1980; Jacod and Protter 1998; Barndorff-Nielsen and Shephard 2002).

  6. We thus take into account the fact that the H perceived at large scales is less affected by noise than the H at small scales.

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Acknowledgements

We thank Elisa Alòs, Fabienne Comte, Christa Cuchiero, Eva Flonner, Gilles Pagès, Andrea Pallavicini, and Mathieu Rosenbaum for useful comments on a preliminary version. We also thank the participants of the 2019 Quantitative Methods in Finance conference, Sydney, for useful comments.

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  1. Research Center, Léonard de Vinci Pôle Universitaire, 92916, Paris La Défense, France

    Matthieu Garcin & Martino Grasselli

  2. Department of Mathematics “Tullio Levi Civita”, University of Padova, via Trieste 63, 35121, Padova, Italy

    Martino Grasselli

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  1. Matthieu Garcin
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Correspondence to Martino Grasselli.

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Garcin, M., Grasselli, M. Long versus short time scales: the rough dilemma and beyond. Decisions Econ Finan 45, 257–278 (2022). https://doi.org/10.1007/s10203-021-00358-3

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