Rough volatility models are continuous time stochastic volatility models where the volatility process is driven by a fractional Brownian motion with the Hurst parameter less than half, and have attracted much attention since a seminal paper titled "Volatility is rough" was posted on SSRN in 2014 claiming that they explain a scaling property of realized variance time series . From our point of view, the analysis is not satisfactory because the estimation error of the latent volatility was not taken into account; we show by simulations that it in fact results in a fake scaling property. Motivated by this preliminary finding, we construct a quasi-likelihood estimator for a fractional stochastic volatility model and apply it to realized variance time series to examine whether the  volatility is really rough.  Our quasi-likelihood is based on a central limit theorem for the realized volatility estimation error and a Whittle-type approximation to the auto-covariance of the log-volatility process. We prove the consistency of our estimator under high frequency asymptotics, and examine by simulations the finite sample performance of our estimator . Our empirical study suggests that the volatility is indeed rough; actually it is even rougher than considered in the literature.
Rough volatility models are continuous time stochastic volatility models where the volatility process is driven by a fractional Brownian motion with the Hurst parameter smaller than half, and have attracted much attention since a seminal paper titled "Volatility is rough" was posted on SSRN in 2014 showing that the log realized volatility time series of major stock indices have the same scaling property as such a rough fractional Brownian motion has. We however find by simulations that the impressive approach tends to suggest the same roughness irrespectively whether the volatility is actually rough or not; an overlooked estimation error of latent volatility often results in an illusive scaling property. Motivated by this preliminary finding, here we develop a statistical theory for a continuous time fractional stochastic volatility model to examine whether the Hurst parameter is indeed estimated smaller than half, that is, whether the volatility is really rough. We construct a quasi-likelihood estimator and apply it to realized volatility time series. Our quasi-likelihood is based on the error distribution of the realized volatility and a Whittle-type approximation to the auto-covariance of the log-volatility process. We prove the consistency of our estimator under high frequency asymptotics, and examine by simulations its finite sample performance . Our empirical study suggests that the volatility is indeed rough; actually it is even rougher than considered in the literature.
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claiming that they explain a scaling property of realized variance time series . From our point of view, the analysis is not satisfactory because the
showing that the log realized volatility time series of major stock indices have the same scaling property as such a rough fractional Brownian motion has. We however find by simulations that the impressive approach tends to suggest the same roughness irrespectively whether the volatility is actually rough or not; an overlooked
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the latent volatility was not taken into account; we show by simulations that it in fact results in a fake
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we construct a quasi-likelihood estimator for a
here we develop a statistical theory for a continuous time
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and apply it to realized variance time series to
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Hurst parameter is indeed estimated smaller than half, that is, whether the
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We construct a quasi-likelihood estimator and apply it to realized volatility time series.
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a central limit theorem for the realized volatility estimation error
the error distribution of the realized volatility
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of our estimator