Is Volatility Rough
Is Volatility Rough
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发表时间：
2019-05
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通讯作者：
M. Fukasawa;Tetsuya Takabatake;Rebecca Westphal
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作者：
M. Fukasawa;Tetsuya Takabatake;Rebecca Westphal
粗糙波动率模型是连续时间随机波动率模型，其中波动率过程由Hurst参数小于一半的分数布朗运动驱动，自从一篇名为《波动性是粗糙的》的开创性论文发表以来，2014年发表在SSRN上的一篇文章显示，主要股票指数的对数实现波动率时间序列与这种粗糙分数布朗运动具有相同的标度特性议案有。然而，我们发现，通过模拟，令人印象深刻的方法往往建议相同的粗糙度，无论波动率实际上是粗糙与否，一个被忽视的潜在波动率的估计误差往往会导致一个虚幻的标度属性。基于这一初步发现，本文提出了一个连续时间分数阶随机波动率模型的统计理论，以检验Hurst参数的估计值是否确实小于一半，即波动率是否真的很粗糙。我们构造了一个拟似然估计，并将其应用于已实现波动率时间序列。我们的拟似然是基于已实现波动率的误差分布和对数波动率过程的自协方差的惠特尔型近似。我们证明了我们的估计在高频渐近下的一致性，并通过模拟来检验其有限样本性能。我们的实证研究表明，波动率确实是粗糙的，实际上它甚至比文献中考虑的更粗糙。
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.