论文标题
从粗糙到多重分裂的波动率：日志S-FBM模型
From Rough to Multifractal volatility: the log S-fBM model
论文作者
论文摘要
我们介绍一个随机度量的家族$ m_ {h，t}（d t）$，即log s-fbm，以至于$ h> 0 $，$ h> 0 $，$ m_ {h，t}（d t）= e^{ω____________{h，t}（t）（t）（t）} d t $ where $ where $ wher $ h $ - 分类的布朗运动。此外，当$ h \至0 $时，一个人具有$ m_ {h，t}（d t）\ rightarrow {\ wideTilde m} _ {t}（t}（d t）$（在弱感上），其中$ {\ widetilde m} _ {t}（d t）$是庆祝的log-norm mult log-norm multaul multemal multaul log-norm multanal toberal量子。因此，该模型允许我们在同一框架内考虑两种流行的多重分数（$ h = 0 $）和粗糙的波动性（$ 0 <H <1/2 $）模型。讨论了日志S-FBM的主要属性，并解决了其估计问题。我们显然表明，从$ \ ln（m_ {h，t}（[t，t，t+τ]））$的缩放属性的直接估计，在固定的$τ$处，可以强烈高估$ h $的价值。我们提出了一种更好的GMM估计方法，该方法在高频渐近状态下被证明是有效的。当应用于大量的经验波动数据时，我们观察到库存指数的值左右$ h = 0.1 $，而单个股票的特征是$ h $的值，这些值可能非常接近$ 0 $，因此MRM很好地描述了。 We also bring evidence that unlike the log-volatility variance $ν^2$ whose estimation appears to be poorly reliable (though used widely in the rough volatility literature), the estimation of the so-called "intermittency coefficient" $λ^2$, which is the product of $ν^2$ and the Hurst exponent $H$, appears to be far more reliable leading to values that seem to be universal for respectively all individual stocks and all stock指数。
We introduce a family of random measures $M_{H,T} (d t)$, namely log S-fBM, such that, for $H>0$, $M_{H,T}(d t) = e^{ω_{H,T}(t)} d t$ where $ω_{H,T}(t)$ is a Gaussian process that can be considered as a stationary version of an $H$-fractional Brownian motion. Moreover, when $H \to 0$, one has $M_{H,T}(d t) \rightarrow {\widetilde M}_{T}(d t)$ (in the weak sense) where ${\widetilde M}_{T}(d t)$ is the celebrated log-normal multifractal random measure (MRM). Thus, this model allows us to consider, within the same framework, the two popular classes of multifractal ($H = 0$) and rough volatility ($0<H < 1/2$) models. The main properties of the log S-fBM are discussed and their estimation issues are addressed. We notably show that the direct estimation of $H$ from the scaling properties of $\ln(M_{H,T}([t, t+τ]))$, at fixed $τ$, can lead to strongly over-estimating the value of $H$. We propose a better GMM estimation method which is shown to be valid in the high-frequency asymptotic regime. When applied to a large set of empirical volatility data, we observe that stock indices have values around $H=0.1$ while individual stocks are characterized by values of $H$ that can be very close to $0$ and thus well described by a MRM. We also bring evidence that unlike the log-volatility variance $ν^2$ whose estimation appears to be poorly reliable (though used widely in the rough volatility literature), the estimation of the so-called "intermittency coefficient" $λ^2$, which is the product of $ν^2$ and the Hurst exponent $H$, appears to be far more reliable leading to values that seem to be universal for respectively all individual stocks and all stock indices.
