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Volatility of volatility estimation: central limit theorems for the Fourier transform estimator and empirical study of the daily time series stylized facts
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{\bf Keywords:} volatility of volatility, non-parametric estimation, central limit theorem, stochastic volatility, Fourier analysis.
{\bf JEL Classification:} C14, C58.
In the last decades, different stochastic volatility models have been proposed to describe the evolution of asset prices, motivated by empirical studies on the patterns of volatilities in financial time series. Further, the availability of high-frequency data has given impulse to devise statistical techniques aimed at the efficient estimation of model parameters in the stochastic volatility framework, e.g., the leverage and the volatility of volatility processes. {The estimation of these model parameters is rather complicated, the main difficulties being due to the fact that some factors are unobservable. In particular, the estimation of the volatility of volatility is a challenging task, because a pre-estimation of the spot volatility is typically required as a first step, due to the latency of the volatility process.}
Unlike the case of the integrated volatility, the non-parametric estimation of the integrated volatility of volatility is a relatively recent topic. BNVe propose a new class of stochastic volatility of volatility models, with an extra source of randomness, and show that the volatility of volatility can be estimated non-parametrically by means of the quadratic variation of the preliminarily estimated squared volatility process, which they name {\sl pre-estimated spot variance based realized variance}. Vetter proposes an estimator of the integrated volatility of volatility which is also based on increments of the pre-estimated spot volatility process and attains the optimal convergence rate in the absence of noise. The common feature of these estimators is that they first reconstruct the unobservable volatility path via some consistent estimator thereof and then compute the volatility of volatility using the estimated paths as a proxy of the corresponding unknown paths. The issue of estimating the volatility of volatility in the presence of jumps is studied in CuTe: first, the authors combine jump robust estimators of the integrated variance and the Fourier-Fej\'{e}r inversion formula to get an estimator of the instantaneous volatility path; secondly, they use again jump robust estimators of the integrated volatility, in which they plug the estimated path of the volatility process, to obtain an estimator of the volatility of volatility. In the same spirit of BNVe,Vetter, LiLiuZhang also propose an estimator of the integrated volatility of volatility by means of a pre-estimation of the spot volatility, but, in order to extend the study to the case when the observed price process contains jumps and microstructure noise, the authors adopt a threshold pre-averaging estimator of the volatility, following Jing.
In this paper, we focus on the estimation of the integrated volatility of volatility via the Fourier estimation method by MM, which does not require the pre-estimation of the spot volatility. An early application of the Fourier methodology to identify the parameters (volatility of volatility and leverage) of stochastic volatility models has been proposed by BaMa2010, where the authors prove a consistency result for the estimator of both the integrated leverage and volatility of volatility in the absence of noise. In the presence of microstructure noise, SanfeliciCuratoMancino study the finite-sample properties of the Fourier estimator of the volatility of volatility introduced in BaMa2010 and show its asymptotic unbiasedness. However, the convergence rate of the estimator is not established, not even in the absence of microstructure noise contamination.
In the present paper we fill this gap. Specifically, after proving that the Fourier estimator of the volatility of volatility by SanfeliciCuratoMancino has a sub-optimal rate of convergence, we define its bias-corrected version and prove that it reaches the optimal convergence rate \textcolor{black}{$n^{1/4}$}. { We also show that the non-corrected estimator with slower rate of convergence displays a smaller asymptotic error variance.} {\color{black} Further, we provide feasible versions of the two CLT's that exploit the product formula for the Fourier coefficients of the volatility of volatility and the fourth power of {the} volatility. The same property of the Fourier coefficients is used in LMM for the estimation of the quarticity.}
These asymptotic results are supported by a simulation exercise, where we also compare the finite-sample performance of the rate-efficient Fourier estimator with that {of the rate-efficient realized estimator based on the pre-estimation of the spot volatility by ASJ.} {\color{black} The comparative study suggests that the Fourier estimator works quite well on the daily horizon, while the performance of the realized estimator appears to be not satisfactory.} This feature may be related to the fact that, differently from the other volatility of volatility estimators, which rely on the pre-estimation of the instantaneous volatility path via a numerical differentiation, the Fourier approach relies only on the reconstruction of integrated quantities, i.e., the Fourier coefficients of the volatility. As it was early observed in MM, this is a peculiarity of the Fourier estimator that renders the proposed method easily implementable and computationally stable.
Finally, we present an empirical exercise where the Fourier estimator is applied to obtain the daily time series of the volatility of volatility of the S&P500 and EUROSTOXX50 indices over, resp., the periods May 1, 2007 - August 6, 2021 and June 29, 2005 - May 28, 2021. As a result, we obtain some novel insight into the empirical regularities that characterize the daily dynamics of the volatility of volatility, which - to the best of our knowledge - up to now had been scarcely explored in the literature. Specifically, we find that the daily volatility of volatility of both the indices spikes in correspondence of periods of financial turmoil (e.g., during the financial crisis of 2008 and the outbreak of the COVID pandemic in 2020). Additionally, we also find that it is usually positively (resp., negatively) correlated with the volatility (resp., the asset return), but appears to be less persistent than the volatility. Finally, we observe that its empirical distribution is satisfactorily approximated by a log-normal distribution in years characterized by higher financial stability, as it is the case for the volatility.
\textcolor{black}{This novel insight appears to be valuable in view of the relevance of the volatility of volatility for scholars and practitioners. Indeed, on the one hand, market operators regularly “trade" the volatility of many financial asset classes via quoted and O.T.C. volatility derivatives (e.g., variance swaps, VIX futures and VIX options), hence the importance of the availability of accurate estimates of the volatility of the “traded" volatility.} {\color{black} On the other hand, the need for efficient estimates of the volatility of volatility arises also in a number of technical tasks, e.g., the calibration of stochastic volatility of volatility models (BNVe, SanfeliciCuratoMancino), the estimation of the leverage coefficient (KX, AFLWY), the inference of future returns Bolls) and spot volatilities (MZ2009). Furthermore, BFR have recently provided empirical support to the dependence between the volatility of volatility of equity assets and structural sources of risk related to firms' characteristics.}
The paper is organized as follows. Section (ref) contains the assumptions and definitions. Section (ref) states the central limit theorems, which are supported by the simulation study in Section (ref). Finally, Section (ref) contains the empirical results and Section (ref) concludes. The proofs are given in Appendix A, while Appendix B contains some auxiliary lemmas on the Fej\'{e}r and Dirichlet kernels.
This section presents the general non-parametric stochastic volatility model which will be considered throughout the paper and defines two estimators of the integrated volatility of volatility based on the Fourier estimation method introduced in MM,MM09. The class considered includes most of the continuous stochastic volatility models commonly used in high-frequency finance and is assumed (to cite one among many others) in Chapter 8.3 of ASJ.
We make the following assumptions.
{\rm (A.I)} The log-price process $p$ and the variance process $v$ are continuous It\^o semimartingales {on $[0,T]$} satisfying the stochastic differential equations
where $v:=\sigma^2$, while $W$ and $Z$ are Brownian motions on a filtered probability space $(\Omega, ({\cal F}_t)_{t\in [0,T]}, P)$ satisfying the usual conditions, possibly correlated \textcolor{black}{(in this regard, note that it is not restrictive to assume a constant correlation $\rho$).}
{\rm (A.II)} The processes $\sigma$, $b$, $\gamma$ and $\beta$ are continuous adapted stochastic processes defined on the same probability space $(\Omega, ({\cal F}_t)_{t\in [0,T]}, P)$, such that for any $p\geq 1$, $$ E\left[\int_0^T \sigma^p(t)dt\right]< \infty \ \ , \ E\left[\int_0^T b^p(t) dt\right]<\infty \ \ , \ E\left[\int_0^T \gamma^p(t)dt\right]< \infty \ \ , \ E\left[\int_0^T a^p(t) dt\right]<\infty. $$ The processes are specified in such a way that \textcolor{black}{the spot volatility and volatility of volatility, resp. $\sigma$ and $\gamma$,} are a.s. positive.
{\rm (A.III)} The process $\gamma$ is a continuous It\^o semimartingale, whose drift and diffusion processes are continuous adapted stochastic processes defined on the same probability space $(\Omega, ({\cal F}_t)_{t\in [0,T]}, P)$.
The assumptions (A.I)-(A.II)-(A.III) are standard in the non-parametric setting and are considered, e.g., in BNVe,ASJ,CuTe,Vetter,LiLiuZhang.
By changing of the origin of time and scaling the unit of time, one can always modify the time window $[0,T]$ to $[0,2\pi]$. Suppose that the asset log-price $p$ is observed at discrete, irregularly-spaced points in time on the grid $\{ 0=t_{0,n} \leq \ldots t_{i,n} \ldots \leq t_{n,n}=2\pi \}$. For simplicity, we omit the second index $n$. Denote $\rho(n):= ~ \max_{0\leq h \leq n-1}|t_{h+1}-t_{h}|$ and suppose that $\rho(n) \to 0$ as $n\to \infty$.
Consider the following interpolation formula $$ p_{n}(t):= ~ \sum_{i=0}^{n-1} p(t_i) I_{[t_i,t_{i+1}[}(t). $$ For any integer $k$, $|k|\leq 2N$, {\color{black} the discretized version of the Fourier coefficient $c_k(dp)$ is denoted by}
{\color{black}where the symbol ${\rm i}$ is the imaginary unit $\sqrt{-1}$}. Further, for any $|k|\leq N$, define the convolution formula
In MM09 it is proved that ((ref)) is a consistent estimator of the $k$-th Fourier coefficient of the volatility process\footnote{Hereinafter, we will follow the relevant econometric literature by using the term volatility as a synonym of variance, thus referring to $\sigma^2(t)$ as the volatility process. Similarly for the volatility of volatility. } and in BaMa2010,SanfeliciCuratoMancino it is shown that it is possible to derive an estimator of the integrated volatility of volatility by exploiting only the knowledge of the Fourier coefficients in ((ref)), without the need of the preliminary estimation of the instantaneous volatility. This feature characterizes the Fourier method for estimating the volatility of volatility. In fact, as far as we know, all other existing methods rely on the pre-estimation of the spot volatility, see ASJ,CuTe,Vetter,LiLiuZhang. In general, these methods entail the pre-estimation of the spot volatility (in the absence or presence of noise contamination) as a first step; then, as a second step, a quadratic variation approach (e.g., the realized volatility formula) is applied to the pre-estimated spot volatility trajectory.
The estimator of the integrated volatility of volatility, defined in SanfeliciCuratoMancino, is given by $2\pi$ times
as $c_0(\gamma^2_{n,N,M})$ is the estimator of $c_0(\gamma^2)={1\over {2\pi}} \int_0^{2\pi}\gamma^2(t) dt$. \textcolor{black}{ SanfeliciCuratoMancino show that the estimator ((ref)) is consistent under the assumptions {\rm (A.I)}-{\rm (A.II)}-{\rm (A.III)} and the conditions $N/n \to 0$ and $M^4/N\to 0$, in the absence of microstructure noise (Theorem 3.2). Moreover, they show that the estimator ((ref)) is asymptotically unbiased in the presence of microstructure noise (Theorem 4.2). However, they do not establish the rate of convergence and the asymptotic normality. Further, note that the conditions on $n, \, N$ and $M$ assumed in Theorem 3.2 by SanfeliciCuratoMancino are only sufficient for the consistency of the estimator ((ref)); indeed, these conditions are not sharp, due to the fact that the focus of the paper is not on the rate of convergence, but on the finite-sample properties of the estimator in the presence of microstructure noise. In the present paper, we show that the convergence rate of the estimator ((ref)) is not optimal (see Theorem (ref)). Moreover, in Theorem (ref) we provide the optimal choices of the cutting frequencies $N$ and $M$ for the estimator ((ref)), in the absence of microstructure noise. In particular, note that Theorem (ref) assumes that the convolution parameter $N$ is equal to the Nyquist frequency $n/2$. In the presence of microstructure contaminations, instead, one needs to choose $N$ much smaller than $n/2$ to filter out the noise present in price observations, see Remark 4.3 in SanfeliciCuratoMancino.}
In order to obtain an estimator with the optimal rate of convergence in the absence of microstructure noise, a bias correction is needed and thus we consider the estimator
where the constant $K$ is determined in ((ref)) and $\widehat \sigma^4_{n,N,M}$ is the Fourier estimator of the quarticity, defined as
The asymptotic normality of the estimator ((ref)) is studied in LMM, while its properties in the presence of microstructure noise are studied in MSquart.
Note that the estimator ((ref)) {differs from} ((ref)) {for the presence of} the bias correction $K \, \widehat \sigma^4_{n,N,M}$. This bias correction, while ensuring a faster rate of convergence, destroys the positivity of the estimator, see also BNHLS. The estimator ((ref)) is instead positive.
In this section we study the asymptotic normality of the Fourier estimators of the integrated volatility of volatility defined by ((ref)) and ((ref)) and prove that the estimator ((ref)) reaches the optimal rate of convergence \textcolor{black}{$n^{1/4}$}, at the cost of a de-biasing term, while the estimator ((ref)) has a smaller asymptotic variance, at the cost of a slower convergence rate.
Note that if $c_N=\pi$ or, equivalently, $N=n/2$ (i.e., the cutting frequency $N$ used for the estimation of the volatility coefficient given the log-prices is equal to the Nyquist frequency), then $\eta(c_N/\pi)=0$ and the asymptotic variance in Theorem (ref) becomes
{\color{black}In order to obtain a feasible CLT from Theorem (ref), a consistent estimator of the conditional variance is needed. We exploit again the Fourier methodology to build a consistent estimator of $${1\over {2\pi}} \int_0^{2\pi} \Lambda(t)dt$$ with
The result is detailed in Proposition (ref). The key ingredients are the following Remark (ref) and the product formula for the Fourier coefficients, as studied in LMM.}
{\color{black} Based on Remark (ref), for any integer $k$, $|k|\leq 2M$, we define
and
where $K$ is computed in ((ref)). They are, resp., consistent estimators of $c_k(\sigma^4)$ and $c_k(\gamma^2)$, for any integer $k$. The following result holds.}
\textcolor{black}{Thus, we have the following feasible CLT.}
It is possible to obtain an estimator of the volatility of volatility without a bias-correction term and a smaller asymptotic variance, but the rate of convergence is slower, precisely $n^{\iota/2}$, with $\iota/2 \in (0,1/5)$. The estimator is simply given by ((ref)) multiplied by $2\pi$ and the following result holds.
In order to build a feasible \textcolor{black}{CLT}, it is enough to apply the same methodology as for Theorem (ref). In particular, under the conditions $N \rho(n) \sim c_N$ and $M\rho(n)^{\iota} \sim c_M$, where $\iota \in (0,2/5)$, a consistent estimator of the asymptotic variance is given by
where
Therefore, the following holds.
In this section we present a simulation study of the finite-sample performance of the rate-efficient estimator ((ref)). The objective of the study is to provide support to the asymptotic result in Theorem (ref), \textcolor{black}{offer insight into the optimal selection of the frequency $M$, assess the robustness of the performance of the estimator to irregular sampling schemes} and illustrate a comparison of its accuracy with that of the rate-efficient realized estimator by ASJ.
We simulated discrete observations from two parametric models which satisfy Assumptions (A.I)-(A.II)-(A.III). The first model that we simulated is the Heston model (see heston):
where $v(t):=\sigma^2(t)$, $\mu \in \mathbb{R}$, $\theta,\alpha, \gamma>0$, and $\rho$ denotes the correlation between the Brownian motions $W$ and $Z$. Under the Heston model, the volatility of volatility is given by $\gamma^2(t)=\gamma^2 v(t)$.
The second model that we simulated is the stochastic volatility of volatility model that appears in BNVee and SanfeliciCuratoMancino. The model is as follows:
where $Y$ is a Brownian motion independent of $W$ and $Z$, and $\mu \in \mathbb{R}$, $\theta,\alpha, \chi, \eta, \xi >0$.
The parameter vectors used for the simulations of the models ((ref)) and ((ref)) were, resp.:
Note that the selection of a negative $\rho$ reproduces the presence of leverage effects. For each model, we simulated \textcolor{black}{$10^4$} trajectories with horizon $T=1/252$, i.e., with horizon equal to one trading day, corresponding to 6.5 hours. For each trajectory, observations were simulated on the equally-spaced grid with mesh equal to $1$ second.
We assessed the finite-sample performance of the estimator ((ref)) for increasing values of the sample size $n$, in order to provide numerical support to Theorem (ref). Specifically, we considered values of $\rho(n)=T/n$ ranging between $5$ minutes and $1$ second. For what concerns the frequency $N$, which is needed for the convolution formula ((ref)), we set $N=\left[c_N \rho(n)^{-1}\right]$ and selected $c_N=T/2$. This selection yields the value of $N$ equal to the Nyquist frequency $[n/2]$ and allows obtaining the smallest variance of the asymptotic error, see ((ref)) in Section (ref). As for the frequency $M$, we set $M=\left[ c_M \rho(n)^{-1/2} \right]$ and optimized the value of the constant $c_M$ based on the (unfeasible) numerical minimization of the mean squared error (MSE). In this regard, we found that the MSE-optimal value of $c_M$ is equal to, resp., $0.05$ and $0.07$ for the models ((ref)) and ((ref)) (see also Subsection (ref), where a feasible procedure to select $M$ is also discussed).
Table (ref) illustrates the finite-sample performance of the estimator ((ref)) under the two different data-generating processes considered. Specifically, Table (ref) illustrates the MSE and the bias for the different values of the sampling frequency $\rho(n) $. As expected, the bias and MSE improve as $n$ is increased for both the data-generating processes considered, thereby providing numerical support to Theorem (ref). In particular, note that the performance of the estimator is still satisfactory for $\rho(n)$ equal to 5 minutes, the sampling frequency typically used in the absence of noise with empirical data. Additionally, it is worth mentioning that the estimator never produced negative volatility of volatility estimates in this simulation study.
As additional support to the results in Thereom (ref), the q-q plots in Figures (ref) and (ref) offer a comparison between the empirical quantiles of the unfeasible standardized estimation error from Theorem (ref) and the theoretical quantiles of a standard normal distribution for different values of $\rho(n)$. Both in the case of the model ((ref)) and the model ((ref)), as $\rho(n)$ becomes smaller, the approximation to the the standard normal distribution improves. In particular, while the approximation of the body of the distribution is satisfactory also for the largest $\rho(n)$ considered, i.e., $5$ minutes, the approximation in the tails becomes accurate for $\rho(n)$ smaller or equal than $5$ seconds.
{\color{black} The careful selection of the frequency $M$, that is, the constant $c_M$, is key to efficiently implement the estimator ((ref)) with finite samples. Given the selection $M=\left[ c_M \rho(n)^{-1/2} \right]$, Figure (ref) shows the sensitivity of the MSE of the estimator to different values of the constant $c_M$ in the range $(0.01,1)$, for different values of $\rho(n)$. Based on Figure (ref), the optimal MSE is achieved when $c_M$ is equal to, resp., $0.05$ and $0.07$ for the models ((ref)) and ((ref)), independently of $\rho(n)$. Further, it appears that the MSE is relatively flat for $c_M$ in the intervals $(0.04,0.06)$ and $(0.06,0.07)$. }
\textcolor{black}{As outlined in the following remark, it is possible to exploit the feasible Theorem (ref) to optimize the selection of $c_M$ with empirical data.}
{\color{black} So far, in the simulation study we assumed that prices were observable on the equally-spaced grid with mesh size $\rho(n)$ equal to $1$ second. However, the setup of Section (ref) allows for an irregular sampling scheme. To assess the robustness of the performance of the estimator ((ref)) to irregular sampling, we considered the case when observation times follow a Poisson process, that is, durations between observations are drawn from an exponential distribution with mean $\lambda$ (see, e.g., book, Chapter $3.3$).
Specifically, we considered three different values of $\lambda$, corresponding to an average duration $\delta$ of $1.25$, $1.5$ and $2$ seconds, and compared the resulting MSE and bias values with the case of regular sampling on the 1-second grid.} {\color{black} For the estimation with the Poisson scheme, we set $N=[n/2]$ and optimized $M$ based on the minimization of the MSE. In this regard, we found that it is MSE-optimal to select a smaller $M$, compared to the regular-sampling case. Specifically, letting $M^*$ denote the optimal selection with regular 1-second sampling, numerical results suggest that it is optimal to select $M=[M^*/2]$. }
{\color{black} Table (ref) reports the resulting bias and MSE, offering a comparison with the case where the a regular 1-second sampling scheme is adopted, corresponding to the last row of Table (ref). The results in Table (ref) suggest that the Fourier estimator ((ref)) may still offer a satisfactory performance with irregular sampling schemes; in particular, it appears that the bias is relatively less affected than the MSE, compared to the regular-sampling case.}
This subsection contains a comparative study of the finite-sample performance of the rate-efficient Fourier estimator ((ref)) and the rate-efficient realized estimator by ASJ (see Remark (ref)). We recall the definition of the latter. Let $\kappa(n)$ denote a sequence of integers such that $\kappa(n)\sim \beta \rho(n)^{-1/2} $, $\beta>0, \, \rho(n):=T/n$. The estimator reads
where $$\displaystyle \widehat{\sigma}_{n }^2(t_i) = \frac{1}{\kappa(n)\rho(n)}\sum_{m=0}^{\kappa(n)-1} \Big( p(t_{i+m}) - p(t_{i+m-1})\Big)^2$$ is the local estimator employed to pre-estimate the spot variance at time $t_i=i{T/ n}$, $i=0,...,n$. The estimator ((ref)) is also studied in Vetter, where the author replaces $\big(\widehat{\sigma}_n^2(t_{i}\big))^2$ with
The rate-efficient realized estimator considered in Vetter thus reads
{\color{black} For the comparison, we replicated the simulation study carried out in Subsection (ref), this time using the realized estimators ((ref)) and ((ref)) to obtain estimates of the daily integrated volatility of volatility. The implementation of realized estimators requires the selection of the tuning parameter $\beta$. After setting $\kappa(n)=[ \beta \rho(n)^{-1/2}]$, we selected $\beta=0.04$ (resp., $\beta=0.06$) in the case of the model ((ref)) (resp., (ref)), based on the unfeasible optimization of the MSE with 1-second samples. Tables (ref) and (ref) summarize the results. By comparing the latter with Table (ref) in Subsection (ref), it is immediate to see that the performance of the realized estimators ((ref)) and ((ref)) is not satisfactory, both in terms of bias and MSE, compared to the case of the Fourier estimator ((ref)). See also SanfeliciCuratoMancino and TosRec for similar considerations on the finite-sample performance of realized volatility of volatility estimators. Moreover, note that the comparison for $\rho(n)$ equal to $5$ minutes is omitted, since the resulting bias and MSE of the realized estimators are larger than $1$ in absolute value.} Finally, simulations suggest that the use of the quarticity estimator in the de-biasing term in ((ref)) does not improve the finite-sample performance.
To the best of our knowledge, the empirical properties of the volatility of volatility of financial assets have been scarcely explored in the literature. The aim of the empirical study presented in this section is thus to provide insight into the existence of stylized facts pertaining to the daily dynamics of the volatility of volatility.
In fact, the numerical evidence presented in Section (ref) suggests that the Fourier methodology allows reconstructing the integrated volatility of volatility with satisfactory accuracy on daily intervals by means of the rate-efficient estimator ((ref)). Accordingly, in this section we use the estimator ((ref)) to obtain the daily volatility of volatility series for two market indices: the S&P500 and the EUROSTOXX50. The periods considered for the analysis are, resp., May 1, 2007 - August 6, 2021 and June 29, 2005 - May 28, 2021.
For the empirical analysis we used the series of $5$-minute trade prices, recorded during trading hours. Specifically, for the S&P500 index, we used the prices recorded between $9.30$ a.m. and $4$ p.m., while for the EUROSTOXX50 index we employed the prices recorded between $9$ a.m. and $5.30$ p.m. Days with early closure were discarded.
The estimation of the daily integrated volatility of volatility was performed via the rate-efficient Fourier estimator ((ref)), without considering overnight returns. Before performing the estimation, we run the test by AX on 5-minute series and found that the assumption of absence of noise could not be rejected at the $5\%$ significance level for both indices. Moreover, following WM14, days with jumps were removed, based on the results of the test by LM08, which was applied at the $1\%$ significance level. Overall, the number of days for which we estimated the volatility of volatility is $3343$ and $3522$ for, resp., the S&P500 and EUROSTOXX50.
\textcolor{black}{For the estimation, given the choice of the sampling frequency $\rho(n)=5$ minutes, we set $N=[c_N\rho(n)^{-1}]$ and $M=[c_M\rho(n)^{-1/2}]$. Then we selected $c_N$ such that $N$ equals the Nyquist frequency (see Subsection (ref)) and $c_M$ based on the feasible procedure illustrated in Subsection (ref). The resulting values of $c_M$ are equal, on average, to $0.054$ and $0.065$ for, resp., the S&P500 and the EUROSTOXX50.} Figures (ref) and (ref) display the reconstructed trajectories of the daily volatility of volatility of the two indices, while Table (ref) compares sample statistics of volatility and volatility of volatility estimates. Daily volatility estimates were obtained via the Fourier estimator by MM, applied to 5-minute returns\footnote{All the analyses appearing in this section and involving the computation of the volatility were also performed using volatility estimates obtained via the 5-minute realized variance and the final outcome was pretty much the same.}. We note that all volatility of volatility estimates obtained are strictly positive.
Figures (ref) and (ref) show that the volatility of volatility spikes during financial crises, while remaining rather low and stable during tranquil periods. Indeed, both daily series reach their three highest peaks in correspondence of, resp., the global financial crisis starting at the end of 2008, the instabilities of the Euro area in the second part of 2011 and the outbreak of the COVID pandemic in the first months of 2020.
Moreover, based on Table (ref), we make the following remarks. First, the volatility of volatility is on average smaller than the volatility in the case of both indices. Secondly, the volatility of volatility appears to be more volatile than the volatility itself, as it displays larger sample standard deviations and maxima for both the estimated series. Finally, the volatility of volatility appears to be much more skewed and leptokurtic than the volatility for both the indices.
An analysis of the empirical regularities displayed by the reconstructed daily series of the volatility of volatility of the S&P500 and the EUROSTOXX50 is illustrated in the next subsection.
The literature on the stylized facts related to the behavior of the volatility of financial assets is very rich (see, for instance, ABDE, PE and corsi, among many others). These include, e.g., clustering, long memory, mean-reversion, log-normality and leverage effects. Nowadays, the volatility can be regarded, in some sense, as a traded asset itself. In fact, it is possible to “trade" the volatility of many financial asset classes via quoted and O.T.C. volatility derivatives (e.g., variance swaps, VIX futures and VIX options). Therefore, it may be of interest to evaluate which typical features of the daily volatility actually apply to the daily volatility of the volatility itself.
We have already observed in the previous subsection that the volatility of volatility shows clusters, being larger in correspondence of crises and smaller and less volatile during periods of economic stability. However, based on the observation of the sample auto-correlation function (see Figures (ref) and (ref)), it appears to be less persistent than the volatility for both the indices considered. As for the mean-reversion property, the Augmented Dickey-Fuller test rejects the hypothesis of a unit root for both volatility of volatility series analyzed, at the $0.01\%$ significance level.
We also examined the year-by-year correlation of the daily volatility of volatility with, resp., the daily volatility and the daily log-return, computed as the difference between the closing and opening log-price. The dynamics of such correlations are summarized in Tables (ref) and (ref), where the values of return-variance correlations, a rough proxy of the leverage effect, are also displayed for comparison\footnote{Note that the correlations appearing in Tables (ref) and (ref) are typically pushed towards zero by the presence of a finite-sample bias, see AFL. For this reason, their true values are likely to be larger, in absolute value. However, obtaining unbiased and efficient estimates of these correlations goes beyond the scope of the exploratory analysis proposed.}. For both the indices, we observe that the yearly correlation between the log-return and the volatility of volatility tends to be negative and to follow the return-variance correlation closely, although being most often smaller in absolute value. This result may suggest the existence of a “second-order" leverage effect: what we observe is in fact that when the asset price decreases, not only the volatility increases, due the asset becoming riskier, but also the volatility of volatility - which can be seen as a proxy of the uncertainty about the amount of risk perceived by market operators, that is, the “volatility of risk" - becomes larger. The yearly correlation between the volatility of volatility and the volatility is instead positive and close to {\color{black} $0.7$} on average for both the indices. This is consistent with the presence of volatility-of-volatility peaks in periods of higher market volatility.
Finally, we test for the Gaussianity of the logarithmic volatility and volatility of volatility estimates, using the Jarque-Bera and Anderson-Darling tests at the $5\%$ significance level. The years in which both tests reject the null hypothesis of Gaussianity are 2008, 2011, 2016 and 2020, that is, the years in the sample that were the most characterized by market turmoil (in order: the global financial crisis, the Euro-area instability phase, Brexit and the outbreak of the COVID pandemic). This happens for both the quantities tested and both the indices analyzed, thus suggesting that the log-normal approximation for the distribution of the volatility and the volatility of volatility is more satisfactory in periods of market stability.
This paper fills a gap in the literature on financial econometrics by deriving the convergence rate of the Fourier estimator of the volatility of volatility. In this regard, we showed that the bias-corrected version of the estimator reaches the optimal rate \textcolor{black}{$n^{1/4}$}, while the estimator without bias-correction achieves a sub-optimal rate, but has a smaller asymptotic variance.
Further, we presented a numerical study that shows that the rate-optimal Fourier estimator of the volatility of volatility performs well in finite samples, even at the relatively small daily estimation horizon, where the competing rate-efficient realized estimator shows a poor performance.
Finally, we applied the Fourier estimator to multi-year samples of S&P500 and EUROSTOXX50 observations and gained some new knowledge about the empirical regularities that characterize the daily dynamics of the volatility of volatility, a topic which so far had been scarcely explored in the literature.