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On the estimation of leverage effect and volatility of volatility in the presence of jumps
Semimartingale processes are widely used in finance. For example, the fundamental asset pricing theorem states that in an arbitrage-free and frictionless financial market, the logarithmic price process of an asset is necessarily to be a semimartingale (DS1994). We denote $\{X_t\}_{0\leq t\leq T}$ as the log-price process of an asset over $[0,T]$ and assume that it is an It$\hat{\text{o}}$ semimartingale on the filtered probability space $(\Omega,\mathcal{F},(\mathcal{F}_t)_{ 0\leq t\leq T},\mathbb{P})$. In general, $X_t$ can be represented as
where $b$, $\sigma$ are adapted and locally bounded c$\grave{\text a}$dl$\grave{\text a}$g processes; $B$ is a standard Brownian motion and $\sigma^2$ is the volatility process; $J$ is a pure jump process. In high-frequency financial econometrics, the path of $X$ over $[0,T]$ can not be fully obtained, and it can only be observed at some discrete time points. We assume that $X$ is observed equidistantly at the discrete time points $t_i^n:=i \Delta_n$, for $i=0, 1,\cdots, n$ with $n=\lfloor\frac{T}{\Delta_n}\rfloor$, where $\Delta_n$ is constant and only depends on $n$. Eventually, we consider the infill asymptotic regime of $\Delta_n\rightarrow0$ for fixed $T$.
To quantify the intensity of the jump process, YJ2009 introduced the following jump activity index (JAI) for a semimartingale process $X$:
where, $\Delta X_{t} = X_{t} - X_{t-}$ is the size of the jump at time $t$. With this definition, it holds almost surely that $0\leq \text{JAI}< 2$, and as JAI increases, the (small) jumps tend to become more frequent. If $\text{JAI}=0$, then the process has finite activity, otherwise, the process has infinite activity, corresponding to the case of $\text{JAI}>0$. Moreover, when $\text{JAI}<1$, the jump is locally summable, thus of finite variation, while of infinite variation if $\text{JAI}>1$. Furthermore, for a L$\acute{\text{e}}$vy process, JAI coincides with Blumenthal-Getoor index. In the special case where $X$ is a stable L$\acute{\text{e}}$vy process, $\text{JAI}$ is also the stable index of the process. We refer to YJ2009 for more details.
With the widely available high-frequency data, the volatility-related quantities have been studied, including the integrated volatility $\int_{0}^{T}\sigma_t^2dt$, the spot volatility $\sigma^2_t$ for any fixed $t\in[0,T]$, and the general volatility functional $\int_{0}^{T} g(\sigma_t^2)dt$ with some function $g$, see AJ2014 for a comprehensive introduction. Without the consideration of the jump part in (ref), it is well known that the integrated volatility can be estimated by the realized volatility (See, e.g. ABDL2003). The presence of jumps brings in bias to the standard realized volatility, and various methods have been proposed to elliminate the effect of jumps. They include thresholding approach in M2011 and MR2011, which filtered the increments through a truncation procedure, and bi-power and multi-power estimator in BSW2006, W2006 and J2008, which employed the product of two or more consecutive increments to reduce the effect of jumps. When the jump is of infinite variation, the above two estimators are still consistent, but the central limit theorem is no longer valid, see JR2014 and JT2014. By using the empirical characteristic function of the high-frequency increments, JT2014 constructed an estimator of the integrated volatility that achieves both the optimal rate of convergence and the efficient asymptotic variance. The spot volatility estimators have been studied by D2010, JP2012, YP2014, LLL2018. Empirically, it was reported in YJ2009 and JKLM2012 that the jump activity of real high-frequency financial data can be intensive as of infinite variation.
In recent years, the variational pattern of the volatility process and its relationship to the log-price process, such as leverage effect and volatility of volatility, have attracted great attention. Leverage effect is defined as the covariance between asset price process and its volatility (See, e.g. C1982, AFL2013, CT2024). The estimation of leverage effect by using the high-frequency data in the continuous setting was investigated by AJ2014 and WM2014. When the jump part is present, AFLWY2017 and KX2017 applied the thresholding technique to remove the increments with jumps, and constructed the leverage effect estimator subsequently by replacing the true spot volatilities with their estimates and properly removing the bias terms caused by the estimation procedure. All these existing estimators have restrictions on the jump activity, for example, the consistency holds only if the jump part is of finite variation. Volatility of volatility quantifies the variation strength of the volatility process, hence further describes the variational pattern of the asset price. The volatility process is not observable and can only be estimated, making the estimation of volatility of volatility more challenging. Estimation of volatility of volatility was considered by AJ2014 and M2015 without the consideration of jumps, and in BV2009 with jumps. For the latter case, only the consistency of the estimators was established, under the condition that the jumps in the log-price process are of finite variation. The main concern in this paper is to investigate the effect of possible infinite variation jumps on the estimation of leverage effect and volatility of volatility. For a purpose of comparison, we collect the conditions on the jump activity index for these aforementioned estimators and our proposed estimators of leverage effect and volatility of volatility in Table (ref), when the optimal convergence rate of $n^{-1/4}$ can be achieved for all the estimators. The finite sample performances of these estimators are presented in Section (ref).
Recall that in the estimation of leverage effect and volatility of volatility, the main challenge is that the volatility path can not be directly observed and has to be estimated. Besides, the presence of jumps brings in bias for the estimation, and the estimation error could be even larger when the jump is intensive. To avoid the influence from the jumps, we apply the jump-robust spot volatility estimator proposed in LLL2018. The estimator was constructed by using the empirical characteristic function of the high-frequency increments, and was shown to be more effective than the thresholding technique and the bi-power estimator in diminishing the effect of jumps, especially for infinite variation jumps. Subsequently, the leverage effect and volatility of volatility estimators are proposed after plugging the estimated volatility curve into their definitions and properly removing the bias caused by the estimation procedure. Under mild conditions, we establish the consistency and asymptotic normality for our proposed estimators. After consistently estimating the volatility functional, the feasible central limit theorems are obtained.
The contribution of this paper is multiple folds. First, from Table (ref), we see that our theoretical results hold for more general jump than the existing ones. In fact, our condition can be further relaxed when the jump part has some special structures as in Assumption (ref). Specifically, for the estimation of volatility of volatility, the consistency holds for any jump activity index smaller than 2, and smaller than $1.5$ is required for the asymptotic normality, as stated in Theorem (ref). In Section (ref), we provide intuitive explanations on how the thresholding technique, bi-power estimator and our method diminish the effect of jumps and why ours outperforms the existing ones. Moreover, we find that the spot volatility estimator proposed in LLL2018 is particularly suitable for the scenario when the increments of estimated volatility are used, such as in the estimation of volatility of volatility, because the bias terms due to the presence of jumps will cancel out. Second, in some existing literature such as WM2014, leverage effect was defined as the quadratic covariation between the log-price process and a general function of volatility process. We extend our theory to this scenario and derive corresponding central limit theorem. Comparing our estimator with the one in WM2014, we see that both estimators achieve the optimal convergence rate, but our estimator has smaller asymptotic variance. Moreover, WM2014 did not consider the presence of jumps. Third, unlike the traditional thresholding technique, our method does not require a parameter-tuning procedure, making it a more convenient alternative for empirical study. Although the spot volatility estimator involves a parameter $u$, our simulation studies show that setting it as a constant is already able to yield satisfactory finite sample performance, both for the estimation of leverage effect and volatility of volatility. Third, as a by-product, we show that the spot volatility estimator in LLL2018 can exactly achieve the optimal convergence rate of $n^{1/4}$, rather than the almost optimal one of $n^{1/4}/\log(n)$ provided therein. Essentially, the estimator inherits two balancing terms controlling the convergence rate, depending on the selection of bandwidth. Proposition (ref) unveils how its asymptotic property changes as the bandwidth varies, which is similar to the analyses of spot volatility estimation in AJ2014. More importantly, only the infeasible central limit theorem for spot volatility estimator was established in AJ2014, while a feasible version is provided in this paper.
The remainder of this paper is organized as follows. Section (ref) presents the model and assumptions. In Section (ref), we give the proposed estimators of spot volatility, leverage effect, volatility of volatility and volatility functional, followed by their asymptotic properties. Several issues are discussed in Section (ref), including some other definitions of leverage effect used in existing literature and the effects of jumps. Simulation studies are conducted in Section (ref). We apply our estimators to a real high-frequency dataset in Section (ref). Section (ref) concludes the paper. All the technical proofs are collected in Appendix (ref).
For the jump part $J$ in the underlying data generating process (ref), we consider
with $L$ and $J'$ being pure jump processes, $\gamma$ is an adapted and locally bounded c$\grave{\text a}$dl$\grave{\text a}$g process. According to JS2003, we can write $L$ and $J'$ as
where $\delta$ is a predictable process on $\Omega \times [0,T] \times \mathbb{R}$; $\mu$ and $\mu'$ are Poisson random measures on $[0,T] \times \mathbb{R}$, with intensity measure $\nu(dt,dx)=dt \otimes \lambda(dx)$ and $\nu'(dt,dx) = dt \otimes \lambda'(dx)$, respectively. Moreover, we suppose that $L$ and $J'$ are independent. With the above definitions, $X$ has the following form:
where $b'_t = b_t+\int_{\{|x| > 1\}}\gamma_t\cdot x\lambda(dx) + \int_{\{|x| > 1\}}\delta(t,x)\lambda'(dx)$. The same model was also used in JT2014 for the estimation of integrated volatility $\int_{0}^{T} \sigma_t^2dt$, and LLL2018 for spot volatility $\sigma_t^2$ with $t\in[0,T]$. In (ref), we separate the jump part $J$ into two pure jump processes, $L$ and $J'$, the reason is as follows. Both of them are possibly of infinite variation and the only difference is that we will specify some structural assumptions on $L$ but not for $J'$. We will show that, with or without such structure, the conditions on jump activity index for the asymptotic properties of our leverage effect and volatility of volatility estimators are different, as summarized in Section (ref).
We further assume that $\sigma$ is a continuous It$\hat{\text{o}}$ semimartingale and can be written as
where $\tilde{b}$, $\tilde{\sigma}$ and $\tilde{\sigma}'$ are adapted and locally bounded c$\grave{\text a}$dl$\grave{\text a}$g processes; $B'$ is a standard Brownian motion independent of $B$. A direct application of It$\hat{\text{o}}$'s lemma implies that the volatility process $\sigma^2$ can be written as
Assumption (ref) includes some local boundedness and smoothness conditions for the driving processes of $X$ and $\sigma$, which are regular in high-frequency literature. Specific examples satisfying condition (ref) include It$\hat{\text{o}}$ semimartingale. Assumption (ref) assumes that the jump process $L$ performs closely to a stable process around zero, via restricting the deviation between the tail functions $\overline{F}^{\pm}(x)$ and $1/x^{\beta}$. With this condition, the characteristic function of $L_t$ can be approximately given by $E[e^{iuL_t}]=e^{-C|u|^\beta t}$. Detailed analysis was given in JT2014 to show that such an assumption can cover tempered stable processes, which include time changed Brownian motion of normal inverse Gaussian process and Carr-Geman-Madan-Yor (CGMY) model. These models are widely used in finance. We also note that, to some extent, the symmetric assumption can be removed for the estimation of volatility by replacing the original increments with the difference between consecutive increments. This point was also discussed in JT2014. From the above two assumptions, we see that the jump activity indexes of $L$ and $J'$ in (ref) are $\beta$ and $r$, respectively.
In this section, we begin with the estimation of spot volatility, based on which we will construct the estimators for leverage effect and volatility of volatility. The properties of consistency and asymptotic normality for these estimators will be established. Furthermore, to make the central limit theorems feasible, we propose a consistent estimator of a general volatility functional. Throughout the paper, for a process $Z$, we define the increments $\Delta_i^nZ = Z_{t_i^n} - Z_{t_{i-1}^n}$, for $i=1,...,n$. We use $\longrightarrow^{p}, \longrightarrow^{L}, \longrightarrow^{L_s}$ to denote the convergence in probability, convergence in law and stable convergence in law\footnote{Its detailed definition and introduction will be given after Proposition (ref) in Section (ref).}, respectively.
For model (ref), a kernel-based spot volatility estimator was proposed in LLL2018, where the empirical characteristic function was used to separate the volatility from the presence of infinite variation jumps. We apply the estimator with uniform kernel $K(x) = 1_{\{0< x\leq 1\}}$\footnote{We do not consider a general kernel function for the clarity of exposition and use the uniform kernel since it has the minimum asymptotic variance and is most widely used.}, that is
where $k_n\in \mathbb{Z}^{+}$ is the number of increments, and $u$ is a positive real number. As given in LLL2018, the threshold of $\frac{1}{\sqrt{k_n}}$ guarantees that the log function is well-defined and it plays no role asymptotically. According to the analyses in LLL2018, due to the presence of jump part $L$ in (ref), the spot volatility estimator at time $t \in [0,T]$ suffers a bias term with the form
Considering an infeasible debias procedure, we have the following limiting theory.
The stable convergence in law in the above conclusion is a kind of limiting result stronger than the convergence in law. Moreover, it also implies the convergence in probability. Specifically, for a sequence of random variables $Z_n$ defined on the probability space $(\Omega,\mathcal{F},\mathbb{P})$ and a random variable $Z$ defined on an arbitrary extension $(\tilde{\Omega},\tilde{\mathcal{F}},\tilde{\mathbb{P})}$ of $(\Omega,\mathcal{F},\mathbb{P})$, $Z_n$ stably converges in law to $Z$ implies
for any bounded continuous functions $f$ and bounded random variables $Y$ on $(\Omega,\mathcal{F})$, where $\mathbb{E}, \tilde{\mathbb{E}}$ denote the expectation with respect to $\mathbb{P}, \tilde{\mathbb{P}}$ respectively, see JS2003, JP2012 and PV2010 for more details.
By taking $k_n = \lfloor \kappa n^{1/2} \rfloor$, (ref) can be written as
The convergence rate of our result is faster than $n^{1/4}/\log{n}$ in LLL2018, and in fact, it is optimal for the estimation of spot volatility. Meanwhile, the minimum limiting variance can be achieved by letting $\kappa = \sqrt{Var(V_{t}|\mathcal{F})/ Var(V'_{t}|\mathcal{F}) }$, with the value of $2\sqrt{Var(V_{t}|\mathcal{F})\cdot Var(V'_{t}|\mathcal{F}) }$. We note that the central limit theorem established in LLL2018 serves as one part of our result, that is (ref). For this result, the asymptotic variance of the spot volatility estimator in LLL2018 is $2 (\sigma_{t})^4$, while ours is $\mathbb{E}[(V_{t})^2|\mathcal{F}]$. This is not a contradiction. In LLL2018, they require $u$ to be a sequence $u_n$ tending to 0 at a proper rate as $n \rightarrow \infty$, and we can obtain the same result by doing so. By applying Taylor's expansion at 0, we can get $\mathbb{E}[(V_{t})^2|\mathcal{F}] \longrightarrow^{p} 2 (\sigma_{t})^4$ if $u \rightarrow 0$. We do not require $u\rightarrow 0$ since it will complicate the asymptotic condition. Besides, such a manipulation also brings in extra approximation error.
The results of (ref), (ref) and (ref) are intrinsically not feasible because the bias terms $ b_{ t,n}(u)$ are not attainable, but these terms are asymptotically negligible under some proper conditions. To be specific, it requires $\sqrt{k_n}b_{ t,n}(u) \longrightarrow^{p} 0$ for (ref), (ref), and $\frac{1}{\sqrt{k_n\Delta_n}}b_{ t,n}(u) \longrightarrow^{p} 0$ for (ref), namely
When the convergence rate in our central limit theorem is optimal, corresponding to $k_n = O(\sqrt{n})$, we see that the bias is always negligible if $\beta<1$. The restriction on $\beta$ can further be relaxed if we let $u \rightarrow 0$. This point can be seen from the analyses in LLL2018, where they demonstrated that the presence of jumps has no effect on the asymptotic normality of the spot volatility estimator if $\beta \leq 1.5$. When the jump activity index is large so that the bias term $b_{ t,n}(u)$ is not asymptotically negligible, a further procedure is needed. We do not consider this issue in this paper since it is not the main purpose.
The concerned leverage effect over $[0,T]$, denoted as $\mathcal{L}_{[0,T]}$, is defined as the quadratic covariance between $X$ and its volatility process $\sigma^2$, namely
According to the definition, since
where $[X, \sigma^2]^n_T:=\sum_{i=1}^{n} (\Delta_i^n X \cdot \Delta_i^n \sigma^2)$, an natural way of estimating leverage effect is directly replacing the spot volatility $\sigma_{t_i^n}^2$ in $[X, \sigma^2]_T^n$ by its estimator $\widehat{\sigma}_{t_i^n}^2$ in (ref). This yields\footnote{We use $\widehat{\sigma}^2_{t_{i-}^n}$, instead of $\sigma_{t_{(i-1)+}^n}^2$, to estimate $\sigma_{t_{i-1}^n}^2$. This avoids the presence of overlapping increments between it and $\widehat{\sigma}_{t_{i+}^n}^2$. The same idea was also adopted by AFLWY2017.}
with
where $u \in \mathbb{R}^{+}$, $I^n_{i+} = \{i+1,...,i+k_n\}$ and $I^n_{i-} = \{i-k_n,...,i-1\}$. We see that $I^n_{i+}$ and $I^n_{i-}$ are two local windows of length $k_n\Delta_n$, just after and before time $t_i^n$. And, in fact, $\widehat{\sigma}^2_{t_{i-}^n}$ is the spot volatility estimator in LLL2018 at time point $t_{i-1}^n$ with kernel function $K(x) = 1_{\{-1\leq x < 0\}}$, namely $t_{i-}^n = t_{i-1}^n$. Moreover, $\widehat{\sigma}^2_{t_{i-}^n} = \widehat{\sigma}^2_{t_{(i-k_n-1)+}^n} $.
A similar estimator of leverage effect was also proposed by AFLWY2017, where they used the classical technique of thresholding to deal with the jumps. They require strictly $r<1$ for the consistency, while $r\leq 1$ is required for our result. Comparing Theorem 3 in AFLWY2017 with Theorem (ref), we see that our restriction for the jump activity is more relaxed. To be specific, we require $r \leq 1$ while they require $r < \frac{1}{2}$. Moreover, the performance of thresholding method largely depends on the proper selection of two tuning parameters ($\alpha$ and $\varpi$ in AFLWY2017), which are also related to the jump activity index ($\varpi \in [\frac{3}{4(2-\beta)},\frac{1}{2})$). Thus, how to set these parameters becomes a critical and complicated problem, especially for empirical applications when the jump activity index is unknown. This is not a problem for our estimator since the parameter $u$ in our estimator does not depend on the jump activity indices $\beta$ and $r$, and the simulation studies in Section (ref) show that fixing $u=1$ can generate satisfactory results.
When $b<\frac{1}{2}$, as discussed in Section (ref), by taking $u\rightarrow 0$, $h_1(u,t)$ in (ref) will tend to $2\sigma_t^4$. As a result, the asymptotic variance $Var(U|\mathcal{F})$ becomes $\frac{4}{\kappa}\int_{0}^{T} \sigma_t^6 dt$, which coincides with the one obtained by AFLWY2017. When $b=\frac{1}{2}$, the convergence rate of $n^{-1/4}$ can be achieved, and it is known to be optimal for the estimation of leverage effect if the presence of microstructure noise is not considered. Meanwhile, the minimum limiting variance can be achieved by letting
For the case of $b>\frac{1}{2}$, the asymptotic variance $Var(U|\mathcal{F})$ is the same as the one in AFLWY2017.
To make the central limit theorem in Theorem (ref) feasible, we propose a consistent estimator of the asymptotic variance $Var(U|\mathcal{F})$ in Section (ref).
In this part, we consider the estimation of the integrated volatility of volatility process $\sigma^2$. According to model (ref), the integrated volatility of volatility is defined as
To construct the estimator, we first discretize $VoV_{[0,T]}$ as
To estimate it, we plug in the spot volatility estimates, such an idea is adopted in the estimation of volatility functional (See, e.g. JR2013). We notice that, the integrand in (ref) appears in the asymptotic variance of the spot volatility estimator $\widehat{\sigma}^2_{t_{i+}^n}$ in Proposition (ref) when $\frac{1}{2} \leq b<1$\footnote{In fact, the integrand is always contained in the exact variance for any $0<b<1$, but when $0<b< \frac{1}{2}$, the asymptotic variance is dominated by $h_1(u,t, \sigma_{t}^2)$. This point can be seen from the proof of Lemma (ref).}. Moreover, similar result holds for $\widehat{\sigma}^2_{t_{i-}^n}$ as well. In addition, we can establish a joint central limit theorem for $(\widehat{\sigma}^2_{t_{i+}^n} , \widehat{\sigma}^2_{t_{i-}^n})$, that is Lemma (ref) in Appendix. Based on this, we have
where we recall that $h_1(\cdot)$ and $h_2(\cdot)$ are defined in (ref). This inspires us to propose the following spot volatility of volatility estimator at $t \in [0,T]$:
where $\widetilde{m}$ is an integer sequence tending to infinity, with $\widetilde{m}k_n\Delta_n \rightarrow 0$. For the integrated version, we can equivalently separate the interval $[0,T]$ into $\lfloor T/(\widetilde{m}k_n\Delta_n) \rfloor$ blocks with each block having length $\widetilde{m}k_n\Delta_n$, and construct the estimator as
In the above, although the processes of $\sigma^2, b$ are not observable, under Assumption (ref), we can get $\sigma^2_{t_{i+}^n} - \sigma^2_{t_{i-}^n} = O_p(\Delta_n^{1/2}) $ and $b_{ t_{i+}^n,n}(u) - b_{ t_{i-}^n,n}(u) = O_p(\Delta_n^{(3-\beta)/2})$, which are asymptotically negligible. And $h_1(u,\sigma_{t_i^n}^2)$ can be estimated by plugging in the volatility estimator $\widehat{\sigma}_{t_{i+}^n}^2$. This yields our integrated volatility of volatility estimator
The same estimator can be obtained for $0<b<\frac{1}{2}$ by a similar analysis\footnote{Although the estimator can be constructed for this case, the following Theorem (ref) shows that such a scenario is not applicable. Moreover, the de-bias procedure may leads to negative estimates of volatility of volatility since the bias term is of relative larger order. Thus, a relative longer selection of $b\geq \frac{1}{2}$ is preferred.}.
We can see from the above results that the selection of $0<b<\frac{1}{2}$ is not applicable. Noticing that the convergence rate is $\sqrt{k_n}$ in this case, and
Its asymptotic variance tends to infinity, if $b<\frac{1}{2}$, making the central limit theorem invalid.
In Theorem (ref), our convergence rate is the same as the ones in AJ2014 and M2015, where the continuous case was considered. By taking $b=\frac{1}{2}$, the optimal convergence rate of $1/n^{1/4}$ can be achieved, and our asymptotic variance is close to the ones in AJ2014 and M2015. Notice that the asymptotic variance depends on the parameter $\kappa$ in a rather complicated nonlinear way, we do not discuss the realization of minimum asymptotic variance. Again, a feasible version of central limit theorem can be obtained with a consistent estimator of the asymptotic variance $Var(W|\mathcal{F})$, which will be presented in Section (ref).
The central limit theorems in Proposition (ref), Theorem (ref) and Theorem (ref) are infeasible in practice since the limiting variances of the estimators, $Var(V_{t}|\mathcal{F})$, $Var(V'_{t}|\mathcal{F})$, $Var(U|\mathcal{F}) $ and $Var(W|\mathcal{F})$, are unknown. Thus, consistent estimators of these terms are required. Notice that $Var(U|\mathcal{F}) $ and $Var(W|\mathcal{F})$ are essentially specific forms of the integrated volatility functional, so we first consider consistent estimation of volatility functional. Based on this, the local estimation can be used for functions of spot volatility. As a result, consistent estimators of $Var(V_{t}|\mathcal{F})$ and $Var(V'_{t}|\mathcal{F})$ can be obtained.
We consider the estimation of the integral over $[0,T]$ of a given function $g$ of the volatility process $\sigma^2$, namely,
with the function $g$ being continuous. A natural idea for constructing the estimator of $I(g)$ is approximating the integral via a Riemann sum based on local estimators of the point-wise volatility. This yields
The same idea was also adopted in JP2012, JR2013, JT2014, and some others.
Now, we are ready to give consistent estimators of $Var(U|\mathcal{F}) $ and $Var(W|\mathcal{F})$. With Theorem (ref), and observing that $VoV_{[0,T]} = 3 \int_{0}^{T} h_2(t, \sigma_{t}^2,(\tilde{\sigma}_{t})^2, (\tilde{\sigma}'_{t})^2) dt $, these inspire us to estimate $Var(U|\mathcal{F})$ by
Similarly, for $Var(W|\mathcal{F})$ in (ref), The terms $\int_{0}^{T} \frac{(h_1(u,t,\sigma_{t}^2))^2}{\kappa^4T^2}dt$, $\int_{0}^{T} \frac{h_1(u,t,\sigma_{t}^2)}{\kappa^2T} h_2(t,\sigma_{t}^2,(\tilde{\sigma}_{t})^2, (\tilde{\sigma}'_{t})^2) dt$, $\int_{0}^{T}\frac{(h'_1(u,t,\sigma_{t}^2))^2(h_1(u,t,\sigma_{t}^2))}{\kappa^6T^3} dt$, $\int_{0}^{t}\frac{(h'_1(u,t,\sigma_{t}^2))^2h_2(t,\sigma_{t}^2,(\tilde{\sigma}_{t})^2, (\tilde{\sigma}'_{t})^2)}{\kappa^4T^2}dt$ can be estimated by $\widehat{H_n^1}$, $\widehat{{H_n}^2}$,$\widehat{{H'_n}^1}$, $\widehat{{H'_n}^2}$, respectively:
And we will show in Theorem (ref) that
where
With the above results, we define
Similarly, the term $Var(V_{t}|\mathcal{F})$ can be estimated by
For the estimation of $Var(V'_{t}|\mathcal{F})$, noticing that $h_2(t, \sigma_{t}^2,(\tilde{\sigma}_{t})^2, (\tilde{\sigma}'_{t})^2))$ is a local version of $VoV_{[0,T]} /3$, thus a local version of $ \widehat{VoV}_{[0,T]}$ can be used, namely
where $m_n$ is an integer sequence tending to infinity, as $n\rightarrow\infty$.
By combining the results in Section (ref)--(ref), together with Proposition 2.5 in PV2010, we can obtain the following feasible versions of Proposition (ref), Theorem (ref) and Theorem (ref), respectively.
Based on the feasible central limit theorems in Corollary (ref), direct applications include constructing confidence intervals for the true values of leverage effect and volatility of volatility, building hypothesis testing problems against the null hypotheses that the leverage effect or the volatility of volatility is at a particular level. For example, one would be interested in testing if the volatility of volatility $VoV_{[0,T]}$ is zero or not, and the formal null and alternative hypotheses can be written as
According to (ref), we can use the test statistic
and we have $\widetilde{T}_n \longrightarrow^{L} \mathcal{N}(0,1)$ under the null hypothesis, and $\widetilde{T}_n$ goes to $+\infty$ at the rate of $n^{\frac{1-b}{2}}$ under the alternative hypothesis. In LLZ2022, with the consideration of both jumps and microstructure noise, they demonstrated that the convergence rate can be further improved. This is because, if the null hypothesis is true, the diffusion terms in the volatility process disappear and the spot volatility becomes a bounded variation process. This should facilitate the estimation of spot volatility process since the volatility is smoother than the usual setting when it is an It$\hat{\text{o}}$ process. It would be our future work to extend the discussion to this scenario and propose a test statistic with a faster convergence rate.
In Section (ref), we define leverage effect as the quadratic covariance between $X$ and its volatility process $\sigma^2$, while there are some alternative definitions used in the existing literature. We will discuss these different definitions and their estimation. After that, we summarize the effect of jumps on the estimation of leverage effect and volatility of volatility.
In KX2017, they defined the leverage effect as a time-varying correlation process with
where $'$ denotes the first derivative with respect to time. For simplicity, we will assume that it is constant over $[0,T]$ with $\rho_t \equiv \rho$. In fact, such a model encompasses the popular Heston model, for which it turns out that $\rho$ is the constant correlation between two Brownian motions within Heston model. With the constancy assumption, we can alternatively define leverage effect, from the correlation perspective, as
It can be estimated by plugging in the corresponding consistent estimators for the denominator and numerator terms, which yields
where $\widehat{\mathcal{L}}_{[0,T]}$ and $\widehat{VoV}_{[0,T]}$ are given by (ref), (ref) respectively, and the integrated volatility estimator $\widehat{IV}_{[0,T]}$ can be found in JT2014.
For the central limit theorem regarding to $\widehat{\mathcal{L}}^{cor}_{[0,T]}$, we need to derive the joint distribution of $\langle X, \sigma^2\rangle_T$, $\langle X, X\rangle_T$ and $\langle \sigma^2, \sigma^2\rangle_T$. This is out of the scope of this paper and will be considered in the future.
In WM2014, they defined leverage effect as the quadratic covariance between $X$ and $F(\sigma^2)$, namely,
where $F(\cdot)$ is a twice continuously differentiable function and is monotone on $(0,\infty)$. According to It$\hat{\text{o}}$'s lemma with (ref), we can get
where $F'$ and $F''$ are the first derivative and second derivative of $F$, respectively. As a result, we have
Similarly to the estimation of $ \mathcal{L}_{[0,T]}$, $\mathcal{L}^{func}_{[0,T]}$ can be estimated by
And we can establish
After comparing the asymptotic variance in the above theorem with the one in WM2014, we conclude that our theoretical one is always smaller, regardless of the convergence rate. Similarly, the feasible version of the theorem can be obtained by estimating the asymptotic variance with the volatility functional estimator, as discussed in Section (ref).
Now, we summarize the effect of jumps on the estimation of leverage effect and volatility of volatility. Recall that in model (ref), our jump process consists of two parts. The first part is a L$\acute{\text{e}}$vy process driven by $L$ satisfying Assumption (ref). The second part has jump activity index $r$ and has a general structure without any further condition. We see from Theorem (ref)-(ref) that, for the first jump part, the restriction is more relaxed: $\beta\leq 1$ $(r\leq 1)$ and $\beta<2$ $(r<4/3)$ are required for the consistency of leverage effect estimator and volatility of volatility estimator, respectively. For the asymptotic normality, the conditions $\beta\leq 1$ $(r\leq 1)$ and $\beta<3/2$ $(r<1)$ are needed, respectively. The reason is that, with condition (ref), the bias $b_{t,n}$ in (ref) can be diminished if we take difference between the spot volatility estimates at two consecutive time points, namely $ \left( \widehat{\sigma}^2_{t_{i+}^n} - \widehat{\sigma}^2_{t_{i-}^n} \right)$ in (ref) and (ref). We can take the extreme case of $\gamma_t \equiv \gamma_0$ with $t\in[0,T]$ for illustration, under which we have $b_{t_{i+}^n,n} \equiv b_{t_{i-}^n,n}$, so the difference between $\widehat{\sigma}^2_{t_{i+}^n}$ and $\widehat{\sigma}^2_{t_{i-}^n} $ can remove the bias terms $b_{t_{i+}^n,n}$ and $b_{t_{i-}^n,n}$.
We provide some intuitive explanations on how the bi-power estimator, thresholding technique and our method diminish the effect from the presence of jumps. Volatility estimators applying these three different methods are (ref), (ref) and (ref) respectively. We consider the special case $\Delta_i^n X = \Delta_i^n B + \Delta_i^n J$ for explanation. As we know, $\Delta_i^n B = O_p(\sqrt{\Delta_n})$ and $\Delta_i^n J = O_p(1)$. Similar to (ref), the jumps can be classified to rare “large" jumps with jump size larger than 1 and relative intensive “small" jumps with jump size not larger than 1. For bi-power estimator, it is formulated with $|\Delta_i^n X||\Delta_{i+1}^n X|$, and as the frequency increases ($n\rightarrow \infty$), at most one increment, say $\Delta_i^n X$, contains jumps. Thus
from which we see that the influence from $|\Delta_i^n J|$ is brought down to $O_p(\sqrt{\Delta_n})$, both for “large" and “small" jumps. The thresholding technique works with $ (\Delta_i^n X)^2\cdot 1_{\{|\Delta_i^n X| \leq \alpha \Delta_n^{\omega} \}}$ for some parameters $\alpha, \omega$, so that the indicator function can remove the “large" jumps with probability one, but it can not detect “small" jumps. In this sense, by Mean Value Theorem, there exists some $\xi_i^n \in [\Delta_i^n B, \Delta_i^n B+ \Delta_i^n J]$ such that
which demonstrates that the effect of “small" jumps remains of order $O_p(1)$. From the above analysis, we can conclude that thresholding is more effective for “large" jumps while bi-power estimator can work better for “small" jumps. Numerical comparison between these two methods can be found in V2011, and it verifies the intuition. As for our estimator, it is constructed based on $\cos(\Delta_i^n X)$, and by Mean Value Theorem, there exists some $\xi_i^{'n} \in [\Delta_i^n B, \Delta_i^n B+ \Delta_i^n J]$ such that
Since $\sin(\xi_i^{'n})$ is bounded, the influence from the jumps is always controlled. Moreover, if $ \Delta_i^nJ$ is relatively small, our method can diminish the small jumps better than thresholding since $|\sin(x)| \leq |x|$ holds for $x\in[-1,1]$. This also inspires us that our method can be improved by using thresholded increments to totally remove the “large" jumps.
We now conduct simulation studies to compare the finite sample performance of our estimators with that of existing ones and verify the theoretical results established in the previous sections.
We first introduce some existing leverage effect and volatility of volatility estimators and present their detailed forms, and they will be compared with our proposed estimators subsequently. For leverage effect, Lev-AJ14, Lev-WM14, Lev-AFLWY17, Lev-KX17, Lev-our are used for the estimators in AJ2014, WM2014, AFLWY2017, KX2017 and this paper, respectively. For volatility of volatility, we name the estimators in AJ2014, M2015, BV2009 and this paper as Vov-AJ14, Vov-V15, Vov-BV09 and Vov-our, respectively. The conditions on the jump activity index for these estimators are compared in Table (ref).
For the estimation of leverage effect, several estimators have been proposed under different settings. Without the consideration of jumps, the spot volatility can be estimated by
for $i=0,...,n-k_n$. By using the increments within non-overlapping time intervals, WM2014 proposed the following leverage effect estimator:
where $\tau_{j}^n:= t_{jk_n}^n$ and $\tau_{j+}^n:=t_{jk_n+}^n$ for $j=0,...,\lfloor n/k_n \rfloor-1$. We name this estimator as “$\text{Lev-WM14}$". In AJ2014, the proposed estimator takes the following form:
We call it “Lev-AJ14" estimator. When the jumps are possibly present in $X$, AFLWY2017 applied the thresholding technique (Refer to Mancini2009, MR2011, and etc.) to filter the jumps and constructed the estimator
with
where $\alpha>0$ and $\omega\in(0,\frac{1}{2})$ are some constants. We name their estimator as “$\text{Lev-AFLWY17}$". For (ref), by following the setting in their simulation part, we take $\omega=0.49$ and $\alpha=5\sqrt{\text{BV}_{n}}$, where $\text{BV}_{n}$ stands for the bipower variation estimator (See, e.g. BN2004) and can be written as
As for volatility of volatility, estimators in AJ2014 (“Vov-AJ14") and M2015 (“Vov-V15") are proposed for the continuous case, and in BV2009 (“Vov-BV09") for the consideration of jumps. These estimators can be written as
where $\widehat{\sigma}^{(4)}_{t_{i}^n} = \frac{1}{3k_n\Delta_n^2} \sum_{j=1}^{k_n} |\Delta_{i+j}^n X|^4$. Besides, we also reformulate the estimator “Vov-V15" by further removing the jumps via thresholding method. This results in the following “Vov-V15-thr" Estimator:
with $\widehat{\sigma}^{(4,thr)}_{t_{i}^n} = \frac{1}{3k_n\Delta_n^2} \sum_{j=1}^{k_n} \left( |\Delta_{i+j}^n X|^4 \cdot 1_{\{ |\Delta_j^n X| \leq \alpha \Delta_n^{\omega} \}}\right)$. For comparison, we name our leverage estimator (ref) and volatility of volatility estimator (ref) as “Lev-our" and “Vov-our", respectively.
We consider the following models for the log-price process $X$ and the volatility process $\sigma^2$:
where $W$ and $V$ are two independent standard Brownian motions; $L$ is a strictly symmetric stable L$\acute{\text{e}}$vy process with Blumenthal-Getoor index $\beta$; and $J_t=\sum_{i=1}^{N_t} Y_{i}$ is a compound Poisson process where $N_t$ is a Poisson process with intensity $\lambda'$ and $Y_i\stackrel{i.i.d}\sim N(0, 0.01^2)$. The continuous part of $X$ in (ref) is a Heston model, while the discontinuous part consists of a possible infinite variation jump $L$ and a finite jump part $J$. We fix $T=1$, for which the time unit is measured in month. For matching the trading scheme in the real financial market and mimic the high-frequency data, we suppose that each month has a total number of 21 trading days, and within each trading day, the number of observations is 130, corresponding to sampling every 3 minutes within a 6.5-hour trading day. Given these considerations, we set $n=2730 (21\times 130)$, and repeat the simulation 1000 times. The same model was also considered by AFLWY2017 and LLL2018, and we use the same parameter setting. Namely, $X_0 = 0$, $\sigma_0^2 = 0.02$, $v=0.05$, $\lambda' = 3$, $\theta=0.02$, $\zeta= 5$\footnote{By ensuring the Feller condition $2\zeta\theta>\eta^2$, the volatility process can be guaranteed to be strictly positive.}. For the values of $\rho, \eta$ and $\gamma$, we consider different selections for robustness check, and they will be specified later. In model (ref), the true leverage effect to be estimated turns to be $\eta \rho \int_{0}^{1} \sigma_t^2dt$, and $\eta^2\int_{0}^{1} \sigma_t^2dt$ for the volatility of volatility. We use Riemann's sum to approximate the integrated volatility $\int_{0}^{1} \sigma_t^2dt$ for all experimental studies.
For achieving the optimal convergence rate for all the estimators given in Section (ref), we fix the setting of $k_n = \lfloor \sqrt{1/\Delta_n}\rfloor$, if not specified.
We first compare the finite sample performance of our estimators with the above mentioned ones, for various values of jump activity index $\beta = 0.5, 1, 1.5$. And we take a relative small value of $u=\frac{(\log(n))^{-1/40}}{\sqrt{\text{BV}_n}}$, which is consistently estimating $u^{\star}=\frac{(\log(n))^{-1/40}}{\sqrt{\int_0^1\sigma_t^2dt}}$, for our adopted spot volatility estimators (ref) and (ref). Such a scheme was also used in JT2014 and LLL2018. For leverage effect, we fix $\eta=0.3$, and vary $\rho=-0.6, -0.4, -0.2$. For each generated sample path, we calculate the relative bias $\frac{\widehat{\text{Lev}} - \text{Lev}}{\text{Lev}}$, where $\widehat{\text{Lev}}$ stands for the value of estimate from a general leverage effect estimator, and $\text{Lev}$ is the true leverage effect. The results of mean (M.), standard deviation (S.D.) and mean squared error (M.S.E.) of the relative biases are exhibited in Table (ref). From the table, we can make the following conclusions. Without the consideration of jumps ($\gamma=0$), Lev-WM14 already perform badly with large M., S.D. and M.S.E., perhaps this is because it uses non-overlapping increments for spot volatility. For the other three, our leverage effect estimator almost has the smallest absolute value of M., but a relatively larger value of S.D., which is in line with our theoretical results that our estimator has larger asymptotic variance. When the jumps are present with $\gamma=0.2$, Lev-WM14 and Lev-AJ14 perform badly, so we do not record their results in this scenario. Comparing our estimator with Lev-AFLWY17, we observe that our estimator has smaller bias but larger S.D., when $\beta=0.5, 1$. For large value of $\beta=1.5$, the bias and S.D. of our estimator are the smallest, yielding the smallest M.S.E., which shows that our estimator works better when the jumps are intensive. The same experiment is done for the volatility of volatility, but we use $n=2730\times 12$, which corresponds to 3-minute data within one year\footnote{We note that all the volatility of volatility estimators do not work for $n=2730$, which inspires that larger amount of data is needed for the estimation of volatility of volatility, compared with leverage effect.}. Its numerical results are presented in Table (ref), for which we fix $\rho=-0.2$, and vary $\eta = 0.1, 0.2, 0.3$. “Vov-BV09" does not work even without the presence of jumps. The conclusions for the volatility of volatility are similar to the ones obtained for leverage effect. Without jumps or with inactive jumps, our estimator has smaller bias but larger S.D.. When the jumps are intensive, our estimator performs the best in the sense of all measures of M., S.D. and M.S.E. Moreover, for most of the numerical experiments, the magnitude of M. is well controlled under $10\%$ for our proposed leverage effect estimator and volatility of volatility estimator.
Next, we verify the central limit theorems established in Theorem (ref) and Theorem (ref), where different values of $\kappa$, $u$ and $\beta$ are considered. We fix the parameters $\rho=-0.4, \eta=0.2, b=0.5$ and randomly generate a sample path of $\sigma^2$ in (ref) first. Fixing this volatility path, a total of 1000 sample paths of $X$ in (ref) are then generated. For each path, we calculate the estimates on the left hand side of (ref) and record the mean (M.) and variance (Var.) in Table (ref).\footnote{We recall that $\kappa_{opt}$ is defined as in (ref) and $u^{\star} = \frac{(\log(n))^{-1/40}}{\sqrt{\int_0^1\sigma_t^2dt}}$ is used in the last experimental study. For the specific generated volatility path in this simulation, we have $\kappa_{opt} \approx 2$ and $u^{\star} \approx 7$.} Moreover, we record the theoretical variance (T-Var.) in (ref)\footnote{For (ref), we have $h_2(t, \sigma_{t}^2,(\tilde{\sigma}_{t})^2, (\tilde{\sigma}'_{t})^2) = (\eta^2 \sigma_t^2)/3$. Thus, the theoretical variance only depends on the path of $\sigma^2$ when $\eta$ is fixed.} for evaluation, and present the results in Table (ref). The same experimental study is done for volatility of volatility by using the same parameters, except that we set $n=2730\times 12$. The results are displayed in Table (ref). From Table (ref)-(ref), we see that, for various indexes of jump activity, irrespective of the selection of $u$, all the values of M. are close to 0 and the values of Var. are close to the corresponding theoretical ones, T-Var.. The simulation results inspire us that a data-driven way of selecting $u$ is not necessary for our estimators, and a manual selection such as $u=1$ can meet our requirement and be applied in practice. Besides, compared with $\kappa=1.5,1$, the minimal values of both Var. and T-Var. are obtained when $\kappa = \kappa_{opt}$ for the estimation of leverage effect, which is in line with our theory.
Then, we turn to the verification of the feasible central limit theorems established in Corollary (ref). For the estimation of leverage effect, since the estimation of volatility of volatility is involved when estimating the asymptotic variance, and we see from the first experiment that a larger sample size is required for guaranteeing the estimation accuracy\footnote{In practice, this target can be achieved by using relative larger amount of historical data.}, thus we take $n=2730\times12$. For the estimation of leverage effect, we fix $u=1$ and estimate $\kappa_{opt}$ in (ref) by plugging in corresponding estimators of the numerator and denominator, which are given in Section (ref), with a pre-specified value $\kappa=1$. We take $b=0.55$, since a small value of $k_n$ may leads to negative estimates of volatility of volatility, as mentioned in Section (ref). All the other parameters remain the same as the ones in the last simulation study. With a pre-generated volatility curve, 1000 sample paths of $X$ are generated, based on which the estimates on the left-hand side of (ref) are calculated. The histograms and Q-Q plots for different intensity levels of jumps are presented in Figure (ref), from which we see that the distribution of leverage effect estimates are close to standard normal distribution. We also try the case of $\beta = 1.5$, when the theoretical requirement $\beta \leq 1$ is not satisfied. We observe that the mean value is still close to 0 and the distribution seems to be symmetric, but the tail part of the distribution is heavier than standard normal distribution. As for the volatility of volatility estimator, we have to admit that the performance is not satisfying when $n=2730\times12$. Perhaps this is because estimating its asymptotic variance accurately requires an even larger sample size $n$. By using the true value of $Var(W|\mathcal{F})$, we exhibit the histograms and Q-Q plots in Figure (ref). When the condition $\beta<\frac{3}{2}$ in Theorem (ref) is satisfied, we see that the distributions of the estimates are close to standard normal distribution. When this condition is violated with $\beta=1.5, 1.8$, there exists a distinctive enlarging positive bias as $\beta$ increases.
In this section, we apply our proposed leverage effect estimator and volatility of volatility estimator to real high-frequency financial data. Moreover, based on the feasible central limit theorems in Corollary (ref), we conduct tests of zero leverage effect and zero volatility of volatility.
The real high-frequency data resources used in this section are obtained from FirstRate database\footnote{\url{https://firstratedata.com}}. Instead of directly using the tick-by-tick transactional price data, we use a relative sparser frequency data to avoid the adverse effect of market microstructure noise. Specifically, we exclude all half-trading days, overnight and weekend returns and extract the 3-minute log-return data during the trading hours (From 9:30am to 16:00pm) by using the conventional previous tick strategy (See, e.g. Z2011), which takes the latest tick-by-tick price previous to the fixed time grids for approximation. We conduct our study over a total number of eight years from January 3, 2011 to December 31, 2018, which is a few years after the 2008 subprime mortgage crisis and before the COVID-19 pandemic. The subjects of our study include SPDR S$\&$P 500 ETF (SPY) tracking S$\&$P 500 index, and its four constituent individual stocks, Apple (APPL), Amazon (AMZN), Intel (INTC), and Microsoft (MSFT). For both the estimation of leverage effect and volatility of volatility, we fix $u=1$, $b=0.55$, $\kappa=2$.
We first adopt the jump activity index (JAI) estimator proposed in JKLM2012 to estimate the jump intensity in each month, where we assume that JAI is constant within this period. Based on the monthly estimates, the values of mean (M.) and standard deviation (S.D.) for each year are recorded in Table (ref). The mentioned JAI estimator is defined as
with $0<\alpha < \alpha'$, $0<\omega<1/2$, and $V(\omega, \alpha,g)_n = \sum_{i=1}^{n} g\left(\frac{\Delta_i^n X}{\alpha \Delta_n^{\omega}}\right)$, where $g(x)$ decreases to 0 as $|x|$ goes to 0. By following the same setting used in the empirical part of JKLM2012, we use $\omega = 1/5$, $\alpha=0.0013$, $\alpha'=2\alpha$, and
with $a=6/5$, $b=7/5$, $p=5$ and $c=a^{p}+pa^{p-1}(b-a)/2$. From Table (ref), we see that most of the JAI estimates are larger than 0.5, some of them are even larger than 1, especially for SPY. The results demonstrate that the jump activity is relative intensive in real high-frequency financial data, which necessitates our consideration in this paper and makes our estimators to be better alternatives for empirical studies.
We then use the yearly data for volatility of volatility, and present the volatility of volatility estimates (VoV.) and its estimated theoretical standard error (T.S.D.), that is $n^{\frac{b-1}{2}}\sqrt{\widehat{Var(W|\mathcal{F})}}$, in Table (ref). Moreover, we conduct the hypothesis testing problem of (ref) at the significance level $5\%$ by using our proposed test statistic (ref). We reject the null hypothesis of zero volatility of volatility if $\widetilde{T}_n >1.64$. Otherwise, if $\widetilde{T}_n \leq 1.64$, we do not reject $H_0$ and conclude that the alternative hypothesis $H_1$ is true. The testing results over 2011 to 2018 are also included in Table (ref). From the table, we see that, for all the years, SPY has the smallest values of both VoV. and T.S.D.. And almost for all the assets investigated, these quantities achieve the largest magnitude in 2018 and are relative smaller in 2017. As for the zero volatility of volatility test, it fails to reject the null hypothesis in 2011, 2015 and rejects it in 2012, 2013, 2017, for all the assets. Interestingly, in 2018, it fails to reject the null hypothesis for SPY, while all of its four constituent individual stocks are in favor of the alternative hypothesis $H_1$.
Similarly, for the estimation of leverage effect, by using the yearly 3-minute log-return data, the estimated leverage effect (LeV.), its estimated theoretical standard error (T.S.D.), namely $\sqrt{\widehat{Var(U|\mathcal{F})}}/\sqrt{n}^{b \wedge (1-b)}$, are recorded in Table (ref). For the following zero leverage effect hypothesis testing problem:
according to (ref), we can use the test statistic
At the significance level of $5\%$, we reject the null hypothesis of zero leverage effect if $|\overline{T}_n| >1.96$. Otherwise, if $|\overline{T}_n| \leq 1.96$, we do not reject $H_0$ and conclude that the alternative hypothesis $H_1$ is true. The testing results over 2011 to 2018 are also included in Table (ref). From the table, we see that almost all of the leverage effect estimates are negative, which is in line with the definition of leverage effect. And the magnitude of Lev. is relative smaller for SPY, compared with other four individual stocks. For almost all of the assets investigated, the quantities of LeV. and T.S.D. achieve the largest magnitude in 2011 and 2018, for which the null hypothesis of zero leverage effect is also rejected. And in 2015, it fails to reject the null hypothesis $H_0$ for all assets. Interestingly, from 2011 to 2018, except for 2015, we successfully reject the null hypothesis and obtain significant negative leverage effect estimates for SPY, while for its four constituent individual stocks, the reject rate is around 50 percent within these years.
We also conduct the estimation of leverage effect by using monthly data and record the estimates in Figure (ref). We see that most of the estimates are negative and close to 0, especially for SPY. In fact, after repeating the above zero leverage effect testing procedure, we obtain that, among all the 96 months tested, the number of month failing to reject the null hypothesis are 78, 85, 82, 83 and 49, for APPL, AMZN, INTC, MSFT and SPY, respectively. This inspires us that a relative larger number of samples is preferred, and using ultra high-frequency data with market microstructure noise, say tick-by-tick data, may partially solve this problem. We leave this issue for future work.
In this paper, we consider the statistical inference of leverage effect and volatility of volatility by using high-frequency data with the presence of jumps. Based on the empirical characteristic function of the increments, we proposed consistent estimators. Feasible central limit theorems have been established under regular conditions, based on consistent estimation of volatility functional.
Some remaining issues deserved for further investigation. First, we did not consider the presence of market microstructure noise in the observations and the phenomenon of irregular or endogenous observation time, which are also stylized features of high-frequency data. Their effects on the estimation of leverage effect and volatility of volatility deserve further exploration. Second, the effect of infinite variation jumps on the statistical inference of general volatility functional remains to be explored. Third, for the leverage effect from a correlation perspective, although we proposed a consistent estimator, its statistical inference problem was not studied. Moreover, we assumed that $\mathcal{L}^{cor}_{[0,T]}$ is constant over the time interval $[0,T]$, which may be too restrictive in theory and unrealistic in application. Thus, on one hand, a rigorous statistical testing procedure is required to verify whether this assumption is true or not. On the other hand, establishing the theoretical results without such an assumption should be considered.