Efficient Integrated Volatility Estimation in the Presence of Infinite Variation Jumps via Debiased Truncated Realized Variations

Efficient Integrated Volatility Estimation in the Presence of Infinite Variation Jumps via Debiased Truncated Realized Variations

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Introduction

Statistical inference for stochastic processes based on high-frequency observations has attracted considerable attention in the literature for more than two decades. Among the many problems studied to date, arguably none has received more attention than that of the estimation of the continuous (or predictable) quadratic variation of an It\^o semimartingale $X=\{X_{t}\}_{t\geq{}0}$. Specifically, if

where $X_0\in\mathbb{R}$, $W=\{W_t\}_{t\geq{}0}$ is a Wiener process and $X^{j}=\{X^{j}_{t}\}_{t\geq{}0}$ is a pure-jump It\^o semimartingale, our estimation target is \[ IV_T=\int_0^T\sigma^2_s ds. \] This quantity, also known as the integrated volatility or integrated variance of $X$, has many applications, especially in finance, where $X$ typically models the log-return process of a risky asset and $IV_T$ measures the overall uncertainty or variability inherent in $X$ during the time period $[0,T]$. When $X$ is observed at evenly spaced times $0=t_0<t_1<\ldots<t_n=T$, in the absence of jumps, an efficient estimator of $IV_T$ is given by the realized quadratic variation $\widehat{IV}_T=\sum_{i=1}^{n}(X_{t_i}-X_{t_{i-1}})^2$ in the so-called high-frequency (or infill) asymptotic regime; i.e., when $\max_{i}(t_i-t_{i-1})\to{}0$ and $T\equiv t_n$ is fixed. In the presence of jumps, $\widehat{IV}_T$ is no longer even consistent for $IV_T$, instead converging to $IV_T+\sum_{s\leq T}(\Delta X_s)^2$, where $\Delta X_s:= X_s-X_{s^-}$ denotes the jump at time $s$. To account for jumps, several estimators have been proposed, among which the most well-known are the truncated realized quadratic variation and the multipower variations. We focus on the first class, which, unlike the second, is both rate- and variance-efficient, {in the Cramer-Rao lower bound sense,} when jumps are of bounded variation under certain additional conditions.

The truncated realized quadratic variation (TRQV), also called truncated realized volatility, was first introduced by mancini2001 and mancini2004 and is defined as

where $\varepsilon=\varepsilon_n>0$ is a tuning parameter converging to $0$ at a suitable rate. Above, $\Delta_{i}^{n}X:=X_{t_{i}}-X_{t_{i-1}}$ is the $i$-th increment of $(X_{t})_{t\geq 0}$ based on evenly spaced observations $X_{t_{0}},\ldots,X_{t_{n}}$ over a fixed time interval $[0,T]$ (i.e., $t_{i}=ih_n$ with $h_{n}=T/n$). It is shown in mancini2009non that TRQV is consistent when either the jumps have finite activity or stem from an infinite-activity L\'evy process. In a semimartingale model with L\'evy jumps of bounded variation, cont2011nonparametric showed that the TRQV admits a feasible central limit theorem (CLT), provided that $\varepsilon_n = ch_n^{\omega}$ with some ${ \omega} \in [\frac{1}{4-Y}, \frac{1}{2})$, where $Y\in [0,1)$ denotes the Blumenthal-Getoor index. In jacod2008asymptotic, consistency was established for a general It\^o semimartingale $X$, and a corresponding CLT is given when the jumps of $X$ are of bounded variation. In that case, the TRQV attains the optimal rate and asymptotic variance of $\sqrt{ h_n}$ and $2\int_{0}^{T}\sigma_s^4ds$, respectively.

However, in the presence of jumps of unbounded variation, arguably the most relevant for financial applications (see, e.g., SahaliaJacod2009, Belomestny:2010, FigueroaLopez:2012, and the results in Table (ref) below), the situation is notably different, and the available literature on TRQV offers an incomplete picture. In cont2011nonparametric, it is shown that when jumps stem from a L\'evy process with stable-like small-jumps of infinite variation, the TRQV estimator $\widehat{C}_n(\varepsilon)$ converges to $IV_T$ at a rate slower than $\sqrt{h_n}$. Further, in mancini2011speed it is shown that when the jump component $J$ is a symmetric $Y$-stable L\'evy process and $\varepsilon_n=h_n^{\omega}$ with ${\omega}\in(0,1/2)$, the decomposition $\widehat{C}_{n}( \varepsilon_n)-IV_T=\sqrt{h_n}Z_n+\mathcal{R}_n$ holds, where $Z_{n}$ converges stably in law to $\mathcal{N}(0,2\int_0^T\sigma_s^4 ds)$, while $\mathcal{R}_n$ is precisely of order $\varepsilon_n^{2-Y}$ in the sense that $\mathcal{R}_n=O_P(\varepsilon_n^{2-Y})$ and $\varepsilon_n^{2-Y}=O_P(\mathcal{R}_n)$ (which decays too slowly to allow for efficiency when $Y>1$). In AmorinoGloter20, a smoothed version of the TRQV estimator of the form $\widehat{C}^{Sm}_{n}(\varepsilon)=\sum_{i=1}^{n}(X_{t_i}-X_{t_{i-1}})^2\varphi((X_{t_i}-X_{t_{i-1}})/\varepsilon)$ is considered\footnote{The authors in AmorinoGloter20, in fact, consider the more general estimation of $\int_0^T f(X_s) \sigma_s^2 ds$ for functions $f$ of polynomial growth, for which $IV_T$ is a special case.}, where $\varphi\in C^{\infty}$ vanishes in $\mathbb{R}\backslash{}(-2,2)$ and $\varphi(x)=1$ for $x\in(-1,1)$. In that case, using the truncation level $\varepsilon_n:=h_n^{ \omega}$, it is shown that $\widehat{C}^{Sm}_{n}( \varepsilon_n)-IV_T=\sqrt{h_n}Z_n+\mathcal{R}_n$ with $\mathcal{R}_n$ such that $\varepsilon_n^{-(2-Y)}\mathcal{R}_n\to c_{Y}\int \varphi(u)|u|^{1-Y}du$, for a constant $c_Y\neq{}0$, and still $Z_{n}\stackrel{st}{\rightarrow}\mathcal{N}(0,2\int_0^T\sigma_s^4 ds)$. By taking $\varphi$ such that $\int \varphi(u)|u|^{1-Y}du=0$, a “bias-corrected" estimator was considered under the additional condition that $Y<4/3$. Specifically, the resulting estimator is such that, for any { $\tilde{\epsilon}>0$}, $\widehat{C}^{Sm}_{n}( \varepsilon_n)-IV_T=o_{P}(h_n^{1/2-{ \tilde\epsilon}})$, “nearly" attaining the optimal statistical error $O_{P}(h_n^{1/2})$. Unfortunately, the construction of such an estimator requires knowledge or accurate estimation of the jump intensity index $Y$, and no feasible CLT was proved when jumps are of unbounded variation even assuming $Y$ is known.

Apart from TQRV-based approaches, efficient estimation of $IV_T$ when the jumps have unbounded variation is intrinsically limited in the general case. In jacod:reiss:2014, it was shown that when the jump intensity index $Y>1$, the best possible convergence rate, in a minimax sense, over certain “bounded" classes of semimartingales, is of order $(n\log n)^{-(2-Y)/2}$. Nevertheless, in principle, a faster convergence rate may be attainable if one constrains the process $X$ to belong to a certain semiparametric class such as when the jumps exhibit a “locally stable”-like behavior. Obviously, the fastest possible rate one can hope to achieve is $n^{-1/2}$, which coincides with the one attained by the realized quadratic variation in the continuous case and is known to be optimal in a minimax sense.

The first (and to-date, only) rate- and variance-efficient estimator of the integrated volatility known in the literature for semimartingales when $Y>1$ was proposed by JacodTodorov:2014, under a locally-stable assumption on jumps, but with some notable additional restrictions: these results require either that the jump intensity index $Y<3/2,$ or that the “small" jumps of the process $X$ are “symmetric"\footnote{JacodTodorov:2014 also constructed an estimator that is rate-efficient even in the presence of asymmetric jumps, but its asymptotic variance is twice as big as the optimal value $2\int_{0}^{T}\sigma_s^4ds$ and, thus, it is not variance efficient.}. Their estimator is based on locally estimating the volatility from the empirical characteristic function of the process' increments over disjoint time intervals shrinking to $0$, but still containing an increasing number of observations. It requires two debiasing steps, which are simpler to explain for a L\'evy process $X$ with symmetric $Y$-stable jump component $ X^j$. The first debiasing step is meant to reduce the bias introduced when attempting to estimate $\log\mathbb{E}(\cos({u} X_{h_n}/\sqrt{h_n}))$ with $\log\big\{\frac{1}{n}\sum_{i=1}^{n}\cos\big({ u} \Delta_i^n X/\sqrt{h_n}\big)\big\}$. The second debiasing step is aimed at eliminating the second term in the expansion $-2\log\mathbb{E}(\cos({ u} X_{h_n}/\sqrt{h_n})) /{ u}^2=\sigma^2+2|\gamma|^{Y} { u}^{Y-2}h_n^{1-Y/2}+O(h_n)$, which otherwise diverges when multiplied by the optimal scaling $h_{n}^{-1/2}= n^{1/2}$. Using an extension of this approach, Jacod and Todorov were able to apply these techniques to a more general class of It\^o semimartingales in jacod2016efficient, even allowing any $Y<2$, though only rate-efficient, but not variance-efficient, estimators were ultimately constructed.

On the other hand, in the special case of L\'evy processes, efficiency across the full range $0<Y<2$ without symmetry requirements has been attained by Mies:2020, again under a locally stable assumption, via a generalized method of moments. Specifically, for some suitable smooth functions $f_{1},f_{2},\dots,f_{m}$ and a scaling factor $u_{n}\rightarrow\infty$, Mies:2020 {proposed to search} for the parameter values $\widehat{\boldsymbol{\theta}}=(\widehat{\theta}_{1},\ldots,\widehat{\theta}_{m})$ such that

where $\widetilde{X}$ is the superposition of a Brownian motion and independent stable L\'{e}vy processes closely approximating $X$ in a certain sense. The distribution measure $\mathbb{P}_{\boldsymbol{\theta}}$ of $\widetilde{X}$ depends on some parameters $\boldsymbol{\theta}=(\theta_1,\ldots,\theta_m)$, one of which is the volatility $\sigma$ of $X$, and $\mathbb{E}_{\boldsymbol{\theta}}(\cdot)$ denotes the expectation with respect to $\mathbb{P}_{\boldsymbol{\theta}}$. Aside from the fact that this method can only be applied to L\'evy processes, it also suffers from other drawbacks. First, its finite-sample performance critically depends on the chosen moment functions $f_{1},\ldots,f_{m}$. Secondly, its implementation is computationally expensive and may lead to numerical issues since it involves solving a system of nonlinear equations (including possible non-existence of solutions to (ref) over finite samples). Moreover, in {addition} to the required use of a numerical solver to determine the {values} of $\widehat{\pmb{\theta}}$ in (ref), the expectations appearing therein need to be numerically approximated since explicit expressions for the moments $\mathbb{E}_{\boldsymbol{\theta}}(f_{j}(u_{n}\Delta^{n}_{i}\widetilde{ X}))$ are typically not available. This fact introduces additional numerical errors that complicates its performance.

In this paper, we consider a new method to estimate the integrated volatility $IV_T=\int_0^T\sigma_s^2ds$ of an It\^o semimartingale whose jump component is given by a stochastic integral with respect to a tempered-stable-like L\'evy process $J$ of unbounded variation. To the best of our knowledge, our method, together with JacodTodorov:2014, are the only efficient methods to deal with jumps of unbounded variation for semimartingales. The idea is natural. We simply apply debiasing steps similar to those of JacodTodorov:2014 to the TRQV of mancini2009non. To give the heuristics as to why this strategy works, consider a small-time expansion of the truncated moments $\mathbb{E}(X_{h_n}^{2k}{\bf 1}_{\{|X_{h_n}|\leq{}\varepsilon_n\}})$ of a L\'evy process $X_t=bt+\sigma W_{t}+X^j_t$ in the asymptotic regime $h_n, \varepsilon_n\to{}0$ with $\varepsilon_n/\sqrt{h_n}\to\infty$. Using a variety of techniques, including a change of probability measure, Fourier-based methods, and small-large jump decompositions, we show the following two expansions:

for certain constants $c_1,c_2,c_3\neq{}0$ that are explicitly computed. Hereafter, {\rm h.o.t.} stands for `higher order terms'. The expansions above are the most precise of their type in the literature and are of great interest in their own right. Based on the first expansion above, it is easy to see that the rescaled bias ${\mathbb{E}[h_n^{-1/2}(\widehat{C}_{n}(\varepsilon)-\sigma^2T)]}$ satisfies

which suggests the necessity of the condition $h_n^{-1/2}\varepsilon_n^{2-Y} =o(1)$ for a feasible CLT for $\widehat C_n(\varepsilon)$ at the rate $\sqrt{h_n}$. However, together with the (necessary) condition $\varepsilon_n/\sqrt{h_n}\to\infty$, this can happen only if $Y<1$, and removal of the first terms in (ref) is consequently necessary for efficient estimation when jumps are of unbounded variation. To that end, note that for any $\zeta>1$,

The above formulas motivate the “bias-corrected" estimator

which is the essence of the debiasing procedure of JacodTodorov:2014. As we shall see, the story is more complicated than what the simple heuristics above suggest. Our main result shows that, for the class of It\^o semimartingales described above, the estimator (ref) is indeed rate- and variance-efficient provided that $Y\in(1,4/3)$. Furthermore, if $Y\in (1,8/5)$, a second bias-correcting step will achieve both rate- and variance-efficiency (for the case $ 8/5\leq Y<2$, see remark at the end of Section (ref)). Even though our main motivation lies in incorporating jumps of unbounded variation, we show that the debiasing steps will still achieve efficiency in the case that $Y<1$ (though, of course, no debiasing is needed in that case because the jumps are of bounded variation).

Though our approach is natural, mathematically establishing its efficiency is highly nontrivial, starting from the new high-order expansions of the truncated moments of L\'evy processes -- which, beyond heuristics, ultimately play a key role in analyzing asymptotics for the debiasing technique -- to the application of Jacod's stable central limit theorem for semimartingales (in particular, the verification of the asymptotic `orthogonality' condition (2.2.40) in JacodProtter). Our Monte Carlo experiments indicate improved performance compared to JacodTodorov:2014 (and also Mies:2020) for the important class of CGMY L\'evy processes (cf. CarrGemanMadanYor:2002) and for a Heston stochastic volatility model with CGMY jumps in the range of values of $Y$ covered by our theoretical framework.

If we limit ourselves to a L\'evy model, our approach is more computationally efficient and numerically stable than that in Mies:2020\footnote{Though the method of Mies:2020 allows for simultaneous estimation of several parameters of the model (such as both $\sigma$ and $Y$).}. For more general semimartingales, our procedure is simpler than that in JacodTodorov:2014 since it does not require an extra debiasing step to correct the nonlinear nature of the logarithmic transformation employed therein nor does it require a symmetrization step to deal with asymmetric jump components. Furthermore, our method does not rely on a `localization' technique in the sense that it does not need to break the data into disjoint blocks where the integrated volatility is locally estimated. The latter step introduces an additional tuning parameter absent from our method.

Let us emphasize that our method is the first variance- and rate-efficient nonparametric method for integrated volatility free of complete symmetry assumptions on small jumps that is capable of exceeding the limit $Y<3/2$ imposed in JacodTodorov:2014,jacod2016efficient. Symmetry is potentially a strong assumption for financial returns as there is a general belief that significant losses are more likely than significant gains. For instance, recently Asmussen:2022 examined several empirical studies from the literature and observed that the majority display negative skewness. Of course, skewness may arise from either large or small jumps, though large jump asymmetry has received most attention in empirical work. Indeed, there are few studies to date that estimate the intensity of small positive and negative jumps separately. An exception is Bianchi, who, using MLE applied to daily S&P500 index data from 1996-2006, obtained the estimates $\hat{C}_+=0.7119$, $\hat{C}_-=0.5412$, $\hat{G}=59.94$, $\hat{M}=59.94$, $\hat{Y}_+=1.0457$, and $\hat{Y}^-=1.1521$ under a pure-jump L\'evy model with L\'evy density $C_{+}e^{-x/G}|x|^{-Y_+-1}{\bf 1}_{x>0}+C_-e^{-|x|/M}|x|^{-Y_{-}-1}{\bf 1}_{x<0}$, which points to asymmetry in small jumps. For further illustration, in Table (ref) below, we fit a L\'evy model $\sigma W_t+X_t^j$, with semi-parametric L\'evy density $(C_+{\bf 1}_{x>0}+C_-{\bf 1}_{x<0})q(x)|x|^{-Y-1}$ (here, $q(x)\stackrel{x\to{}0}{\longrightarrow}1$). We use Mies' method of moments (cf. Mies:2020) for different stocks and frequencies\footnote{As pointed out in Mies:2020 (see also the paragraph after (ref) above), numerical issues can arise related to feasibility of the estimating equations associated with the method. We are only presenting the results when the algorithm successfully finishes and yields reasonable values.} over a 1-year period in 2005. It is clear that $\hat{C}_+$ is different from $\hat{C}_-$, sometimes by a relatively large value, indicating further evidence for asymmetry in small jump behavior. Furthermore, all values of $\hat{Y}$ are larger than $1$, indicating the presence of a jump component of unbounded variation.

Finally, let us also remark that our result opens the doors to attain rate- and variance-efficient estimators free of symmetry requirements beyond the mark $8/5$ or in more general semiparametric models with successive Blumethal-Getoor indices by considering further debiasing steps. These directions will be investigated in further work.

The rest of this paper is organized as follows. Section (ref) introduces the framework and assumptions as well as some known preliminary results from the literature. Section (ref) introduces the debiasing method and main results of the paper. Section (ref) illustrates the performance of our method via Monte Carlo simulations and compares it to the method in JacodTodorov:2014. The proofs of the key results are deferred to appendix sections.

Setting and background

In this section, we introduce the model, main assumptions, and some notation. We consider a 1-dimensional It\^o semimartingale $X=(X_{t})_{t\in\mathbb{R}_{+}}$ of the form (ref), defined on a complete filtered probability space $(\Omega,\mathscr{F},(\mathscr{F}_{t})_{t\in\mathbb{R}_{+}},\mathbb{P})$. Since it has no impact on the value of the increments of $X$, for simplicity throughout we assume $X_0=0$. We assume the jump component $X^{j}$ can be decomposed into a sum of an infinite-variation process $X^{j,\infty}$ and a finite variation process $X^{j,0}$ given, for $t\in\mathbb{R}_+$, as

where $\chi=\{\chi_t\}_{t\geq{}0}$ is an adapted process satisfying appropriate integrability conditions, $J:=(J_{t})_{t\in\mathbb{R}_{+}}$ is an independent pure-jump L\'{e}vy process with L\'{e}vy triplet $(b,0,\nu)$, $\mathfrak p_0,\mathfrak p_1,$ are Poisson random measures on $\mathbb{R}_+\times \mathbb{R}$ with intensities $\mathfrak q_i(ds,dz)=ds\otimes \lambda_i(dz)$, where the $\lambda_i$'s are $\sigma$-finite measures on $\mathbb{R}$, and $\mathfrak p_1$ is assumed independent of $J$. The specific conditions on $\nu$, $\lambda_i$, and on the coefficient processes $\delta_i$, $\chi$, and $\sigma$ are given below.

The L\'{e}vy measure $\nu$ is assumed to admit a density $s:\mathbb{R}_{0}\rightarrow\mathbb{R}_{+}$ of the form

Above, $\mathbb{R}_{0}:=\mathbb{R}\backslash\{0\}$, $C_{\pm}>0$, $Y\in(1,2)$, and $q:\mathbb{R}_{0}\rightarrow\mathbb{R}_{+}$ is a bounded Borel-measurable function satisfying the following conditions:

These processes are sometimes called “stable-like L\'evy processes" and were studied in FigueroaLopezGongHoudre:2016,FigueroaLopezOlafsson:2016 and many other works. In simple terms, condition (i) above says that the small jumps of the L\'evy process $X$ behave like those of a $Y$-stable L\'evy process with L\'evy measure

The condition $Y\in(1,2)$ implies that $J$ has unbounded variation in that sense that $\sum_{i=1}^{n}|J_{t_i}-J_{t_{i-1}}|\to{}\infty$, a.s., as the partition $0=t_0<t_1<\dots<t_n=T$ is such that $\max\{t_i-t_{i-1}\}\to{}0$.

As discussed in Section (ref), in view of jacod:reiss:2014, the locally $Y$-stable aspect of Assumption (ref) is crucially important for our results, and similar assumptions have been made by other authors (e.g, Mies:2020,JacodTodorov:2014). Though not completely general, the class is still relevant in applications, as many of the models proposed in the literature (especially, in finance) fall within this class. Nevertheless, from a theoretical point of view, it remains to be seen as to what the broadest assumptions may be under which one can still attain estimation efficiency.

As in FigueroaLopezGongHoudre:2016, it will be important for our analysis to apply a density transformation technique Sato:1999 to “transform" the process $J$ into a stable L\'evy process. Concretely, we can change the probability measure from $\mathbb{P}$ to another locally absolutely continuous measure $\widetilde{\mathbb{P}}$, under which $W$ is still a standard Brownian motion independent of $J$, but, under $\widetilde{\mathbb{P}}$, $J$ has L\'evy triplet $(\tilde{b},0,\tilde{\nu})$, where $\widetilde{\nu}(dx)$ is given as in (ref) and $\widetilde{b}:=\bar b+\int_{0<|x|\leq 1} x(\tilde{\nu}-\nu)(dx)$. The key assumption above is (ii), which would allow us to decompose the log-density process \[ U_t:=\ln \frac{d\widetilde{\mathbb{P}}\big|_{\mathscr{F}_{t}}}{d\mathbb{P}\big|_{\mathscr{F}_{t}}}, \] as a sum of a bounded variation process and two spectrally one-sided $Y$-stable L\'evy processes.

Finally, we give the conditions on $\sigma$ and the coefficient processes $b$, $\delta_0,\delta_1$, and $\chi$ in (ref) and (ref).

Above, the parameters $r_0,r_1$ control the degree of activity in the nuisance finite-variation jump terms. Two such terms are included to allow for a broader range of finite-variation jump activity in our model setup. Note that $Y/2$ is always bigger than $Y/(2+Y)$, which shows that when the bounded variation jump component is independent from the other processes, we can incorporate a wider range of jump activity. These restrictions effectively guarantee that the bias introduced by finite variation components are negligible in comparison to leading bias terms arising from the locally-stable jumps in $X$.

Main results

In this section, we construct an efficient estimator for the integrated volatility $IV_T=\int_0^T \sigma_s^2ds$ based on the well-studied estimator TRQV (ref).

Throughout, we assume the process $X=\{X_{t}\}_{t\geq{}0}$ is sampled at $n$ evenly spaced observations, $X_{t_1}, X_{t_2}, \ldots, X_{t_n}$, during a fixed time interval $[0,T]$, where for $i=0,\dots,n$, $t_{i}=t_{i,n}= ih_{n}$ with $h_n=T/n$, and, for simplicity, assume that $T=1$. As usual, we define the increments of a generic process $V=\{V_t\}_{t\geq{}0}$ as $\Delta_{i}^{n}V:=V_{t_{i}}-V_{t_{i-1}}$, $i=1,\dots,n$. We often use the shorthand notation: \[ V_i^n=V_{t_{i}},\quad \mathbb{E}_{i}[\cdot]=\mathbb{E}[\cdot|\mathcal{F}_{t_i}]. \]

As mentioned in the introduction, in the presence of jumps of unbounded variation, the TRQV estimator $\widehat{C}_n(\varepsilon)$ is not efficient since it possesses a bias that vanishes at a rate slower than $n^{-1/2}$, the rate at which the “centered” TRQV

admits a CLT. To overcome this, our idea is to apply the debiasing procedure of JacodTodorov:2014 to the TRQV. As mentioned in the introduction, our procedure is simpler than JacodTodorov:2014 since it does not require an extra debiasing step to account for the logarithmic transformation nor does it require a symmetrization step to deal with asymmetric L\'evy measures. Furthermore, our method does not have to be applied in each subinterval of a partition of the time horizon, which introduces another tuning parameter.

Before constructing our estimator, we first establish the asymptotic behavior of TRQV (ref) with a fully specified centering quantity $A(\varepsilon,h)$ rather than the inexplicit centering $\mathbb{E}_{i-1}\left[\big(\Delta_{i}^{n}X\big)^{2}{\bf 1}_{\{|\Delta_{i}^{n}X|\leq\varepsilon\}}\right]$ of (ref). It also characterizes the structure of the bias $A(\varepsilon,h)$ in the threshold parameter $\varepsilon$ that will ultimately be exploited in our debiasing procedure. Below and throughout the rest of the paper, we use the usual notation $a_{n}\ll b_n$, whenever $a_n/b_n\to0$ as $n\to\infty$.

The next theorem establishes the stable convergence (in particular, the convergence rate) of the difference $\widetilde{Z}_n(\zeta\varepsilon) -\widetilde{Z}_n(\varepsilon)$, for some $\zeta>1$, which is the second main technical result we use to deduce the efficiency of our debiased estimator.

We are now in a position to introduce our proposed estimator. To this end, we will exploit the structure of the bias term $A(\varepsilon,h)$ in $\varepsilon$. The idea is simple. Suppose that a function $f(x)$ takes the form $a+bx^{\alpha}$ for any $a\in\mathbb{R}$ and $\alpha,b\neq{}0$. Then, it is easy to see that, for any $\zeta>1$, \[ f(x)-\frac{(f(\zeta x)-f(x))^2}{f(\zeta^2x)-2f(\zeta x)+f(x)}=a+bx^\alpha-\frac{b^2x^{2\alpha}(\zeta^\alpha-1)^2}{bx^{\alpha}(\zeta^{2\alpha}-2\zeta^\alpha+1)}=a, \] hence, recovering $a$ without requiring knowledge of $b$ and $\alpha$. These heuristics suggest the following debiasing procedure to successively remove each term appearing in $A(\varepsilon,h)$. For any $\zeta_1,\zeta_2>1$, in a first step, we compute

and, in the second step,

The next theorem is the main result of the paper. It establishes the rate- and variance-efficiency of the two-step debiased estimator $ \widetilde C_n ''(\varepsilon, \zeta_2,\zeta_1)$ provided $Y\in (0,1)\cup (1,8/5)$.

It is customary to use power thresholds of the form $\varepsilon_n = c_0 h_n^{ \omega}$, where $c_0>0$ and ${ \omega}>0$ are some constants\footnote{ Though, it is shown in gong2021 that in the case of a L\'evy process $X$ with jump component $J$ as in Section (ref), the MSE-optimal threshold $\varepsilon_n^{*}$ is such that $\varepsilon_n^{*}\sim\sqrt{(2-Y)\sigma^{2}h_n\ln(1/h_n)}$, as $n\to\infty$.}. In that case, the assumption $h_n^{\frac{4}{8+Y}}\ll \varepsilon_n\ll h_n^{\frac{1}{4-Y}\vee { \frac{2}{4+Y}}}$ in Theorem (ref) becomes

Monte Carlo performance with CGMY jumps

In this section, we study the performance of the two-step debiasing procedure of the previous section in two settings: the case of a L\'evy process with a CGMY jump component $J$ (cf. CarrGemanMadanYor:2002) and a Heston stochastic volatility model with again a CGMY jump component. Following JacodTodorov:2014, we also consider variants of our debiasing procedure that make use of the sign of the bias terms that lead to further improved finite-sample performance.

Constant volatility

We start by considering simulated data from the model (ref) and (ref), where the coefficient processes $\sigma$, $b$, and $\chi$ are constants, {$\delta_0=\delta_1\equiv 0$ (no bounded variation components)}, and $\{J_t\}_{t\geq{}0}$ is a CGMY process, independent of the Brownian motion $\{W_{t}\}_{t\geq{}0}$, with L\'evy measure having a $q$-function, in the notation of (ref), of the form:

and $C_{+}=C_{-}=C$. Thus, the conditions of Assumption (ref) are satisfied with $\alpha_{+}=-M$ and $\alpha_{-}=G$. For simplicity we take $b=0$ and $\chi=1$, and adopt the parameter setting

which are similar to those used in FigueroaLopezOlafsson:2016\footnote{FigueroaLopezOlafsson:2016 considers the asymmetric case $\nu(dx)=C_{{\rm sgn} (x)}\bar{q}(x)|x|^{-1-Y}\,dx$ with $C_+=0.015$ and $C_-=0.041$. Here, we take $C=(C_++C_-))/2$ in order to simplify the simulation of the model. The parameter values of $C_+$, $C_-$, $G$, and $M$ used in FigueroaLopezOlafsson:2016 were taken from Kawai, who calibrated the tempered stable model using market option prices.}, and take $\sigma=0.2$, $0.4$, and $Y=1.25$, $1.35$, $1.5$, $1.7$, respectively. We fix $T=1$ year and $n=252(6.5)(60)$, which corresponds to a frequency of $1$ minute (assuming $252$ trading days and a $6.5$-hour trading period each day).

In a fashion similar to JacodTodorov:2014, for the threshold $\varepsilon = c_0h^{\omega}$, we take $c_0 = \sigma_{BV}$, where \[ \sigma_{BV}^2:= \frac{\pi}{2} \sum_{i=2}^n |\Delta_{i-1}^nX||\Delta_i^nX|, \] which is the standard bipower variation estimator of $\sigma^2$ first introduced by Barndorff. {For the value of ${ \omega}$ we take ${ \omega} = \frac{5}{12}$, which, as mentioned above, satisfies the condition (ref) for any $Y\in(1,8/5)$.} We compare the performance of the following estimators:

We further compare the simulated performance of the above estimators to the estimators proposed in JacodTodorov:2014 {and Mies:2020}. Specifically, we use the equation (5.3) in the paper JacodTodorov:2014: $$

$$ where $\widehat C_{\text{JT}}$ denotes their ``nonsymmetrized'' two-step debiased estimator (see Eq.~(3.1) in therein). For the tuning parameters, we take $$ \zeta=1.5, \quad u_{n}=(\ln(1/h_{n}))^{-1/30}/\sigma_{BV}, \quad p_0=0.2, $$ where the values of $\zeta$ and $u_n$ were those suggested by \cite{JacodTodorov:2014} and the value of $p_0$ was chosen for favorable estimation performance. Note that, since a L\'evy model has constant volatility, it is not necessary to localize the estimator and, hence, we treat the 1-year data as one block, which corresponds to taking $k_n=252(6.5)(60)$ in the notation of \cite{JacodTodorov:2014}. For the moment estimator proposed in \cite{Mies:2020}, we use the same moment functions and the parameter settings as suggested by \cite{Mies:2020}. We denote this estimator $\widehat C_{M,4}$. We also examine the performance of another moment estimator, denoted $\widehat C_{M,3}$, that is computed under a similar algorithm to \cite{Mies:2020} but with $3$ different moment functions suggested in \cite{gong2021}. We remark that that the moment functions used in the construction of $\widehat C_{M,3}$ do not satisfy the strict constraints imposed in \cite{Mies:2020}, and therefore the asymptotic efficiency of the estimator $\widehat C_{M,3}$ has not been established. We refer to \cite{gong2021} for more details about the computations of $\widehat C_{M,3}$ and $\widehat C_{M,4}$.

The sample means, standard deviations (SDs), the average and SD of relative errors, the mean squared errors (MSEs), and median of absolute deviations (MADs) for each of the estimators described above are reported in Tables (ref)-(ref). In addition, we show the results corresponding to `case-by-case favorably-tuned' versions of $\widetilde C_{n,pb}''$, and $\widehat C_{\text{JT}, 53}$; i.e., their tuning parameters were chosen for achieving the `best' performance for each pair $(Y,\sigma)$ based on a grid search; we distinguish these estimators and their corresponding tuning parameters by the superscript $^*$. That is, $\widetilde C_{n,pb}''$ is based on the choices $(\zeta_1,\zeta_2,p_1,p_2)=(1.2,1.2,.65,.75)$ across all considered values of $Y$ and $\sigma$. With these values of the tuning parameters, $\widetilde C_{n,pb}''$ exhibits generally good performance overall. However, for each given fixed pair $(Y,\sigma)$, its counterpart $\widetilde C_{n,pb}''^*$ is tuned to have superior performance for those particular values of $Y$ and $\sigma$. For instance, as shown in Table (ref), when $Y=1.25$ and $\sigma=0.2$, the estimator $\widetilde C_{n,pb}''$ attains an MSE of $1.45\times 10^{-7}$, whereas the choice of parameters $(\zeta_1^*,\zeta_2^*,p_1^*,p_2^*)=(1.35,1.20,0.5,0.85)$ leads to an MSE of $4.70\times 10^{-8}$ for $\widetilde C_{n,pb}''^*$.

We provide a broad summary of our simulation results. For the estimators using the case-by-case tuned parameters $(\zeta_1^*,\zeta_2^*,p_1^*,p_2^*)$, based on both MSE and MAD, $\widetilde C_{n,nb} ''^*$ outperforms every other estimator considered in each setting, except when $Y=0.8$ when the performance between $\widetilde C_{n,nb} ''^*$ and $\widehat C_{\text{JT}, 53}^*$ is comparable (c.f. Figure (ref), top row) but $\widehat C_{\text{JT}, 53}^*$ has a slight edge in both MAD and MSE when $\sigma=0.4$ and a slight edge in MAD when $\sigma=0.2$. Using the parameters $(\zeta_1,\zeta_2,p_1,p_2)=(1.2,1.2,.65,.75)$, compared to the method of Mies:2020, $\widetilde C_{n,nb} ''$ outperforms $\widehat C_{\text{M},4}$ in all cases, as measured by MSE and MAD in Tables (ref)-(ref). When $\sigma=0.2$ and $Y=1.25$, or when $\sigma=0.4$ and $Y=1.25, 1.35, 1.5, 1.7$, $\widetilde C_{n,nb} ''$ outperforms $\widehat C_{\text{M},3}$ (though, as noted earlier, computation of $\widetilde C_{n,nb} ''$ is much faster and more numerically stable than that of either $\widehat C_{\text{M},3}$ or $\widehat C_{\text{M},4}$). Next, compared with JacodTodorov:2014, when $\sigma=0.2$ and $Y=1.25, 1.35,1.5$, or when $\sigma=0.4$ and $Y=1.25$ or $1.35$, $\widetilde C_{n,nb} ''$ has superior performance compared to $\widehat C_{\text{JT}, 53}$ as measured by MSE and MAD. Generally, $\widetilde C_{n,nb} ''$ significantly outperforms $\widetilde C_{n,pb}'$. Table (ref) also shows that, though when $\sigma=0.2$ and $Y=1.5$, $\widetilde C_{n,nb} ''$ has larger MSE and MAD than $\widehat{C}_n$ and $\widetilde C_{n,pb}'$, it still performs better than $\widehat C_{\text{JT}, 53}$: the MSE and MAD of $\widetilde C_{n,nb} ''$ in these cases are approximately {22% and 85%} of those of $\widehat C_{\text{JT}, 53}$, respectively. Tables (ref) and (ref) show that, when $Y=1.7$, which is not covered by our theoretical framework (see Remark (ref)), $\widetilde C_{n,nb} ''$ has slightly larger MSE and MAD than $\widehat C_{\text{JT}, 53}$. Overall, we conclude that our debiasing procedure outperforms $\widehat C_{\text{JT}, 53}$ when $Y = 1.25, 1.35, 1.5$, and outperforms $\widehat C_{\text{M},4}$ in all parameter settings considered, and the estimation performance of $\widehat C_{\text{JT}, 53}$ and $\widetilde C_{n,nb} ''$ is comparable when $Y=0.8$.

We also study the asymptotic approximation for the sampling distribution of $\widetilde C_{n,nb} ''$ and $\widehat C_{\text{JT}, 53}$ based on the case-by-case optimally-tuned estimators $\widetilde C_{n,nb} ''^*$ and $\widehat C_{\text{JT}, 53}^*$. Figure (ref) shows their normalized simulated sampling distributions, $\sqrt{n}(\hat\sigma - \sigma^2)/\sqrt{2\sigma^4}$, and the theoretical asymptotic normal distribution, $\mathcal{N}(0,1)$. Compared to $\widehat C_{\text{JT}, 53}^*$, the simulated distribution of $\widetilde C_{n,nb} ''^*$ yields a better match with the asymptotic normal distribution, especially for $Y\leq 1.5$. Note that, in general, the values of $\widetilde C_{n,nb} ''^*$ are much less spread out than those of $\widehat C_{\text{JT}, 53}^*$. Though the simulated distributions of $\widehat C_{\text{M},3}$ and $\widehat C_{\text{M},4}$ are not shown in Figure (ref), we can see that $\widetilde C_{n,nb} ''^*$ performs much better than $\widehat C_{\text{M},3}$ and $\widehat C_{\text{M},4}$ as seen in Tables (ref)-(ref).

{

}

Stochastic volatility

In this section, we apply our two-step debiasing procedure to estimate the daily integrated variance under a stochastic volatility model with a CGMY jump component and compare it with the estimator of Jacod and Todorov JacodTodorov:2014.

Specifically, we consider the following Heston model:

where $\{W_{t}\}_{t\geq 0}$ and $\{B_{t}\}_{t\geq 0}$ are two {correlated} standard Brownian motions {with correlation $\rho$} and $\{J_{t}\}_{t\geq 0}$ is a CGMY L\'{e}vy process independent of $\{W_{t}\}_{t\geq 0}$ and $\{B_{t}\}_{t\geq 0}$. The parameters are set as

The values of $\kappa$, $\xi$, and {$\rho$} above are borrowed from ZhangMyklandAitSahalia:2005. The CGMY parameters are the same as those in the previous section.

We consider $1$-min observations over a one-year ($252$ days) time horizon with $6.5$ trading hours per day. We break each path into $252$ blocks (one for each day) and estimate the integrated volatility $IV = \int_{t}^{t+1/252}V_{s}ds$ for each day ($t=0,1/252,\dots,251/252$). As suggested and used in JacodTodorov:2014, to improve the stability of the estimates, the estimated bias terms in (ref) and (ref) are split into two components each: $\big(\widehat C_n(\zeta_1\varepsilon)-\widehat C_n(\varepsilon)\big)$ and $\big(\widetilde C_{n,pb} '(\zeta_2\varepsilon, \zeta_1, p_1)-\widetilde C_{n,pb} '(\varepsilon, \zeta_1, p_1)\big)$. These are computed using the data in each day, and the factors $\eta_1$, $\eta_2$, which only depend on $Y$, are computed using the data during the whole time horizon. In practice, one would precompute $\eta_1$ and $\eta_2$ using historical data over 1 year and use those values to compute the daily integrated volatility afterward. The precise formulas for our estimators are described below:

For the estimator of {JacodTodorov:2014}, we use equation (5.3) therein with tuning parameter $k_{n}=130$ (number of observation in each block), $\xi=1.5$, and $u_{n}=(-\ln h_{n})^{-1/30}/\sqrt{BV}$ (these values were suggested in JacodTodorov:2014). Here, $BV$ is the bipower variation of the previous day. The resulting estimator is denoted as $\overline C_{\text{JT}, 53}$. To assess the accuracy of the different methods, we compute the Median Absolute Deviation (MAD) around the true value, $IV_t=\int_{t}^{t+1/252}V_{s}ds$, and the MSE, i.e. the sample mean of $(\widehat{IV}_t - IV_t)^2$, for 5 arbitrarily chosen days over $1000$ simulation paths.

The results are shown in Table (ref). The last column of Table (ref) shows the sample means of the MSE and MAD over the 252 days for the different estimators. When $Y=0.8,1.25, 1.35, 1.5$, all MSEs and MADs of $\overline C_{n,nb} ''$ are smaller than those of $\overline C_{\text{JT}, 53}$, with the MSE of $\overline C_{n,nb} ''$ typically half of that of $\overline C_{\text{JT}, 53}$ or smaller. For the case $Y=1.7$ (outside the scope of our theoretical framework), the MSE and MAD of $\overline C_{n,nb} ''$ is slightly larger than those of $\overline C_{\text{JT}, 53}$.

This behavior can also be observed in Figure (ref), which shows the true daily integrated volatility (dashed red line) for one fixed simulated path compared with the estimates corresponding to $\overline C_{n,nb} ''$ (solid black line) and $\overline C_{\text{JT}, 53}$ in JacodTodorov:2014 (dotted blue line). From the figure, we conclude that for this specific stochastic volatility model, our debiasing method achieves significant improvement when $Y\leq{}1.5$. For $Y=1.7$, both estimators $\overline C_{n,nb} ''$ and $\overline C_{\text{JT}, 53}$ are very close and significantly overestimate the true daily integrated volatility for this simulated path. This changes from path to path, though $\overline C_{n,nb} ''$ and $\overline C_{\text{JT}, 53}$ are typically close when $Y=1.7$.