Debiased Kernel Estimation of Spot Volatility in the Presence of Infinite Variation Jumps

Debiased Kernel Estimation of Spot Volatility in the Presence of Infinite Variation Jumps

Introduction

Spot volatility estimation is a central problem in financial econometrics, underpinning numerous critical applications ranging from short-horizon risk management to derivatives pricing and hedging. Formally, the core task is to estimate the coefficient $\sigma_\tau$ of the Brownian component of an It\^o semimartingale $X$ at a fixed point of time $\tau$ given high-frequency observations. For recent developments, we refer to LIU2018,figueroa2020optimal,Figueroa-Lopez_Wu_2024,mancino2022asymptotic, as well as the monographs jacod2012discretization,ait2014high for in-depth treatments.

Widely observed empirically in high-frequency financial data, jumps are a salient feature of price dynamics, separate from fluctuations captured by diffusive movements in $X$, and must be carefully accounted for when estimating volatility and related functionals of $X$. When jump activity is high, however, standard estimation methods face inherent limitations, as estimation difficulty increases with the degree of jump activity of the underlying process. Three qualitatively distinct regimes can be distinguished: jumps of finite activity, infinite activity jumps of bounded variation, and infinite activity jumps of unbounded variation, the latter posing the greatest challenge, and the setting of the present work.

A fundamental approach to handling jumps is through truncation, introduced by Mancini2004. Establishing jump-robustness for truncation-based methods classically amounts to two steps: suppose $\hat\theta_n(\Delta_1^nX,\dots,\Delta^n_n X)$ is an estimator of some functional $\theta$ of the continuous component $X^c$ of $X$, where $\Delta_i^nU:=U_{t_{i}}-U_{t_{i-1}}$ denotes the $i^{th}$ increment of a generic process $U$ sampled at times $0<t_0<\dots<t_n=T$ during a finite time period $[0,T]$. In the first step, one shows that the (infeasible) estimator $\hat\theta^{\text{Cont}}_n:=\hat{\theta}_n(\Delta_1^nX^c,\dots,\Delta_n^n X^c)$ is consistent and satisfies a central limit theorem (CLT) with, say, rate $r_n$:

Second, it is shown that, by choosing an appropriate threshold level $v_n\searrow{}0$, the truncated or thresholded version $\hat\theta^{Trunc}_n:=\hat{\theta}_n(\Delta_1^nX\mathbf{1}_{\{|\Delta_1^n X|\leq v_n\}},\dots,\Delta_n X^n\mathbf{1}_{\{|\Delta_n^n X|\leq v_n\}})$ is close enough to $\hat\theta^{Cont}$ so that

An emblematic example of this approach arises in the estimation of the integrated variance or volatility $IV:=\int_0^T \sigma^2_sds$, where one passes from the quantity

to its truncated version

By (essentially) employing the steps (ref)-(ref), it has been established that in the bounded variation case mancini2009non,cont2011nonparametric, an efficient CLT (see, e.g., jacod:reiss:2014) is possible: \[ \sqrt{n}\left(\widehat{IV}^{\text{Trunc}}-IV\right)\stackrel{\mathcal{D}}{\longrightarrow}\mathcal{N}\left(0,2\int_0^T\sigma_s^4ds\right). \] However, with unbounded-variation jumps, this approach alone falls short. It was proved by cont2011nonparametric,mancini:2011 that $\sqrt{n}\big(\widehat{IV}^{\text{Trunc}}-IV\big)\stackrel{P}{\longrightarrow}\infty$ in the presence of an infinite variation symmetric stable L\'evy process, precisely because the bias of $\widehat{IV}^{\text{Trunc}}$ -- and thus the difference $(\widehat{IV}^{\text{Trunc}} - \widehat{IV}^{\text{Cont}})$ -- vanishes too slowly.

Two methodological advances have since moved the field forward. The first is due to Jacod and Todorov Jacod2014,jacod:todorov:2016, who proposed an approach for integrated variance estimation based on the empirical characteristic function of the rescaled increments $\Delta_i^nX/\sqrt{\Delta_n}$, where $\Delta_n$ is the time span of the increment. After applying a suitable logarithmic transformation and debiasing procedure, this approach admits a CLT with optimal rate and variance, even in the case of unbounded variation jumps. However, with this method, full (rate and variance) efficiency is attainable only under specific conditions that require either the jump activity index $Y$ to satisfy $Y<3/2$, or under additional restrictive jump symmetry assumptions. The second major advance was introduced more recently by BONIECE2024, who showed that a debiasing method similar to Jacod2014 applied to the realized truncated variations (ref) leads to a fully efficient estimator for any $Y\leq 8/5$ without any symmetry conditions.

A natural next question is how these insights extend to the problem of spot volatility estimation. A standard and widely used approach to constructing spot volatility estimators is to apply kernel smoothing to an estimator of integrated variance, \[ \hat\sigma^2_\tau=\int K_b(t-\tau)d \widehat{IV}_t, \] where $\widehat{IV}_t$ denotes an estimator of $IV_t=\int_0^t \sigma_s^2ds$ and $K_b(u):=K(u/b)/b$ for a chosen kernel $K$ and a bandwidth $b>0$. For instance, using the realized quadratic variation $\widehat{IV}_t=\sum_{i=1}^{[nt]}(\Delta_i ^nX)^2$, this reduces to

as proposed in kristensen2010nonparametric,fan2008spot.

Spot volatility estimators differ markedly from their integrated volatility counterparts, with their local nature altering the efficiency picture in subtle but important ways: on the one hand, optimal convergence rates are of order $n^{-1/4}$ (compared with $n^{-1/2}$ in the integrated volatility case); on the other hand, kernel-based estimators enjoy a degree of built-in robustness against jumps, since “large” jumps are relatively rare and are unlikely to fall in a narrow neighborhood of $\tau$ where $K_b$ places most of its mass. For instance, jacod2012discretization shows that in the case of uniform kernels (i.e., $K(u)={\bf 1}_{[0,1)}(u)$ or $K(u)={\bf 1}_{(-1,0]}(u)$) the estimator (ref) admits a rate-optimal CLT even in the case of unbounded variation jumps, provided $Y<4/3$\footnote{The case of jumps of bounded variation corresponds to $Y<1$, while the jumps are of unbounded variation when $Y>1$.}. However, high levels of jump activity can still degrade performance, and the rate becomes suboptimal for $Y\geq 4/3$: in general, for any $a\in(0,1/2]$ such that $Y<2/(1+a)$, one can find a bandwidth $b_n$ such that $n^{a/2}(\hat\sigma_\tau^2-\sigma_\tau^2)\stackrel{\mathcal{D}}{\longrightarrow}\mathcal{N}(0,V_\sigma)$, which yields a rate that is slower than $n^{-\frac{2-Y}{2Y}}$. Thus, if, for instance, $Y\leq 3/2$, the rate is slower than $n^{-1/6}$. Surprisingly, following the two-step approach (ref)-(ref), even when thresholding is applied to (ref), i.e.,

there is no apparent asymptotic benefit: the optimal rate of $n^{-1/4}$ remains attainable only if $Y<4/3$, and the best possible rate when $Y\geq 4/3$ remains bounded by $n^{-\frac{2-Y}{2Y}}$, which is strictly slower.

Given these limitations, one may wonder if estimation performance can be improved by modifying the kernel $K$. This question was addressed in Figueroa-Lopez_Wu_2024, which extended the approach in jacod2012discretization to two-sided kernels of unbounded support. Although the convergence rate remains unchanged, they established that the asymptotic variance can be substantially reduced. For example, using the kernel $K(u)=.5e^{-|u|}$ leads to an asymptotic variance that is one-quarter of that obtained by a uniform kernel in the suboptimal rate regime $(Y\geq 4/3)$. Moreover, in the optimal rate regime $(Y< 4/3)$, this kernel was shown to be variance-optimal, highlighting the utility of unbounded kernels in general. However, we note that the results in both jacod2012discretization and Figueroa-Lopez_Wu_2024 are based on the two-step approach (ref)-(ref), which we depart from in the present work.

These findings motivate a fundamental question: can one develop methods that attain better rates -- possibly optimal -- beyond the boundary $Y< 4/3$? A recent step in this direction was taken by LIU2018, who developed a localized variant of the characteristic function approach of Jacod2014. Assuming the infinite-variation jump component is of the form $\int_0^t\chi_{s}dL_s$, where $L$ is a strictly $Y$-stable symmetric L\'evy process, and while restricting to the class of compactly-supported kernels that are continuously differentiable, they establish that their estimator satisfies

for any $Y\leq{}3/2$, but with a suboptimal convergence rate (i.e., $\sqrt{\frac{\Delta_n}{b_n}}\gg n^{-1/4}$). By applying an additional debiasing step similar to that in Jacod2014, they further showed that associated tuning parameters can be chosen so that (ref) holds for any $Y<2$, however, still with a suboptimal rate and under the jump symmetry condition described above. Consequently, the boundary $Y<4/3$ has long marked the limit of efficiency in spot volatility estimation, and the regime $Y\geq 4/3$ has remained largely underexplored without symmetry assumptions.

In this work, we push this efficiency boundary substantially forward by introducing appropriately debiased versions of the estimator (ref). First, by exploiting high-order expansions of the truncated moments of $\Delta_i X$, rather than relying on the coarser two-step framework (ref)-(ref), which does not explicitly account for higher-order bias, we show the estimator (ref) actually achieves rate-optimality for any $Y\leq{}3/2$ (namely, $n^{1/4}(\hat\sigma_\tau^2-\sigma_\tau^2)\stackrel{\mathcal{D}}{\longrightarrow}\mathcal{N}(0,V_\sigma)$) under standard assumptions and without additional adjustment. Second, by introducing debiasing steps similar to those in Jacod2014,BONIECE2024, we construct fully efficient estimators that cover the range $0<Y<20/11$. This extension is practically relevant, as empirical studies indicate that values of $Y$ are typically between $1.5$ and $1.8$ for liquid stocks (see BONIECE2024 for a brief supporting empirical analysis and further references). Our method stands in sharp contrast to the results in LIU2018, offering rate-optimal estimators over a wider range of $Y$, broader applicability through more flexible model assumptions, reduced asymptotic variances enabled by the use of unbounded kernels, and simpler estimators that are easier to implement. To validate our results, we perform a comprehensive simulation study and find that our methods significantly outperform those of LIU2018 in terms of both bias and variance over finite samples. Our findings further demonstrate that the advantages of kernels with unbounded support extend to the debiasing framework developed here, providing substantial improvements over widely-used uniform kernels, which are often the default choice in practice, especially in inference problems that require estimation of spot volatility as a preliminary step.

The paper is organized as follows. Section (ref) introduces the setting, assumptions, and preliminary results. Section (ref) presents the main results and their proofs. Section (ref) contains simulations assessing the performance of our estimators. Some additional technical proofs are collected in Appendix (ref).

Setting and background

Consider a 1-dimensional It\^o semimartingale $X = \big(X_t\big)_{t\in \mathbb{R}_{+}}$ defined on a complete filtered probability space $\big(\Omega,\mathscr{F},(\mathscr{F})_{t\in \mathbb{R}_{+}},\mathbb P\big)$ that can be decomposed as

where $W:=(W_t)_{t \in \mathbb{R}_{+}}$ is a Wiener process, $\sigma$ is an adapted càdlàg process, $J:=(J_t)_{t \in \mathbb{R}_{+}}$ is an independent pure-jump Lévy process with Lévy triplet $(b,0,v)$, and $\mathfrak{p}$ is a Poisson random measure on $\mathbb{R}_+\times\mathbb{R}$ with an intensity $\mathfrak{q}(ds,dz) = ds \otimes \lambda(dz)$, such that the $\lambda$ is a $\sigma$-finite measure on $\mathbb{R}$, possibly dependent with $J$. The Lévy measure $v$ is assumed to admit a density $s:{\mathbb{R}\backslash\{0\}}\rightarrow \mathbb{R}_+$ of the form

{We have the following standing assumptions:}

We assume we have at our disposal $n$ evenly-spaced discrete observations $X_{t_i}$ during a fixed time interval $[0,T]$. For simplicity, we further assume $T=1$ and, thus, $t_i=i/n$. Denote $\Delta_n = 1/n$, $\mathcal{F}_{i}^n:=\mathcal{F}_{t_i}$, and $\Delta^n_iA = A_{i\Delta_n} - A_{(i-1)\Delta_n}$ for any process $A$, where we often omit the superscript $n$. Our estimation target is {$c_\tau=\sigma_\tau^2$} for a fixed time $\tau\in(0,T)$. We consider the spot volatility estimator

where $K$ is a kernel function, and $m_n$ controls the bandwidth {$b_n:=m_n\Delta_n$ such that} $m_n \rightarrow \infty$ and $m_n\Delta_n\rightarrow 0$, and $K_b(x):=K(x/b)/b$. We assume the following conditions on $K$, which are standard:

{The estimator (ref) without the denominator factor, truncation, or any debiasing was studied in fan2008spot and kristensen2010nonparametric (see also additional references in the introduction). The denominator, motivated by the standard Nadarayan-Watson nonparametric regression estimator, and noted in passing in kristensen2010nonparametric, serves a dual role: it provides edge correction in finite samples and simplifies several proofs, notably leading to more relaxed technical assumptions in the optimal bandwidth regime (see Remark (ref) below). In contrast, when the denominator is omitted, our results remain valid, but stricter conditions on the bandwidth $b_n = m_n \Delta_n$ and its relationship with $Y$ are required; however, in the suboptimal rate regime (see Remark (ref), case (ii), below), the denominator can be dispensed of entirely without any loss of generality. The truncated version in (ref) was studied in figueroa2020optimal and Figueroa-Lopez_Wu_2024, also without debiasing. Finally, Assumption (ref) permits kernels of unbounded support, which, as discussed in the introduction, enables estimators that are more variance efficient in both the optimal and suboptimal convergence regimes.}

Notation: Throughout, {$A_n\lesssim B_n$ and $a_n\ll b_n$ mean $A_n = O_P(B_n)$ and $a_n=o(b_n)$, respectively.} For any process $V$, when unambiguous we write $\Delta_iV$ in place of $\Delta_i^nV$.

Main results

The following moment expansions play key roles in our analysis. The proofs are based on arguments in analogous results in BONIECE2024 (see {Lemmas} 2, 3, 4 and 5 therein) and are given in Appendix (ref) for completeness.

To analyze the asymptotic {behavior} of the estimator $\hat c_n(m_n, v_n)$ {defined in Eq. (ref)}, we consider the following decomposition:

Our next result is central to our analysis and underpins our main debiasing procedure. It establishes the joint asymptotic behavior of the terms in (ref), accounting for appropriate bias terms, and provides the framework for separating qualitatively distinct asymptotic regimes for $m_n$. In addition, on its own, it yields feasible CLTs for $\hat c_n(m_n,v_n)$ in certain settings that sharpen existing results in the literature (see Remark (ref)).

The next result is the second key ingredient needed for our debiasing method. For an arbitrary fixed $\zeta>1$, it establishes the convergence rate of the difference $\tilde Z_n(m_n, \zeta v_n)-\tilde Z_n(m_n, v_n)$, i.e. of $\hat c_{n,1}(m_n,\zeta v_n)-\hat c_{n,1}(m_n, v_n)$ subject to a bias correction, which we exploit in our debiasing procedure.

Finally we are ready to formally construct our debiasing estimators. Our debiasing steps follow the same idea as in Jacod2014. Write $\vec \zeta_k=(\zeta_1,\dots,\zeta_k)$, where $\zeta_1,\dots,\zeta_k>1$ are fixed constants. Let

for each $k\geq 1$, with $\widetilde c^{(0)}_n(m_n, v_n) = \hat c_n(m_n, v_n) $. Our next theorem is the second main result of this paper; it establishes a CLT for the feasible estimator $\widetilde c^{(2)}_n(m_n,v_n,\vec \zeta_2)$.

Simulation

In the numerical experiments below, we evaluate the performance of our debiased estimators under a Heston-type volatility model in two different settings. First, for comparison, we replicate the parameter setting and sampling scheme used in [15], which incorporates a symmetric stable jump component, although this configuration is arguably unrealistic for most financial applications (e.g., the overall annualized volatility is set to 1). Second, we consider a parameter setting that better reflects the empirical features of financial data. In this case, we also truncate the increments of the stable L\'evy process, as is commonly done in applications, and investigate the impact of different choices of the threshold parameter $v_n$.

Simulation with Stable Jump Component

Following LIU2018, we consider the following model:

where $B$ and $B'$ are correlated standard Brownian motions with correlation $\rho$, $J$ is a strictly symmetric stable L\'evy process \footnote{The rstable package in R was used to simulate the $Y$-stable portion of the jump component in all cases.} with Blumenthal-Getoor index $Y$ and scale parameter $1$ (note LIU2018 includes an additional finite-activity component which we omit for simplicity). In this section, we take the same parameter values as in LIU2018: \[ X_0=1,\quad V_0=0,\quad \rho=0,\quad b_t\equiv 1, \quad\kappa=0.03,\quad\theta=1,\quad \xi=1.5. \]

In this simulation, we compare the performance of our estimator $\hat c(m_n,v_n)$ defined in (ref), $\tilde c^{(1)}(m_n,v_n,\vec\zeta_1)$ and $\tilde c^{(2)}_{n,\tau}(m_n,v_n,\vec\zeta_2)$ defined in (ref) to the estimators $\hat \sigma^2_{\tau,n}(u_n,h)$ and $\tilde \sigma^2_{\tau,n}(\lambda,u_n,h)$ introduced in LIU2018, which are based on the empirical characteristic approach of Jacod2014. Specifically, $\hat \sigma^2_{\tau,n}(u_n,h)$\footnote{LIU2018 defines $\hat \sigma^2_{\tau,n}(u_n,h) = \bar \sigma^2_{\tau,n}(u_n,h)- \frac{2h}{u_n^2\Delta_n}(\sinh(\bar \sigma^2_{\tau,n}(u_n,h)))^2$ for simulation. The difference lies in the reciprocal placement of $h$ and $\Delta_n$ in the bias correction term. We interpret the version in LIU2018 as a likely typo and instead follow the formulation consistent with Jacod2014.} and $\tilde \sigma^2_{\tau,n}(\lambda,u_n,h)$ are defined as

where

and $\tilde B_{\tau,n}(\lambda,u_n,h)$ is a bias-correction term\footnote{The formula {of $\tilde B_{\tau,n}(\lambda,u_n,h)$} in LIU2018 forces each summand in the denominator to be nonpositive. However, in our experiments, we found that omitting control of the sign of these terms significantly improved the performance of the estimators. }

which aggregates spot estimators across $t$ for improved finite-sample performance. This approach was also used in Jacod2014.

For comparison, for our one- and two-step debiased estimators $\tilde c^{(1)}_{m,\tau}(m_n,v_n,\vec\zeta_1)$ and $\tilde c^{(2)}_{n,\tau}(m_n,v_n,\vec\zeta_2)$, we also adopt the debiasing adjustments of LIU2018 and Jacod2014 described above. Specifically, denoting $\tilde{c}^{(k)}_{n,\tau}(m_n,v_n,\vec \zeta_k)$ the estimator $\tilde{c}^{(k)}_{n}(m_n,v_n,\vec \zeta_k)$ at time $\tau$, we consider

where $A^{(k)}_{{n}}:=A^{(k)}_{n}(m_n,v_n,\vec \zeta_k)$ is given by $$A_n^{(k)} =\pm\frac{\sum_{i=1}^{\lfloor\frac{1}{m_n\Delta_n}\rfloor}\big(\widetilde c^{(k-1)}_{n,im_n\Delta_n}(m_n,\zeta_kp_k v_n,\vec \zeta_{k-1})-\widetilde c^{(k-1)}_{n,im_n\Delta_n}(m_n, p_kv_n,\vec \zeta_{k-1})\big)}{\sum_{i=1}^{\lfloor\frac{1}{m_n\Delta_n}\rfloor}\big(\widetilde c^{(k-1)}_{n,im_n\Delta_n}(m_n,\zeta_k^2 p_kv_n,\vec \zeta_{k-1})-2\widetilde c^{(k-1)}_{n,im_n\Delta_n}(m_n,\zeta_k p_kv_n,\vec \zeta_{k-1})+\widetilde c^{(k-1)}_{n,im_n\Delta_n}(m_n,p_k v_n,\vec \zeta_{k-1})\big)},$$ where the sign of the bias terms are chosen as in BONIECE2024 and $c^{(0)}_{n,\tau}(m_n,v_n,\vec \zeta_0) = \hat c(m_n,v_n)$ at $\tau$, i.e., $A_n^{(1)}$ is positive and $A_n^{(2)}$ is negative, so as to ensure compatibility with the theoretical sign of the terms in $A(v_n,m_n)$.

To evaluate estimation performance, we simulate $M$ independent paths and aggregate various pathwise error measures computed along each simulated path for each of the estimators considered above. Denoting the true value of the spot variance at $t_i$ in {the} $j$-th path as $\sigma^2_{t_i,j}:=V_{t_i,j}$ and a given estimator as $\hat \sigma^2_{t_i,j}$, we consider an estimate of the pathwise root mean squared error (RMSE) on the time grid $\{l_i = i\floor {n/100}\}_{i=10,\dots,90}$ as follows: $$\widehat{RMSE} := \sqrt{\frac{1}{M}\sum_{{j=1}}^M MSE_j}\,,$$ where $MSE_j := \frac{1}{81} \sum_{i=10}^{90}(\hat \sigma^2_{l_i,j}-\sigma^2_{l_i,j})^2$. Similarly, we also consider the mean absolute relative error $\widehat{ARE}$ and the mean relative error $\widehat{RE}$: $$\widehat{ARE} := \frac{1}{M}\sum_{j=1}^M AE_j \;\text{ and }\;\widehat{RE} := \frac{1}{M}\sum_{j=1}^M E_j,$$ where $AE_j := \frac{1}{81} \sum_{i=10}^{90}|\hat \sigma^2_{l_i,j}-\sigma^2_{l_i,j}|/\sigma^2_{l_i,j}$ and $E_j := \frac{1}{81} \sum_{i=10}^{90}(\hat\sigma^2_{l_i,j}-\sigma^2_{l_i,j})/\sigma^2_{l_i,j}$.

For our estimators $\hat c_n(m_n,\Delta_n)$, $\tilde c^{(1)}_{m,\tau}(m_n,v_n,\vec\zeta_1)$ and $\tilde c^{(2)}_{n,\tau}(m_n,v_n,\vec\zeta_2)$, we use the exponential kernel $K(x) = \exp({-|x|})/2$ as well as the two-sided uniform kernel $G(x) = \mathbf{1}_{-1\leq |x|\leq 1}/2$, and set $m_n = \Delta_n^{-1/2}$, $v_n = \sqrt{BV} \Delta_n^{5/12}$, where $BV = \frac{\pi}{2}\sum_{i=2}^n|\Delta_iX||\Delta_{i-1}X|/T$ is the bipower estimator for the integrated volatility. For the estimators $\hat \sigma^2_{\tau,n}(u_n,h)$ and $\tilde \sigma^2_{\tau,n}(\lambda,u_n,h)$ of LIU2018, we take the kernel $K_3(x) = \frac{15}{16}(1-x^2)^2\mathbf{1}_{|x|\leq 1}$ as specified on LIU2018\footnote{$K_3$ is the best-performing kernel among those considered in LIU2018 that satisfies the assumptions of their results (in particular, that is continuously differentiable as assumed by their main theorem).} We also use the best-performing bandwidth as reported in their simulations, $h=\Delta_n^{0.51}$ and $u_n = \frac{\Delta_n^{0.0025}}{\sqrt {BV}}$\footnote{Instead of using bipower variation for \( u_n \), the authors of LIU2018 employed a scaled local bipower variation defined as $BV_\tau = \frac{\pi}{2(i_2 - i_1)\Delta_n} \sum_{i = i_1}^{i_2} |\Delta_i X||\Delta_{i+1} X|,$ where $ i_1 = \left\lfloor\frac{\tau - h}{\Delta_n} + 1\right\rfloor \vee 1 $ and $ i_2 = \left\lfloor\frac{\tau + h}{\Delta_n} + 1\right\rfloor \wedge n $. However, in our experiments, this choice lead to large RMSE when aggregating errors across various \( \tau \), particularly for \( Y = 1.2 \). In contrast, the estimators demonstrated greater stability with our proposed choice of \( u_n \).}. The remaining tuning parameters $\vec\zeta_1$, $\vec\zeta_2$, $\lambda$, $p$, $p_1$, $(p_1,p_2)$ are selected via a grid search that minimizes $\widehat {RMSE}$\footnote{Specifically, to each point in the grid, we simulated $M=100$ independent paths to estimate the $\widehat{RMSE}$ for the associated parameter settings.}. The grid for $\zeta$ and $\lambda$ ranged from 1.1 to 1.9 with increments of 0.05, while the grid for $p$ spanned from 0.1 to 0.9 with increments of 0.05. For reproducibility, where applicable, we also report the auxiliary tuning parameter values found in our grid search for each considered estimator.

As in LIU2018, we take $T=1$ month and $n = 8580$, roughly corresponding to a 5-minute sampling frequency (assuming 24 hours of trading). For additional comparison, we also include the results for $n = 42900$, corresponding to roughly 1-minute frequency data in this setup. Using $M = 1000$ iterations, we report the results in Tables (ref) and (ref). We consider the values $Y\in\{0.8,1.2,1.6,1.75\}$, which correspond to the cases where, in principle, no debiasing is needed ($Y\in\{0.8,1.2\}$), one step of debiasing is needed ($Y=1.6$), and two steps are needed $(Y=1.75)$ to attain asymptotic efficiency {according to our theory, though we may observe very different finite sample behavior}.

As shown in Tables (ref) and (ref), the two-step debiasing estimator $\tilde{c}_n^{(2)}$ consistently delivers the best performance, with the lowest RMSE and ARE, and generally well-controlled error. The advantage of two-step debiasing becomes especially pronounced as $Y$ increases: While the performance of $\hat \sigma^2_n$ and $\widetilde \sigma^2_n$ deteriorates sharply, the one- and two-step debiased estimators exhibit markedly greater robustness against increased jump activity. Notably, two-step debiasing yields improvement even where it is not strictly needed for asymptotically efficient estimation ($Y=0.8,1.2$). The pattern is similar at both 5-minute and 1-minute frequencies, where the two-step debiasing method is the only one that remains stable across all considered settings, offering reductions in RMSE often by a factor of 2 to 5 relative to the considered benchmarks.

Across kernel choices for $\tilde c_n^{(i)}$, the exponential kernel systematically outperforms the uniform kernel across all values of $Y$ and at both 1- and 5-minute sampling frequencies. This finding is consistent with the established theoretical optimality of exponential kernels in related settings where debiasing is not required Figueroa-Lopez_Wu_2024, and more broadly underscores the advantages of using unbounded kernels in spot volatility estimation. Overall, the two-step estimator with the exponential kernel achieves the strongest and most reliable performance among all configurations considered.

Simulation with truncated stable jump component

In this section, we consider a parameter setting that more closely reflects the empirical features of financial data. We also take a truncated stable L\'evy process as the jump component, which is common in applications. We also study the effect of the truncation level $v_n$.

For this experiment, we fix a time unit of 1 year (i.e., $t$ is measured in years). The increments of the observed process (corresponding to an asset's log return over $(t_{i-1},t_i]$) are given by

where $J$ is a symmetric $Y$-stable L\'evy process with scale parameter $0.5$ and $V$ follows the same Heston model (ref) with parameter values as in BONIECE2024: $$\rho = -0.5,\quad \kappa = 5,\quad\xi=0.5,\quad\theta=0.16.$$ In particular, we note the annualized average volatility is $\sqrt{.16}=.4$, which is more realistic for financial data. Assuming 252 trading days per year, and a 6.5 hour trading day, we set $\Delta_n = (252*6.5*12)^{-1}$, corresponding to a sampling frequency of 5 minutes. We consider a time horizon of 3 months ($T = \frac{1}{4}$), yielding a sample size of $ n = T/\Delta_n=4914$. In our experiments, we examine performance in the cases $Y=1.6$ and $Y=1.75$, corresponding to 1 and 2 steps of debiasing required for efficiency, and use an exponential kernel for our estimators $\hat c_n(m_n,\Delta_n)$, $\tilde c^{(1)}_{m,\tau}(m_n,v_n,\vec\zeta_1)$ and $\tilde c^{(2)}_{n,\tau}(m_n,v_n,\vec\zeta_2)$. The tuning parameters are similar to the previous section, with $\vec\zeta_1$, $\vec\zeta_2$, $\lambda$, $p$, $p_1$, and $(p_1,p_2)$ being selected via a grid search that minimizes $\widehat{RMSE}$ over 100 iterations (using the same grid as described in previous section), but here we take $v_n = \sqrt{BV}v_0$, for different $v_0$ to assess the sensitivity of our methods to the threshold choice. We report the performance for our estimators across various threshold choices $v_0$ in Table (ref) $(Y=1.6)$ and in Table (ref) $(Y=1.75)$ as measured by $\widehat{RMSE}$, $\widehat{ARE}$ and $\widehat{RE}$ using $M=1000$ paths. In each table, we also include the performance of the estimators of LIU2018 for comparison. We also report the auxiliary tuning parameter values found in our grid search.

As shown in Tables (ref) and (ref), for both $Y=1.6$ and $Y=1.75$ the two-step estimator $\tilde c_n^{(2)}$ again provides the most reliable performance, delivering the lowest or near-lowest RMSE and ARE across all $v_0$ considered. The advantage of debiasing is most pronounced at the moderate threshold choices $v_0=\Delta_n^{20/48}=\Delta_n^{5/12}$ and $v_0=\Delta_n^{21/48}$, where the two-step estimators offer reductions in RMSE over $\hat \sigma^2_n$ and $\tilde \sigma^2_n$ by roughly a factor of 2, and these improvements are observed consistently for both $Y=1.6$ and $Y=1.75$. Remarkably, the choice $v_0=\Delta_n^{5/12}$ is the same as suggested by the theory for two-step debiasing in the integrated volatility case (see BONIECE2024). Broadly, these findings highlight the benefit of two-step debiasing beyond specific threshold choices, offering reliable nontrivial reductions in MSE across a variety of jump activity levels.