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Efficient Sampling for Realized Variance Estimation in Time-Changed Diffusion Models
\footnotetext[1]{Corresponding author. Faculty of Economics and Business, Goethe University Frankfurt, 60629 Frankfurt am Main, Germany, [email removed].} \footnotetext[2]{Institute of Economics, University of Freiburg, Germany; email: [email removed]} \footnotetext[3]{University of St. Gallen, Switzerland} \footnotetext[4]{Institute of Economics, University of Freiburg, Germany; email: [email removed]} \footnotetext[5]{KOF Swiss Economic Institute, ETH Zürich, Switzerland; email: [email removed]} \footnotetext[6]{Department of Computer and Information Science, University of Konstanz, Germany; email: [email removed]}
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The estimation and forecasting of the variance of daily stock returns plays a major role in risk management, portfolio optimization and asset pricing. Accurate estimates of the daily variation of asset prices are commonly obtained by using intraday information as in the realized variance (RV) estimator introduced by \citet*{andersen1998a} and \citet*{andersen2001a, andersen2001b}. Together with \citet*{barndorff2002a} and \citet*{meddahi2002}, they show that under the assumption that the logarithmic price process follows a standard continuous-time diffusion model, \textcolor{black}{RV is an unbiased and consistent estimator of the quadratic variation, which coincides with the integrated variance (IV) in the absence of jumps \citep*{barndorff2008, Barndorff2011SubsamplingRK, Andersen2012}.}
Despite the theoretically appealing approaches of subsampling zhang2005, realised kernels barndorff2008 and pre-averaging PodolskijVetter2009 for robustifying the RV estimator to market mircrostructure noise (MMN), the standard RV estimator at low frequencies such as sampling every five minutes is still regularly employed in empirical work, see e.g.\ LiuPattonSheppard2015, bollerslev2018risk, bollerslev2020good, bollerslev2022zero, bates2019crashes, bucci2020realized, reisenhofer2022harnet, alfelt2023singular, Patton2023Bespoke among many others.\footnote{More fundamentally, the bibliographic review of Hussain2023 analyses 2920 papers and summarizes that “5-minute interval data appear to be the most favored choices in terms of high-frequency data usage.”} Reasons for the standard RV's ongoing popularity include its simple and intuitive implementation, the fact that low(er) frequencies can be used at which MMN is not a major concern, that its convergence rate is substantially faster compared to the previously mentioned approaches, and that it still performs comparably well in empirical studies LiuPattonSheppard2015.
While most of the literature focuses on sampling returns equidistantly in calendar time such as every five minutes, financial markets do not tick in calendar time. Instead, their intraday trading activity and tick variance (price variance of adjacent transactions or quotes) is time-varying, which might provide important information about the market's pulse and especially its riskiness. In this paper, we study the theoretical and empirical benefits of using intraday returns sampled in intrinsic time scales to efficiently estimate the daily IV through the RV estimator. These time scales accelerate the clock time when the trading or price variations are intense, and they slow the time down when the markets are calm. In particular, we differentiate between the time scale driven by the trading activity (Transaction Time Sampling - TTS), the intraday price volatility (Business Time Sampling - BTS), and observed absolute price changes (Hitting Time Sampling - HTS). For TTS and BTS, we distinguish their implementation into intensity and realized/jump-based versions, where the latter use the observed amount of trades on a given day whereas the former rely on estimated intensities. In contrast to e.g., bandi2008 who derive an optimal sampling frequency given equidistant sampling points, we focus on the “inverse” question of how to optimally allocate the sampling points under a given frequency. \textcolor{black}{By optimal or efficient, we mean a sampling scheme that, among a class of unbiased schemes, attains the smallest mean squared error (MSE), which in this case equals its estimation variance.}
Summarizing our main contributions, we show that using HTS, which samples such that the absolute returns are (approximately) equal, provides a theoretical lower bound for the efficiency of the RV estimator in finite samples in terms of its \textcolor{black}{MSE}. Furthermore, the newly introduced realized BTS (rBTS) scheme, which samples according to a combination of the observed ticks and the (estimated) variance at these ticks, arises as most efficient when restricting attention to sampling schemes that do not use the observed high-frequency prices for the construction of the sampling points. This restriction is motivated by the empirical presence of MMN, which has a particularly severe impact on HTS as its sampling times are obtained directly from the noise-contaminated price observations. In our simulations and the empirical application, both HTS and rBTS exhibit an excellent and overall comparable performance, and clearly dominate the classically used sampling in calendar or tick time. While HTS dominates for very low frequencies where MMN is (almost) absent, rBTS arises as most efficient when the sampling frequency exceeds the 5 minute level.
Our theory builds on the assumption of a price process that follows a stochastic diffusion that is time-changed with a jump (e.g., doubly stochastic Poisson or Hawkes) process. We call this the tick-time stochastic volatility (TTSV) model. It is a joint stochastic model for the asset prices together with their transaction (or quote) arrival times. The prices in this model follow a pure jump process that accommodates the time-varying trading intensity and tick variance processes within its diffusive component. The spot variance becomes the product of these two time-varying components that behave empirically different for stock markets as portrayed in Figures (ref) and (ref) below.
The TTSV model is a simple and transparent framework to study statistical (finite sample) properties of the RV estimator with respect to various choices of sampling schemes. This is achieved by having the trading intensity and tick variance as two separately evolving processes that jointly govern the price variability. The separate modeling of trading intensity and tick variance particularly allows for a comparative theoretical analysis of sampling according to calendar time, tick time in the sense of observed ticks or trading intensity, business time as measured by (realized) intraday volatility and hitting time by homogenizing absolute price changes.
A theoretical alternative is to work under discretized diffusion models as e.g.\ employed in Jacod2017, Jacod2019, Jacod2018, DaXiu2021, Li2022remedi, where a continuous diffusion process is augmented with a process separately modeling the arrivals of the transactions. Similar to the TTSV model, the resulting price process is a pure jump process with price changes at the explicitly modeled arrivals of the transactions. We illustrate in Appendix (ref) that these discretized diffusions are closely related to the TTSV model. While similar (finite sample or asymptotic) efficiency results might be derived by relying on discretized diffusions, the TTSV model is attractive due to its simplicity and transparency in distinguishing between trading intensity and tick variance. Some of our results (in particular, Theorem (ref) (b) and (c)), however, require strong independence conditions on the underlying TTSV processes, which could possibly be weakened when working with discretized diffusions. The TTSV model, however, also allows for the analysis of the price-dependent HTS scheme (opposed to e.g., Li2022remedi, Jacod2017 and ait2014high) and it yields the novel realized BTS scheme due to the explicit modeling of the trading times, hence refining the (asymptotic) efficiency results of Barndorff2011SubsamplingRK.
Although the idea of intrinsic time sampling is not new to the literature, especially with regard to its empirical benefits \citep*{clark1973,oomen2005,oomen2006,hansen2006,Andersenal2007,Andersenal2010,aitsahalia2011}, its theoretical advantages over the classical calendar time sampling (CTS) scheme are still largely unexplored, especially in finite samples. Exceptions are \citet*{oomen2005,oomen2006}, who study the statistical properties of RV under intrinsic time sampling schemes, however, based on a compound Poisson price assumption \citep*{press1967}, whose volatility pattern is solely driven by the trading intensity (see also \citet*{griffin2008}). Hence, this model misses a substantial source of daily return variation, i.e., the one due to the tick variance, as illustrated in Figures (ref) and (ref) below. Furthermore, Barndorff2011SubsamplingRK show that (intensity) BTS arises as an asymptotically efficient deterministic sampling scheme for (subsampled) realized kernel estimators. Our results however also apply to finite sampling frequencies and allow for sampling based on observed ticks and prices (instead of being deterministic) and can hence accommodate the HTS and realized BTS schemes. Fukasawa2010RV analyses the asymptotic MSE of the RV estimator under endogenous sampling schemes, assuming a continuous semi-martingale for the price process that is observed whenever the price changes by a fixed quantity. Fukasawa2010RV shows that, asymptotically, HTS is most efficient. In this light, Theorem (ref) (a) can be interpreted as a finite-sample analogue of his result, albeit established in a different setting. FukasawaRosenbaum2012, robert2012volatility and Vetter2017 provide further asymptotic results under endogenous sampling times.
Pure jump processes, as the TTSV model, have already proven to be valuable alternatives to continuous diffusion models to describe financial prices, as they not only capture empirically observed random trading times and price discontinuities, but also offer a flexible framework to address MMN contamination or to price derivatives; see e.g., \citet*{press1967}, \citet*{carr2004}, \citet*{engle2005}, \citet*{oomen2005,oomen2006}, \citet*{Liesenfeld2006} and \citet*{ShephardYang2017}. These processes can be further framed and generalised within stochastic time-changed structures, which are mathematically and empirically very effective, but have received so far only moderate attention in the financial econometrics literature \citep*{clark1973,carr2004}.
The decomposition of spot variance in trading intensity and tick variance has already been addressed by \citet*{jones1994}, \citet*{ane2000}, \citet*{plerou2001}, \citet*{gabaix2003}, \citet*{Dahlhaus2014}, \citet*{dahlhaus2016}, among others, when studying the intraday trading behaviour in relation to the intraday clock volatility pattern in order to measure spot variance or to test for normality of intraday returns sampled in transaction time scales. They find that, while the intraday trading is highly correlated with the intraday spot variance, the tick variance affects the spot variance as well, although it has a flatter intraday shape. Our empirical observation on stock markets complements these findings and reveals that the intraday tick variance and the trading intensity follow mirrored “J” patterns (also see \citet*{Admati1988}, \citet*{oomen2006} or \citet*{dong2014}), which jointly result in the well known “U” shape of the intraday spot variance, as documented by Harris1986, Wood1985, andersen1997 and Bauwens2001.
We validate our theoretical results in extensive simulations, where we also examine the impact of a leverage effect through an asymmetric Hawkes-type process and different specifications of MMN on the bias and the MSE of the RV estimator. Our empirical results show that, as predicted by our theory, the HTS scheme provides the most efficient RV estimates in the absence of noise. However, the HTS scheme is most sensitive to noise as its sampling times directly rely on absolute changes of the noisy price process. In contrast, the rBTS scheme is more robust to noise and is superior for the typical sampling frequencies between 1 and 5 minutes under noisy price processes. The rBTS scheme also clearly dominates a classical implementation of (intensity) BTS, different implementations of TTS and the baseline case of CTS.
The empirical application considers 27 liquid stocks traded at the New York Stock Exchange (NYSE). It provides clear empirical evidence for the benefits of using HTS and realized BTS for increasing the statistical quality of the RV estimator in terms of MSE and QLIKE loss in both an in-sample estimation and out-of-sample forecast environment based on the Heterogeneous AutoRegressive (HAR) model of Corsi2009. For the in-sample evaluation, we follow the method of \citet*{Patton2011RV} that facilitates the empirical comparison of competing RV estimators, in our case computed from the different sampling schemes. The empirical results particularly stress the practical relevance of the HTS and the realized BTS scheme by showing their superiority in a model-free environment.
The remainder of the paper is structured as follows. In Section (ref), we introduce the TTSV model and derive theoretical efficiency results for finite sampling frequencies for the RV estimator. Section (ref) presents a comprehensive simulation study that analyses the performance of RV under different sampling schemes and Section (ref) provides an empirical application to real data. We conclude in Section (ref). \textcolor{black}{Appendix (ref) provides proofs for our main results.}
The supplemental material contains a comparison to discretized diffusions in Appendix (ref), additional finite sample theory in a setting where sampling can use information from the end of the trading day in Appendix (ref), and a specific comparison to the results of oomen2006 in Appendix (ref). \textcolor{black}{All proofs---other than those in Appendix (ref)---are collected in Appendix (ref).} Appendix (ref) discusses generalizations of some theoretical results to mildly dependent processes and Appendix (ref) contains additional empirical results.
This section introduces some preliminaries in Section (ref) and presents the TTSV model in Section (ref). Sections (ref) and (ref) establish finite sample efficiency results for the RV estimator, \textcolor{black}{which is complemented by additional theory in Appendix (ref) that allows for employing information from the entire trading day.}
Throughout the paper, all random objects are defined on a filtered probability space $\left(\Omega,\mathcal{F}, \mathbb{F},\mathbb{P}\right)$ with filtration $\mathbb{F} = \{\mathcal{F}_t\}_{t\geq0}$ that we specify in Assumption (ref) below. If not stated otherwise, all (in)equalities of random variables are understood to hold almost surely. Let $\{P(t)\}_{t\geq0}$ denote the stochastic process representing the logarithmic price process of an asset, which we assume to be a continuous-time stochastic process that is right-continuous with left limits. We sometimes abuse notation and simply write $P(t)$, which we also do for other stochastic processes. \textcolor{black}{We denote the quadratic variation of the process $P(t)$ over $[0,T]$ by $[P]_T$.}
For $0 \le s \le t$, we define the logarithmic return over the interval $[s,t]$ by
Then, the (model free) spot (or instantaneous) variance of the logarithmic price $P$ at time $t$ is\footnote{We consider spot variance in calendar time (instead of some intrinsic time) as this conveniently allows to link it to the trading intensity and tick variance as later formalized in Proposition (ref).}
In this paper, we are interested in estimating the \textcolor{black}{price variability} within a given time period $[0,T]$, where we focus on the case of $T$ being one trading day, i.e., the daily return is given by $r_{\mbox{\scriptsize daily}}:=r\left(0,T\right)=P\left(T\right)-P\left(0\right)$. Here, \textcolor{black}{this price variability} is measured by the integrated variance (IV) associated with the logarithmic price process $P(t)$ over the interval $[0,T]$ barndorff2002a, andersen2006. \textcolor{black}{Formally, the IV is defined as}
Proposition (ref) below provides a more formal justification for the IV as our object of interest given that \textcolor{black}{in expectation}, it equals the variance of the daily asset return.
We primarily focus on the specific choice of a sampling scheme for sparsely sampled intraday returns for estimating IV. Given a filtration $\mathbb{G} = \{\mathcal{G}_t\}_{t\geq0}$ with $\mathcal{G}_t \subset \mathcal{F}_t$, a $\mathbb{G}$-adapted stopping time sampling scheme $\boldsymbol{\tau}$ is a sequence of \textcolor{black}{increasing} $\mathbb{G}$-adapted stopping times on $[0,T]$,
such that $\tau_{j-1} \le \tau_j$ for all $j\in\mathbb{N}$. We require $\tau_0=0$ and that for almost all $\omega \in \Omega$ there exists an $n(\omega)\in \mathbb{N}$ such that $\tau_{n(\omega)}(\omega) = T$ \textcolor{black}{and that $\tau_{j-1} < \tau_j$ for all $j \le n(\omega)$.} We give specific examples how $\boldsymbol{\tau}$ can be chosen in Section (ref).
Given the sampling times $\boldsymbol{\tau}$, the corresponding intraday returns are
where we associate to a sampling scheme $\boldsymbol{\tau}$ the (random) number of intraday returns $M = M(\boldsymbol{\tau}) = \inf\{n:\tau_n=T\}$. Based on the $M \in \mathbb{N}$ intraday returns $r_{j}$ from the grid $\boldsymbol{\tau}$, we follow andersen1998a, among many others, and define the realized variance (RV) estimator as
where we stress the dependence on the employed sampling scheme with the argument $\boldsymbol{\tau}$.
We model the ticks and log-prices based on a diffusion $B$ with stochastic tick variance $\varsigma$, where $B$ is time-changed by a jump process $N$ (e.g., Poisson- or Hawkes-type) that models the individual ticks. We refer to this as the Tick-Time Stochastic Volatility (TTSV) model,
for $t\in [0,T]$, \textcolor{black}{where $U(r) = B(N(r))$}. Formally, we build the model on the following assumption:
The TTSV model provides a joint model for the tick arrivals $N(t)$ together with the log-price process $P(t)$ that can capture both, time-varying, stochastic trading intensity and tick variance patterns. At the same time, $P(t)$ is a semi-martingale as a time-changed diffusion model Monroe1987, Liptser2012. In fact, Proposition (ref) shows that $P$ is an actual martingale, complying with the regularly imposed assumption of efficient markets DelbaenSchachermayer1994.
In the TTSV model, we assume to observe the jump times $N(t)$ together with the logarithmic prices at these times. We treat the jump times $N(t)$ as transaction times, whereas they could also be other measures of interest such as quote arrivals, volume-related quantities or aggregates of these measures. The “intensity” processes $\lambda(t)$ and $\varsigma(t)$ are latent, and can for example be modeled as standard It\^{o} diffusions, or Hawkes process type intensities; see dahlhaus2016 for a range of possible specifications.\footnote{The price process in ((ref)) could further be augmented with a finite-variation predictable mean component \citep*{andersen2003}. However, we follow oomen2006 (see also hansen2006, aitsahalia2011, among others) and set it to zero for simplicity.}
In the general form of Assumption (ref), the transaction times $N(t)$ can follow a general jump process with intensity $\lambda(t)$, which implies that $\mathbb{E} \big[ N(t)-N(s) \mid \mathcal{F}_s \big] = \mathbb{E} \big[ \int_s^t \lambda(r) dr \mid \mathcal{F}_s \big]$ holds a.s.\ for all $0 \le s \le t \le T$, i.e., the expected number of arrivals in the period $[s,t]$ is characterized by the accumulated intensity $\int_s^t \lambda(r) dr$; see bauwens2009 for details. Besides doubly stochastic (and non-homogeneous) Poisson processes that are characterized by independent arrivals, Assumption (ref) also allows more general intensity-based models such as autoregressive intensity processes hamilton2002 or self-exciting Hawkes processes hawkes1971, which can additionally capture the observed dependence and memory of the trade arrivals on financial markets.
Assumption (ref) also allows for capturing “leverage effects” as the jump intensity $\lambda$ and the tick-volatility $\varsigma$ can depend on (the sign of) past price changes. Part (c) of Assumption (ref) governs the price changes at the observed jumps. It essentially rules out anticipative dependence of the calendar-time processes $\lambda$ or $\varsigma$ on $B$, in the sense that the path of the intensities following a jump point is independent of the next increment of the Brownian motion. Assumption (ref) further contains moment conditions, which ensure that the IV and the integrated quarticity (IQ) in Theorem (ref) below are finite.
In the following, we provide a detailed empirical motivation of the TTSV model: The jump process $N(t)$ models the ticks (i.e., the transaction or quote times) through its arrival times $t_i, i \ge 0$, that satisfy $t_i\in[0,\infty)$ and $t_i<t_{i+1}$ for all $i=1,\dots,N(T)$. As illustrated by the blue points and black lines in the upper panel of Figure (ref), the sample path of $N(t)$ is a right-continuous step function with jumps of magnitude one at the arrival times $t_i$ such that $N(t) = i$ for $t \in [t_i,t_{i+1})$. The stochastic intensity process $\lambda(t)$ of $N(t)$ is motivated by the empirical observation that the amount of trading varies drastically throughout the day. E.g., at the NYSE, there is a much higher trading activity just before market closure than throughout the rest of the day. Figure (ref) shows the log-prices of the IBM stock traded on the NYSE on May 1, 2015 between 9:45am and 9:48am and between 15:57pm and 16:00pm. We see that there are drastically more trades in the \textcolor{black}{afternoon} than in the morning, which is caused by many traders closing their position due to various reasons, including settlement rules of exchange markets Admati1988. Figure (ref) shows a non-parametric estimate of the trading intensity $\lambda(t)$ for the IBM stock (details are provided in the figure caption), which confirms this finding.
As $N(t)$ is piecewise constant between its arrival times $t_i$, it holds for all $0 \le s < t \le T$ that
where the index $i$ in $U_i$ corresponds to the $i$'th observed tick $t_i$. As graphically illustrated with the blue dots and black lines in the lower panel of Figure (ref), this implies that the log-price $P(t)$ exhibits jumps of magnitude $\varsigma(t_i) U_i$ at the arrivals of $N(t)$, and it is constant in between.
The stochastic tick volatility $\varsigma(t)$ is essential for the model as one observes empirically varying tick volatility patterns throughout the day on financial markets. E.g., Figure (ref) shows that at the NYSE, the tick variance of the log-price changes is much higher in the morning than in the \textcolor{black}{afternoon}, which is illustrated more formally by the nonparametric estimate of the tick variance $\varsigma^2(t)$ in Figure (ref). This finding is mainly caused by traders who trade overnight information in the beginning of the day, which triggers large oscillations in the transaction prices and thus, a high tick volatility that calms down until lunch time Dahlhaus2014.
Conditionally on an arrival $t_i$, the price change $\varsigma(t_i) U_i$ is normally distributed with mean zero and variance $\varsigma^2(t_i)$, hence justifying the term tick variance. Generalizing the conditional Gaussianity of $\varsigma(t_i) U_i$ in (ref) might be an interesting avenue for future research. Nevertheless, due to the stochastic nature of the processes $N(t)$, $\lambda(t)$ and $\varsigma(t)$, the unconditional distribution of the log-prices in the TTSV model is much more general than Gaussian.
Proposition (ref), which is similarly stated in dahlhaus2016, shows that in the TTSV model, the spot variance at time $t$ conveniently decomposes into the (right-hand limits of the) trading intensity $\lambda(t)$ and the tick variance of the price jumps $\varsigma^2(t)$, hence combining the two different sources of intraday variation as illustrated, for example, in Figure (ref).
Together with the general definition of IV in (ref), Proposition (ref) shows that the IV of the log-price following the TTSV model is given by
The use of IV as the measure of (daily) return variability in the TTSV model is further motivated by the following result.
Hence, under the TTSV model, the variance of the daily return equals the expected IV, which shows that (estimates of) the IV can be interpreted as a measure of daily return variation, similar to classical diffusion processes andersen2003.
For our purposes of analyzing the efficiency of alternative sampling schemes, the TTSV model is particularly useful as it disentangles the time-varying trading activity via the trading intensity $\lambda(t)$, and the time-varying tick variance through $\varsigma^2(t)$. As their intraday dynamics differ markedly in empirical data as shown in Figures (ref) and (ref), the separate model components for $\lambda(t)$ and $\varsigma(t)$ are crucial for some of the results of this paper.
The TTSV model is closely related to many classical models. For deterministic arrival times $t_1, \dots, t_N$ and a constant tick volatility $\varsigma(t)$, it nests a simple Gaussian random walk in transaction time. Furthermore, the compound Poisson process used by \citet*{oomen2005,oomen2006} arises when $N(t)$ follows a doubly stochastic Poisson process and when $\varsigma(t)$ is constant. While this setup allows for modeling tick arrivals as a separate component, it models all time variation in volatility through fluctuations in the arrival intensity. This restriction to a constant tick volatility is a clear limitation.
Lastly, a standard modelling choice is the continuous-time diffusion barndorff2002a (without drift and jump terms)
which is, compared to the TTSV model, not based on a time-change. In order to explicitly model the stochastic tick arrivals within these diffusion models, Fukasawa2010RV, Jacod2018, Jacod2017, Jacod2019 apply discretization schemes, where the tick arrivals (or alternatively, the sampling points) are modeled as random times at which one observes (a possibly generalized version of) the diffusion in (ref). Similar to the TTSV model, the observed prices are then modeled as a pure jump process with random arrival times, however, with the conceptual difference that the former applies a time-change with a jump process while the latter uses discretization.
We provide a detailed comparison of the TTSV model to these discretization schemes in Appendix (ref). While both modeling approaches have their individual merits and limitations, we use the TTSV model in this paper for the following reasons: First, the TTSV model offers an inherent and transparent decomposition of the spot variance into the empirically relevant components of sampling intensity and tick variance, which directly enables the derivation of particularly insightful results for classically used sampling schemes. Second, the simplicity of the TTSV model facilitates the derivation of finite sample MSE results---albeit partly under strong independence assumptions. While such results may also be attainable with discretized diffusion models, we conjecture that doing so would be considerably more involved. Third, as illustrated in Appendix (ref), the novel realized BTS scheme does not arise as naturally within the discretized diffusion framework.
In this section, we derive the bias and MSE of the RV estimator based on general sampling schemes $\boldsymbol{\tau}$ with a \textcolor{black}{fixed (expected) amount of intraday returns}. \textcolor{black}{Our main target is to find an optimal sampling scheme that is efficient in the sense of attaining the smallest MSE among a class of unbiased sampling schemes.}
As the RV estimator is unbiased for any $\mathbb{F}$-adapted sampling scheme, there is no theoretical distinction between different sampling schemes $\boldsymbol{\tau}$ in terms of a bias. We, however, continue by showing that the choice of $\boldsymbol{\tau}$ entails a difference in the estimation efficiency. To this end, we derive a closed-form expression for the finite-sample MSE of the RV estimator depending on the sampling grid $\boldsymbol{\tau}$.
Theorem (ref) provides a finite sample result for the MSE of any $\mathbb{F}$-adapted sampling scheme $\boldsymbol{\tau}$ under general dependence assumptions that for example, allow for Hawkes-type processes including a leverage effect; see the discussion after Assumption (ref). In (ref), the MSE is bounded from below by $\mathbb{E}[\operatorname{IQ}(0,T)]$, which merely depends on the underlying process but is invariant to the employed sampling scheme. Most important for our purposes is the term $\frac{2}{3} \mathbb{E}\left[ \sum_{j=1}^M r^4(\tau_{j-1}, \tau_j) \right]$, which depends on the fourth power of the returns, sampled according to $\boldsymbol{\tau}$. \textcolor{black}{ By applying the Cauchy-Schwarz inequality, this term is minimized by a sampling scheme that aims at homogenizing the absolute values of the intraday returns---as e.g., HTS. As the MSE expression (ref) in Theorem (ref) is only shown to hold for any $\mathbb{F}$-adapted sampling scheme $\boldsymbol{\tau}$, it is unclear how a feasible and $\mathbb{F}$-adapted scheme could be set up in practice that minimizes (ref) exactly, especially as the TTSV price process is discontinuous.\footnote{\textcolor{black}{A trivial---but clearly not $\mathbb{F}$-adapted---approach to minimizing (ref) for a given $M$ would be to allocate $\boldsymbol{\tau}$ among all observed tick times so as to minimize the sum of the fourth power of the resulting returns. However, such a sampling scheme would presumably not yield an unbiased RV estimator, rendering the MSE expression (ref) inapplicable. Moreover, it would be computationally very demanding, particularly on days with many ticks and for large values of $M$.}} We will later consider feasible and $\mathbb{F}$-adapted sampling schemes that aim at making intraday returns as homogeneous as possible---either in terms of their magnitude or in quantities related to their second moment---depending on the setting. }tting. }
Theorem (ref) applies to a very general class of sampling schemes that can access the history of all the processes driving the prices in the TTSV model. In the following, we also consider subclasses of sampling schemes that use less information about the price process and, in particular, are not allowed to depend directly on the observed prices. The intuitive reason is that the actual price observations are affected by MMN, which distorts the MSE result in Theorem (ref). As we will see in our simulations, this distortion is particularly severe for sampling schemes as HTS that directly rely on the observed high-frequency prices.
Therefore, we define the following two restricted filtrations that determine the precise information that (alternative) sampling schemes can use:
where $\mathcal{F}^{\lambda,\varsigma, N}_t = \sigma \big( \lambda(s), \varsigma(s), N(s); \; 0 \le s \le t \big)$ and $\mathcal{F}^{\lambda,\varsigma}_t = \sigma \big( \lambda(s), \varsigma(s); \; 0 \le s \le t \big)$. By considering sampling schemes adapted to the filtrations $\mathbb{F}^{\lambda,\varsigma, N}$ or $\mathbb{F}^{\lambda,\varsigma}$, we ensure that the possibly noisy price observations do not directly determine the sampling times. In the \textcolor{black}{$\mathbb{F}^{\lambda,\varsigma, N}$-adapted} case, we allow for a dependence of the sampling times on the realized tick pattern of the particular day. We refer to the case of $\mathbb{F}^{\lambda,\varsigma, N}$-adapted sampling as “realized” or “jump-based” sampling and to the case of $\mathbb{F}^{\lambda,\varsigma}$-adapted as “intensity-based” sampling.
We continue to investigate the MSE for the specific classes of sampling schemes introduced above. For this, we first state the two following corollaries, which express the MSE for sampling schemes $\boldsymbol{\tau}$ that are adapted to the reduced filtrations $\mathbb{F}^{\lambda,\varsigma, N}$ and $\mathbb{F}^{\lambda,\varsigma}$. The first corollary states that the MSE depends on the realized IV (rIV), which we define as
and interpret as a jump-process based and hence “realized” version of the classical IV given in (ref) and (ref).
The MSE formula from Corollary (ref) provides intuition on the relative efficiency of $\mathbb{F}^{\lambda,\varsigma, N}$-adapted sampling schemes: Invoking $\mathbb{E}[R(\boldsymbol{\tau})]=0$, a condition that holds under independence assumptions that are formalized in Theorem (ref) below, the Cauchy-Schwarz inequality directly implies that the MSE can be minimized by specifying $\boldsymbol{\tau}$ such that $\operatorname{rIV}(\tau_{j-1}, \tau_{j})$ is as homogeneous as possible (in expectation). Notice that the additional requirement in Corollaries (ref) and (ref) that the $U_i^2$ are independent of the entire paths of $\lambda$, $\varsigma$ and $N$ still allows for leverage effects, as the jump process and the tick variance can depend on the past sign of $U_i$. In Appendix (ref), we provide informal theoretical arguments that, under process dependencies that decay fast enough over time (as in Hawkes processes), the remainder term $\mathbb{E}[R(\boldsymbol{\tau})]$ is approximately equal for all sampling schemes, given that sparse sampling is employed. We note here already that our simulations confirm this finding.
Corollary (ref) shows that restricting attention to $\mathbb{F}^{\lambda,\varsigma}$-adapted sampling schemes $\boldsymbol{\tau}$ leads to a similar formula as in Corollary (ref). However, efficiency is now characterized by homogeneity of $\operatorname{IV}(\tau_{j-1}, \tau_j)$ (opposed to the realized IV in Corollary (ref)), and the result is subject to the further remainder term $\widetilde{R}(\boldsymbol{\tau})$.
The following theorem summarizes these results by imposing conditions under which the remainder terms ${R}(\boldsymbol{\tau})$ and $\widetilde{R}(\boldsymbol{\tau})$ vanish in expectation.
Roughly speaking, all three parts of Theorem (ref) suggest homogenizing the sampled returns. These parts mainly differ by the quantity that is homogenized, which will naturally be contained in the filtration the sampling schemes are adapted to. \textcolor{black}{It is important to note that in all three parts of Theorem (ref), adaptiveness to a certain filtration is required. This makes it unclear how the condition of homogenizing returns can be satisfied exactly in practice, rendering these lower bounds infeasible in implementation. In Section (ref), we therefore consider feasible sampling schemes that satisfy the homogeneity conditions approximately.}
Theorem (ref) (a) establishes that the most general finite sample efficiency is achieved when sampling times are chosen such that the absolute return values coincide throughout a trading day, hence pertaining to the HTS scheme. Parts (b) and (c) examine settings where the price information is not used for the construction of the sampling times. These restricted settings are practically relevant, as the observed high-frequency returns are regularly contaminated by MMN, which can make their use in constructing the sampling times problematic as will be illustrated in our simulations.
On a technical level, the additional independence assumptions in parts (b) and (c) ensure that the remainder terms $R(\boldsymbol{\tau})$ and $\widetilde{R}(\boldsymbol{\tau})$ from Corollaries (ref) and (ref) vanish in expectation. As exemplified in Appendix (ref), we conjecture that these remainder terms have a minor dependence on the employed sampling schemes, suggesting that the efficiency results of parts (b) and (c) also continue to hold for processes with mild dependencies, as reflected in our simulations.
While Theorem (ref) describes idealized conditions for efficient sampling, the following Section (ref) discusses their practical implementation.
Most practically relevant sampling schemes $\boldsymbol{\tau}$ that aim to homogenize a certain quantity, as formalized through Theorem (ref), can be specified based on a (weakly) increasing and possibly stochastic accumulated sampling intensity process $\{\Phi(t)\}_{t \in [0,T]}$. For example, for the classical CTS scheme, $\Phi(t) = t$ equals the identity. In contrast, different variants of transaction- and business-time sampling are based on combinations of the accumulated trading intensity, tick variance and the observed tick arrivals. If $\Phi$ is differentiable on $(0,T)$, its derivative is denoted by $\phi$ and has the interpretation of a sampling intensity.
Given an accumulated sampling intensity process $\Phi$, the sampling times $\tau_j$, $j=0,\dots, M$ are chosen as the generalized inverse of $\Phi$,
for some possibly stochastic threshold $\delta > 0$. This ensures that we sample equidistantly in the accumulated sampling intensity with $\tau_0 = 0$ and $\tau_{M} = T$.\footnote{If $\Phi(t)$ is continuous, (ref) implies that $\Phi(\tau_j) - \Phi(\tau_{j-1}) = j \delta - (j-1)\delta = \delta$ is constant for all $j=1,\dots, M$. For the discontinuous versions of $\Phi(t)$ (such as sampling every $K \in \mathbb{N}$ transactions), this only holds approximately.} We then obtain the prices at sampling times $\tau_j$ with the “previous tick method” that is consistent with the TTSV modeling assumption, as illustrated with the red squares in the lower panel of Figure (ref).
In this paper, we focus on the following common sampling schemes that arise by choosing different measures for the sampling intensity:
While CTS is deterministic, iTTS and iBTS are $\mathbb{F}^{\lambda,\varsigma}$-adapted, and rTTS and rBTS are $\mathbb{F}^{\lambda,\varsigma, N}$-adapted, at least given that a deterministic threshold $\delta$ is used. For a practical implementation of iTTS, iBTS, and rBTS, we have to estimate the intensity processes $\lambda$ and/or $\varsigma$, which we do by averaging over past trading days.
The above sampling schemes $\boldsymbol{\tau}$ result in $M = M(\boldsymbol{\tau}) = \Phi(T)/\delta$ sampled returns per day, which is in general a stochastic quantity. In practice, it is, however, often desirable to fix $M$ for the following reasons: First, fixing $M$ allows for a convenient comparison across sampling schemes. We will do this later on in simulations and the empirical application. Second, as argued in zhang2005, among many others, the value of $M$ is the main driver of the bias of the RV estimator in the presence of MMN. By fixing $M$, we particularly “stabilize” the effect of noise on the RV estimator, as this prevents the RV from being more affected by noise on higher volatility days than on lower volatility days.
In empirical work, one often deviates from the stopping time assumption and fixes $M$ by choosing $\delta = \Phi(T)/M$. In practice, when estimating RV at the end of a trading day, the information $\Phi(T)$ is observable or can be estimated. Formally, the sampling schemes are no longer adapted to the filtrations $\mathbb{F}^{\lambda,\varsigma}$ or $\mathbb{F}^{\lambda,\varsigma, N}$, but rather to their enlargements by $\sigma(\Phi(T))$, where $\Phi(T)$ corresponds to the given sampling scheme. While the theoretical results of Section (ref) do not formally apply to that setting, we show in simulations (see Figure (ref)) that the effect is negligible. Moreover, Appendix (ref) derives finite sample theory with results analogous to cases (b) and (c) of Theorem (ref), where the sampling times are allowed to depend on information up to time $T$.
We finally describe the HTS scheme that is already analyzed in Fukasawa2010RV, Vetter2017, FukasawaRosenbaum2012, and which is not based on an accumulated intensity process:
Reconsidering our main result, Theorem (ref), we see that HTS is tailored to the most general case (a), where the absolute return values should coincide. Similarly, rBTS aims at homogenizing rIV, which is the most efficient among the $\mathbb{F}^{\lambda,\varsigma, N}$-adapted sampling schemes, and iBTS homogenizes IV, which is the most efficient among the $\mathbb{F}^{\lambda,\varsigma}$-adapted sampling schemes.
It is important to note that Theorem (ref) suggests idealized sampling schemes, which are, however, not necessarily feasible due to the discontinuity of the underlying processes in the TTSV model as well as in practice. For HTS, this leads to a common “overshooting” effect, where the absolute returns are only guaranteed to be larger than $\delta$. This overshooting effect is particularly pronounced for small values of $\delta$ and for days with little trading activity; see Figure (ref). \textcolor{black}{Although other $\mathbb{F}$-adapted schemes---such as sampling whenever the price process crosses an equidistant grid, ignoring repeated crossings of the same grid level---could also homogenize absolute returns, we find their performance similar to HTS and therefore do not pursue them further.}
For HTS, it is unfortunately not possible to fix the number of samples $M$, which is often desirable, as argued above.\footnote{Even with a large number of values for $\delta$ and trial and error, it might be impossible to obtain certain values of $M$ given an observed price path.} Through Theorem (ref), it is only feasible to fix the expected number of samples $\overline{M}$ by choosing $\delta^2 = \mathbb{E}[\operatorname{IV}(0,T)] \big/ \overline{M}$, at least in the absence of MMN, by ignoring the overshooting effect, and by estimating $\mathbb{E}[\operatorname{IV}(0,T)]$, e.g., by a standard RV estimator based on CTS returns.
Figure (ref) shows the price path of IBM on May 1, 2015, with estimates of the sampling times $\boldsymbol{\tau}$ with $M=26$ under the four sampling schemes CTS, rTTS, rBTS and HTS, presented in the four panels. The figure reveals a substantial variation of the sampling times across the sampling schemes: While the sampling points are equidistant in time for CTS, we sample more often in the afternoon with rTTS, but more often in the morning with rBTS and HTS. In particular, the empirically observed difference between rTTS and rBTS highlights the importance and necessity of a refined price model, such as the TTSV model, that can separately accommodate the different intraday patterns of the trading intensity and tick variance.
We now compare the statistical properties of the RV estimator in (ref) based on different sampling schemes in simulations under general (leverage-type) process and noise specifications. In addition to validating our theoretical derivations, the aim of the simulation study is to analyze the impact of MMN on the sampling schemes and to quantify the efficiency gains of intrinsic time sampling.
We simulate $D=5000$ days with $T=23400$ (seconds) from the TTSV price process
where we distinguish the following two settings.
In the first specification, which we denote as the “independent TTSV process”, $N(t)$ is a doubly stochastic Poisson process independent of $B$. For the underlying intensities, we use the diffusive specifications,
for $t \in [0,T]$, where $B_1$ and $B_2$ (and $B$) are independent Brownian motions. The processes $\lambda(t)$ and $\varsigma(t)$ in (ref)--(ref) consist of deterministic components $\lambda_\text{det}(t)$ and $\varsigma_\text{det}(t)$ that are the same for every simulated day and give the processes a common characteristic shape, and the multiplicative stochastic diffusions $\lambda^{*}(t)$ and $\varsigma^{*}(t)$ that add some day-by-day randomness. We obtain the deterministic components $\lambda_\text{det}(t)$ and $\varsigma_\text{det}(t)$ as averages of their estimates using the estimators of dahlhaus2016, computed over all trading days of the IBM stock in the year 2018. The factor $c_\lambda \in \{2000, 8000, 32000\} / \int_0^T \lambda_\text{det}(t) \mathrm{d}t$ in (ref) allows to control the amount of expected ticks per day to equal $\{2000, 8000, 32000\}$, while its inclusion in (ref) preserves the expected IV, making it invariant to the choice of $c_\lambda$.
The components $\lambda^{*}(t)$ and $\varsigma^{*}(t)$ are Ornstein-Uhlenbeck processes driven by independent Brownian motions $B_i(t)$, $i=1,2$. Their exponential transformations ensure the positivity of $\lambda(t)$ and $\varsigma(t)$, and the coefficients $\bar \lambda^{*}$ and $\bar \varsigma^{*}$ are the daily averages (over all $t \in [0,T]$) of $\exp (0.01 \lambda^{*}(t))$ and $\exp(0.005 \varsigma^{*}(t))$, respectively, such that the exponential functions have unit mean and serve as multiplicative noise. We use Euler discretizations with 23400 steps to simulate the diffusions in (ref)--(ref).
For the second specification, which we denote as the “Hawkes-type TTSV process”, $N(t)$ is a Hawkes process with intensity $\lambda(t)$, which, along with the tick variance, is defined as follows
These intensities extend the specifications in (ref)--(ref) by incorporating dependent Brownian motions $B_1$ and $B_2$ with a correlation of $0.3$ and, more importantly, by including summands corresponding to self-exciting Hawkes-type intensities with an additional leverage specification hawkes2018hawkes, laub2021elements. For the sequence of jump time $t_1,t_2,\dots$ of the process $N$, and $\Delta P(t_{k}) = P(t_{k}) - P(t_{k-1})$, we set
where $\bar{\lambda}_\text{det}$ and $\bar{\varsigma}_\text{det}$ are the daily averages (over all $t \in [0,T]$) of $\lambda_\text{det}(t)$ and $\varsigma_\text{det}(t)$, respectively. Here, past price changes have a self-exciting effect on the intensities that declines exponentially with the time elapsed since that observation, $t-t_k$. Consistent with the classical leverage effect, positive price changes $\Delta P(t_{k}) > 0$ at the previous ticks $t_k$ have a different (weaker) impact than negative price changes $\Delta P(t_{k}) \le 0$.
As above, the constant $\widetilde{c}_\lambda \in \{2000, 8000, 32000\} \cdot (1-\eta) / \int_0^T \lambda_\text{det}(t) \mathrm{d}t$, with $\eta = 0.5 (0.05 \bar{\lambda}_\text{det} + 0.1 \bar{\lambda}_\text{det})/(0.25 \bar{\lambda}_\text{det})$, controls the expected number of ticks per day; see laub2021elements for details. As we are not aware of a closed-form formula for the expected $\varsigma(t)$ to account for the self-exciting effect stemming from the latter sum in (ref), we choose $\widetilde{c}_\varsigma \approx 0.855, 0.837, 0.741$ for the settings of $2000, 8000$, and $32000$ expected ticks, respectively. These choices ensure that all simulation processes have approximately the same expected IV while maintaining control over the expected number of ticks. For the Hawkes-type intensities in (ref)--(ref), we employ the simulation method described in dassios2013exact.
The parameters of the two simulation processes above are chosen to mimic real financial data, while also providing sufficient daily variation (across different days) in the simulated intensities $\lambda(t)$ and $\varsigma(t)$, as can be seen from the three exemplary sample paths of $\lambda(t)$, $\varsigma^2(t)$, $\sigma^2(t)$ and $P(t)$ for both processes shown in Figure (ref).
For both simulation processes, we contaminate the log-price process with either i.i.d.\ or ARMA(1,1) noise with and without a diurnal heteroskedasticity component. Given the randomly simulated trading times $t_1, \dots, t_{N(T)}$, we set
where $v_i$ is independent of all other processes. For the i.i.d.\ noise, we let $v_i \stackrel{i.i.d.}{\sim} \mathcal{N}(0, \sigma^2_v)$ for $i=1,\dots, N(T)$, where $\sigma_v = c_{N} \cdot 1.2 \cdot 10^{-4}$. Here, the factor $1.2 \cdot 10^{-4}$ corresponds to the magnitude of the average tick standard deviation (for the standard setting of 8000 expected ticks per day), and the pre-factor $c_{N} \in \{0, 0.25, 0.5, 1\}$ governs the relative noise level ranging from no noise $c_{N} = 0$ to a high noise setting $c_{N} = 1$, where the noise variance equals the average tick variance. In the results below, we refer to the factor $c_N$ by writing “$100 \cdot c_N \%$ noise”. We emphasize that our “$100 \%$ noise” setting is consistent with the findings and simulation setups of Jacod2017 and Li2022remedi.\footnote{In more detail, our $100 \%$ i.i.d.\ noise setting employs a noise standard deviation of $\sigma_v = 1.2 \cdot 10^{-4}$ for values of $\sqrt{\operatorname{IV}} \approx 1.1 \cdot 10^{-2}$. In contrast, Jacod2017 use the much higher estimated noise standard deviation from their Figure 9 of approximately $5.6 \cdot 10^{-4}$ for Citigroup data in the year 2011 in relation to values of $\sqrt{\operatorname{IV}}$ of around $10^{-2}$. Moreover, Li2022remedi obtain noise standard deviation estimates of approximately $\{0.7, 1.1\} \cdot 10^{-4}$ (obtained as the square root of the autocovariance function at lag 0) for the Coca-Cola stock in the year 2018, where the pre-factors $\{0.7, 1.1\}$ refer to two different noise estimators.}
For the ARMA noise process, we let $v_i =\varepsilon_{i} + 0.5 v_{i-1} + 0.5 \varepsilon_{i-1}$, where $\varepsilon_{i} \sim \mathcal{N}(0, \sigma^2_{\varepsilon,i})$, and $\sigma^2_{\varepsilon,i}$ is either constant or follows a diurnal V-shaped piecewise linear function. The latter assigns double the variance at market opening and closing compared to the middle of the trading day, following Kalnina2008 and Jacod2017. For each of the five choices in $c_{N} $, we specify $\sigma^2_{\varepsilon,i}$ such that the average standard deviation of $v_i$ over the day equals $c_{N} \cdot 1.2 \cdot 10^{-4}$ to make it comparable in magnitude to the i.i.d.\ noise setting.
For all sampling schemes except HTS, we fix the value of $M$ by using information on the respective accumulated intensity $\Phi(T)$ at the end of each trading day in (ref). While this formally violates the stopping-time condition (ref) in Theorems (ref) and (ref), we illustrate in Figure (ref) that the results are invariant to this violation. As fixing $M$ is not possible for the HTS scheme, we fix $\delta$, for which we choose a sequence of 17 values ranging from approximately $0.00022$ to $0.0054$. These values yield reasonable sampling frequencies allowing for a comparison with the other sampling schemes. Note that for HTS and a fixed $\delta$, the number of samples per days is random and can vary substantially across trading days.
While the CTS and rTTS schemes can be implemented straightforwardly, the iTTS, iBTS and rBTS schemes require the intensities $\lambda(t)$, $\varsigma^2(t) \lambda(t)$, and $\varsigma^2(t)$, respectively. For this, we use rolling averages over the past $50$ trading days of the nonparametric estimators $\widehat{\lambda}(t)$, $\widehat{\lambda}(t) \, \hat{\varsigma}^2(t)$ and $\hat{\varsigma}^2(t)$, respectively, which are proposed in dahlhaus2016, who also show consistency of these estimators under i.i.d.\ noise.
Figure (ref) shows the relative bias, i.e., the bias standardized by the respective daily value of IV, of the RV estimator for the considered sampling schemes, a range of $M$ values, and for the two process specifications\footnote{For the Hawkes-type TTSV-process, we compare the estimated RV values against the realized IV, which can easily be computed as $\operatorname{rIV}(0,T) = \int_0^T\varsigma^2(r)dN(r) = \sum_{0 \leq t_i \leq T} \varsigma^2(t_i)$. In contrast, $\operatorname{IV}(0,T) = \int_0^T \varsigma^2(r) \lambda(r) dr$ is much more difficult to approximate in our simulations due to the combination of a continuous time diffusion with the Hawkes-type jumps with exponential decays defined in (ref)--(ref). Note that $\mathbb{E}[\operatorname{rIV}(0,T)] = \mathbb{E}[\operatorname{IV}(0,T)]$.} described above. Results are shown for four magnitudes of i.i.d.\ noise and values of $c_\lambda$ and $\widetilde{c}_\lambda$ that yield 8000 expected ticks per day.
For the specification without noise, we can confirm the unbiasedness of the RV estimator of Theorem (ref) for all sampling schemes and both process specifications. For an increasing amount of noise, the RV estimator exhibits the usual positive bias that grows with the sampling frequency. Notably, the HTS sampling scheme reacts more strongly to increasing noise levels, even for the lowest considered sampling frequencies, where the other sampling schemes are (almost) unbiased. Importantly, the results hold equivalently for both the independent and the Hawkes-type TTSV processes, thereby illustrating the broad applicability of Theorem (ref).
We continue to shed light on the increased bias under noise of the HTS scheme: Using the notation $r(s,t) = P(t) - P(s)$ and $\widetilde{r}(s,t) = \widetilde{P}(t) - \widetilde{P}(s)$, heuristic arguments for the RV estimator under noise, $\widetilde{\operatorname{RV}}(\boldsymbol{\tau})$, yield
In the following, we ignore the asymptotically vanishing $\mathcal{O}_P\big(M^{-1/2}\big)$ term arising from a standard central limit theorem for the (noise-free) RV estimator. Then, (ref) indicates that the bias is driven by two terms: the variance of the noise differences at the sampling points and the covariance between the sampled (noise-free, efficient) returns and the noise differences.
Figure (ref) displays the bias for the four sampling schemes under the independent TTSV process with 8000 expected ticks per day and 25% or 100% i.i.d.\ noise. \textcolor{black}{The colored lines represent the empirical bias obtained from the simulations, i.e., these lines match the respective lines from the second and fourth plot in the upper panel of Figure (ref).} The shaded gray areas correspond to the two approximation terms from (ref), which help explain the sampling-scheme-dependent differences in bias. We estimate these terms from the simulated data according to the formulas in (ref). While the variance term is of a similar magnitude for all sampling schemes, the HTS scheme stands out \textcolor{black}{as the only scheme} with a notably large positive covariance term---the main cause of HTS’s elevated bias, as we explain in the following.
For CTS, rTTS, and rBTS, the efficient returns $r(\tau_{j-1}, \tau_j)$ are independent of the noise terms as the sampling points do not depend on the noise on the given day. \textcolor{black}{In contrast, HTS determines the next sampling time $\tau_j$ as the first time point $t \ge \tau_{j-1}$, where the absolute noisy price change, $\big| \widetilde{r}(\tau_{j-1}, t) \big| = \big| r(\tau_{j-1}, t) + (v_{N(t)} - v_{N(\tau_{j-1})}) \big|$ exceeds $\delta$.}\footnote{As our price process in (ref) (such as real prices at financial markets) generates discrete price paths that are only observed at the realizations of $N$, the absolute values of the HTS returns slightly overshoots the threshold $\delta$ as can be seen in Figure (ref). As shown in Theorem (ref) that applies to arbitrary $\mathbb{F}$-adapted sampling schemes, this should not be the underlying reason for the increased bias of HTS.} \textcolor{black}{ Hence, given a fixed previous sampling point $\tau_{j-1}$, HTS is particularly likely to sample at time points $\tau_{j}$ for which the two quantities $r(\tau_{j-1}, \tau_j)$ and $(v_{N(\tau_j)} - v_{N(\tau_{j-1})})$ share the same sign, and hence accumulate in the noisy return $\widetilde{r}(\tau_{j-1}, \tau_j)$. This behavior results in a positive covariance term in (ref) and in our simulations, we observe associated correlations ranging between $0.15$ and $0.5$ for HTS. }
Figure (ref) presents the relative RMSE of the RV estimator\footnote{ The relative RMSE over the trading days $d=1,\dots,D$ is formally given as
ensuring that the square root and the normalization are taken “outside” of the MSE. This way, the plots indeed analyze the MSE while presenting results in a conveniently interpretable scale.} for the different sampling schemes. As in Figure (ref), we show results for both simulation processes and four noise levels in the subplots. In the absence of noise, HTS clearly yields the lowest RMSE across all sampling frequencies and both process specifications, as implied by Theorem (ref). Furthermore, rBTS and rTTS also improve upon the classically used CTS scheme, in line with part (b) of Theorems (ref). As the noise level increases, the RMSE rises across all sampling schemes and frequencies, reflecting the growing bias illustrated in Figures (ref)--(ref).
The pronounced bias for the HTS scheme leads to the finding that, as the sampling frequency increases, rBTS yields RV estimates with lower RMSE than HTS. The crossing point at which rBTS becomes more efficient than HTS primarily depends on the noise magnitude and ranges from $M \approx 780$ to $M \approx 39$, corresponding to sampling frequencies between 30 seconds and 10 minutes. Similar to the bias, the MSE results are very similar for the independent and the Hawkes-type TTSV processes, hence illustrating the broad applicability of Theorem (ref). This observation also supports the insight from Appendix (ref) that the remainder terms in Corollaries (ref) and (ref) are approximately equal across sampling schemes, even under mild dependence.
Appendix (ref) contains additional simulation results summarized as follows: First, Figure (ref) analyzes the effects of a varying expected number of $\{2000, 8000, 32000\}$ trades per day while keeping the expected IV unchanged. Under noise, HTS performs worse as the number of ticks increases, which is mainly explained by an increased relationship of the noise relative to $\varsigma(t)$: More ticks are generated through a higher level of $\lambda(t)$, which results in a lower $\varsigma(t)$ as the expected IV is held constant. Second, Figure (ref) illustrates that our results are robust to the standard and diurnal ARMA noise specifications. Third, Figure (ref) confirms parts (b) and (c) of Theorem (ref), i.e., that the realized TTS and BTS sampling variants outperform the intensity variants, and that using the true (oracle) intensities yields slightly better RV estimation performance than using their estimated counterparts. Fourth, Figure (ref) shows that employing stopping-times for rTTS and rBTS, as opposed to fixing $M$ (see Section (ref)), produces essentially the same RMSE results.
We start to illustrate the gains in estimation accuracy that HTS and rBTS entail for the RV estimator in Section (ref), and continue to analyze different sampling schemes in a forecasting environment in Section (ref).
In this application, we assess the estimation accuracy of the RV estimator for the different sampling schemes using data on 27 liquid stocks from the NYSE TAQ database.\footnote{We use the 27 stocks with the ticker symbols AA, AXP, BA, BAC, CAT, DIS, GE, GS, HD, HON, HPQ, IBM, IP, JNJ, JPM, KO, MCD, MMM, MO, MRK, NKE, PFE, PG, UTX, VZ, WMT, and XOM.} We filter the raw prices according to barndorff2009. Based on the filtered prices, we compute the five sampling schemes CTS, rTTS, iBTS, rBTS, and HTS as described in Section (ref). We use all trading days from January 1, 2012, to March 31, 2019, for evaluating the estimation accuracy and up to 50 trading days before January 1, 2012, to estimate the intensities required for the iBTS and rBTS methods. We estimate the underlying trading intensity and tick variance with the non-parametric and noise-robust estimators of dahlhaus2016 and average the estimated intensities over the past 50 trading days in a rolling fashion.
For the above sampling schemes, we choose a fixed number of $M \in \{13,26,39,78,130,260,390\}$ log-returns per day, which correspond to intrinsic time sampling frequencies of $390/M$ minutes. As in the simulations, fixing $M$ is done using the information on $\Phi(T)$ available at the end of each trading day. For HTS, however, fixing the threshold $\delta$ leads to a random number of samples $M_\delta$ per day, which can vary considerably. To address this variability, we proceed as follows: For each $M$, asset, and trading day, we select the HTS result corresponding to the threshold $\delta$ for which the realized $M_\delta$ is closest to the given $M$. For $\delta$, we use 29 equally spaced values for $\log_{10}(\delta)$ between $-3.7$ and $-2.3$. Table (ref) shows that averaging $M_\delta$ over time and assets before matching to $M$ does not meaningfully change the results for HTS.\footnote{Table (ref) also shows results when we (i) match monthly averages by averaging $M_\delta$ over all days within each month before matching to the $M$-grid; (ii) use all-time averaging over all trading days in the sample; and (iii) apply all-time and asset-wise averaging across all days and assets.}
We evaluate the competing RV estimators with the data-based ranking method of Patton2011RV, which addresses the challenge that the estimation target, IV, is not observable, even ex post. Specifically, we use the subsequent trading day’s IV estimate as a proxy, assuming it is unbiased but noisy. By using a future RV estimator as the proxy, the method of Patton2011RV “breaks” the correlation between the estimation errors of the RV estimators under consideration and the proxy. In practice, one should use an unbiased proxy that is unlikely to be affected by MMN. While choosing a potentially inefficient estimator still gives an asymptotically valid test, its power might be lower LiuPattonSheppard2015, HogaDimi2022. To balance these points, we set the proxy to the next day`s RV computed from 5 minute CTS returns throughout our analysis. Using different reasonable choices for the proxy such as sampling frequencies of 1, 10, or 15 minutes, \textcolor{black}{or daily squared returns (see Figures (ref) and (ref)),} does not meaningfully change our results. We test for significance of the pairwise loss differences with respect to a benchmark estimator to be specified below (which is in general different from the proxy) by using the DieboldMariano1995 test, with inference drawn by using the stationary bootstrap of PolitisRomano1994 that is shown to be valid in this setting by Patton2011RV.
Table (ref) summarizes the results by reporting the percentage of significantly positive and negative loss differences (at the $5\%$ level) compared to the baseline sampling schemes, aggregated across the 27 assets and the seven considered sampling frequencies. We use CTS and rBTS as the baseline schemes for comparison in the two panels: CTS as the most commonly employed sampling method in the literature, and rBTS to enable a direct comparison to HTS, as motivated by our simulation results. We deliberately compare estimators with the same sampling frequency across sampling schemes as a direct comparison of sampling schemes is the main focus of the paper. The table shows results based on both the MSE and QLIKE loss functions.
Detailed results for each asset and sampling frequency are given in Figures (ref) and (ref), comparing to CTS and rBTS as the baseline schemes, respectively. The upper panels show RMSE and the lower panels QLIKE results. Black (red) points indicate that the considered estimator is significantly better (worse) than the benchmark at the 5% level; absence of a point denotes an insignificant difference. The color intensity indicates the magnitude of the relative improvements in RMSE (capped at $\pm 20\%$) or in QLIKE (capped at $\pm 50\%$).
When comparing the more elaborate (rTTS, iBTS, rBTS, HTS) sampling schemes against the baseline CTS scheme in Figure (ref) and the left panel of Table (ref), we observe far more significantly positive than negative loss differences. This pattern is even more pronounced for the QLIKE loss function, relating to the known fact that evaluation results are often more stable for QLIKE than for MSE loss Patton2011. Figure (ref) further shows that the increases are particularly pronounced at lower sampling frequencies, which are still regularly used in empirical work such as in LiuPattonSheppard2015, bollerslev2018risk, bollerslev2020good, bollerslev2022zero, bates2019crashes, bucci2020realized, reisenhofer2022harnet, alfelt2023singular, Patton2023Bespoke. Consistent with our simulation findings, the most frequent and substantial improvements can be observed for the HTS (at lower frequencies) and the rBTS schemes.
Figure (ref) and the right panel of Table (ref) show that rBTS consistently outperforms CTS, rTTS, and iBTS, with efficiency gains again being more pronounced under the QLIKE loss. The direct comparison between rBTS and HTS reveals that, in line with our simulation results, HTS dominates rBTS at lower sampling frequencies below 5 minutes ($M \le 78$), where noise has a negligible effect. In contrast, rBTS outperforms HTS at frequencies above 5 minutes ($M > 78$) for most of the considered stocks.
\textcolor{black}{ To assess how our sparsely sampled RV estimators perform compared to a state-of-the-art noise-robust benchmark, Figure (ref) compares them to the pre-averaging RV of Jacod2009, computed from all tick-level data with non-zero price changes and with a bandwidth of $0.5 \sqrt{m_\text{ticks}}$, where $m_\text{ticks}$ is the daily number of ticks. Because the pre-averaging RV is independent of the sampling frequency $M$, all sampling-based RV estimators (for different $M$) are compared to a single pre-averaging estimator in Figure (ref). The resulting presentation therefore differs slightly from Figures (ref) and (ref). For the evaluation proxy, we use daily squared returns, since other choices---either a sparsely sampled CTS RV in Figure (ref) or the pre-averaging RV in Figure (ref)---can bias the results. As noted above, using daily squared returns reduces the test’s power but avoids this undesired sensitivity.}
\textcolor{black}{ Figure (ref) shows that our sparsely sampled RV estimators slightly outperform the pre-averaging estimator, particularly the rTTS and rBTS variants at sampling frequencies between $M = 78$ and $M = 390$. Although the HTS estimator exhibits some advantages at very low frequencies ($M < 78$) over the other sampling schemes in Figures (ref) and (ref), RV at these frequencies does not outperform the pre-averaging benchmark in Figure (ref). Our overall findings with respect to the pre-averaging RV estimator are consistent with the empirical study of LiuPattonSheppard2015, who find that the classical RV estimator is difficult to outperform in practice.}
In summary, our empirical analysis confirms our theoretical and simulation-based findings. First, the more elaborate sampling schemes (rTTS, iBTS, rBTS, HTS) that take into account intraday variation clearly outperform CTS. Second, rBTS and HTS perform best within this class\textcolor{black}{, and can also outperform the noise robust pre-averaging estimator using all tick level data.} Third, their relative effectiveness depends on the sampling frequency: HTS excels at (very) low frequencies, while rBTS proves to be more robust at higher ones. The empirical superiority of the HTS and especially the rBTS schemes further underscores the practical value of the TTSV modeling framework, which enables the convenient derivation of the rBTS scheme.
We next assess how the gains in estimation accuracy of HTS and rBTS translate into improved forecast performance following the empirical analysis of LiuPattonSheppard2015. To this end, we use the Heterogeneous AutoRegressive (HAR) model of Corsi2009,
that models RV on day $d$ as a linear function of the past daily, weekly and monthly averages of RV with error term $\varepsilon_d$ and parameters $(\beta_0, \beta_D, \beta_W, \beta_M)$ that are estimated by ordinary least squares.
For each combination of asset, sampling scheme, and sampling frequency, \textcolor{black}{and for the tick-level pre-averaging RV estimator}, we use the HAR model in (ref) to generate one-step-ahead forecasts by estimating the parameters in (ref) with a rolling window consisting of 803 trading days for model estimation starting on January 1, 2012. This results in an evaluation period of 1000 trading days ranging from March 28, 2015 to March 29, 2019. We evaluate the resulting forecasts with the MSE and QLIKE loss functions. \textcolor{black}{As the associated estimation target, we use daily squared returns as in LiuPattonSheppard2015, to have a fair evaluation target for all estimators.}
\textcolor{black}{Figure (ref) reports results aggregated over time and across assets for each sampling scheme and frequency individually. For both the MSE and QLIKE loss function, we report the average ranks of the respective sampling schemes, the proportion of comparisons where each sampling scheme is considered best, and the inclusion rates of the model confidence set (MCS) of hansen2011model using the implementation of bernardi2018model.}
\textcolor{black}{We find that HTS performs best at very low sampling frequencies (below $M = 78$), achieving the lowest average ranks and highest winning rates. The MCS inclusion rates are high across all sampling schemes and frequencies, which is unsurprising given the procedure’s low power, making the differences difficult to interpret. For the higher frequencies between $M=78$ and $M=390$, no sampling scheme consistently outperforms the others, which can be explained by the substantial “empirical noise” that is added in such a forecasting exercise, compared to the estimation results from Section (ref). }
In this paper we provide finite-sample theory as well as empirical results for the statistical quality of the classical RV estimator when the intraday returns are sampled in intrinsic time. This approach accounts for intraday trading (transaction time sampling -- TTS), volatility patterns (business time sampling -- BTS), or absolute price changes (hitting time sampling -- HTS). For BTS, we propose the novel realized BTS variant that samples according to a combination of the observed transactions and the estimated tick variance. The intrinsic time scales leverage the rich information content of high-frequency data by adopting a perspective that differs from traditional equidistant clock-time sampling, reflecting the irregular evolution of market activity and risk.
We find that, in the absence of market microstructure noise, the HTS scheme theoretically provides the most efficient RV estimates in finite samples. However, the rBTS scheme emerges as most efficient in a restricted setting where sampling must occur independent of the observed intraday prices. This restricted setting and consequently the rBTS scheme is motivated through the increased sensitivity of the HTS scheme to market microstructure noise, which we find empirically causes its performance to deteriorate rapidly when \textcolor{black}{(intrinsic)} sampling frequencies exceed five minutes. In contrast, the rBTS scheme is an attractive and robust alternative at all sampling frequencies.
The theoretical framework for our analysis builds on a joint model for the ticks (transaction or quote times) and prices, which we call the tick-time stochastic volatility (TTSV) model: The prices follow a continuous-time diffusion that is time-changed by a jump process that explicitly models the ticks. As a result, prices form a pure jump process with time-varying and stochastic jump intensity capturing the empirical fact that price observations arrive randomly and at irregular intervals throughout the day. Furthermore, the model includes a stochastic tick variance process---representing the variance of price jumps between adjacent ticks---that also varies over time and displays a mirrored intraday pattern relative to the trading intensity.
The TTSV model is particularly useful for theoretically disentangling the effects of intrinsic time sampling for several reasons. First, it captures the natural spot variance decomposition into trading intensity and tick variance that is especially informative when comparing business and tick time sampling variants. Second, it enables the derivation of theoretical finite-sample results in contrast to, for example, Barndorff2011SubsamplingRK who provide asymptotic arguments in favor of the intensity version of BTS. Third, by explicitly modeling the observed ticks through a jump process, the TTSV model naturally encompasses the novel realized BTS scheme, which performs well in our empirical application, demonstrating that its effectiveness reflects genuine practical improvements beyond the TTSV framework.
An interesting theoretical alternative is to accommodate the tick arrivals through discretization instead of a time-change, as recently proposed by Jacod2017, Jacod2019, DaXiu2021, Li2022remedi among others. \textcolor{black}{While the TTSV framework enables convenient finite-sample derivations, we conjecture that the corresponding asymptotic analysis tends to be more complex and demands stronger assumptions compared to the discretization approaches in Jacod2017, Jacod2019, DaXiu2021, Li2022remedi.} Furthermore, advancing the theoretical analysis of noise-robust estimators such as subsampling, realized kernel, or pre-averaging RV, particularly in combination with rBTS and HTS sampling, offers promising paths for future research.
Replication material is available under \href{https://github.com/TimoDimi/replication_RVTTSV}{https://github.com/TimoDimi/replication_RVTTSV}. \\ While the simulations can be fully replicated, we have to exclude the data files for the empirical application as these cannot be made publicly available.
We would like to thank the editor, the associated editor and the two referees for very valuable and constructive comments that have substantially improved the results of the paper. We are further thankful to Dobrislav Dobrev, Christian Gouri{\'e}roux, Andrew Patton, Davide Pirino, Winfried Pohlmeier, Angelo Ranaldo, Roberto Ren{\`o}, Richard Olsen, Philipp Sibbertsen, George Tauchen and the participants at the SoFiE Conference 2019, QFFE Conferences 2019 and 2022, and the Conference on Intrinsic Time in Finance 2022 for helpful comments. All remaining errors are ours. We thank Sebastian Bayer and Christian Mücher for help in preparing the TAQ data. T. Dimitriadis gratefully acknowledges financial support from the German Research Foundation (DFG) through grant number 502572912. R. Halbleib gratefully acknowledges financial support from the DFG through the grant number 8672/1.
\addcontentsline{toc}{section}{References}
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