Common price and volatility jumps in noisy high-frequency data

Common price and volatility jumps in noisy high-frequency data

Abstract

We introduce a statistical test for simultaneous jumps in the price of a financial asset and its volatility process. The proposed test is based on high-frequency data and is robust to market microstructure frictions. To test for and estimate volatility jumps locally at price jump arrival times, we design and analyze a nonparametric spectral estimator of the spot volatility process. A simulation study and an empirical example with NASDAQ order book data demonstrate the practicability of the proposed methods and highlight the important role played by price volatility co-jumps.

Keywords: high-frequency data; microstructure noise; nonparametric volatility estimation; volatility jumps.
JEL classification: E58, C14

1 Introduction

In recent years the broad availability of high-frequency intra-day financial data has spurred a considerable collection of works dedicated to statistical modeling and inference for such data. Semimartingales are a general class of time-continuous stochastic processes to model dynamics of intra-day log-prices in accordance with standard no arbitrage conditions. We consider an Itô semimartingale log-price model allowing for stochastic volatility, price and volatility jumps as well as leverage. Due to the market microstructure of financial data recorded at high frequencies, as effects of transaction costs and bid-ask bounce, log-prices are not directly well fitted by semimartingales. Instead, a noisy observation model turned out to be more suitable. Taking microstructure frictions into account substantially changes statistical properties and involved mathematical concepts of estimators.
One core research topic in statistics, finance and econometrics of high-frequency data is inference on the (integrated) volatility, bringing forth the seminal contributions by Andersen and Bollerslev (1998), Andersen et al. (2001), Barndorff-Nielsen and Shephard (2002), Aït-Sahalia et al. (2005) and much more literature devoted to this aspect. As the volatility takes a leading role in the model, it is important to set up accurate stochastic volatility models, see Eraker et al. (2003) among others. Uncertainty and risk in the evolution of intra-day prices is usually ascribed to two distinct sources: First, the volatility process of the continuous semimartingale part that permanently influences observed returns and, second, occasional jumps in prices. In asset pricing (Duffie et al. (2000), Todorov (2010)), macro and monetary economics (Winkelmann et al. (2016)) and risk management (Liu et al. (2003)) information about jumps is of key importance. While the literature on price jumps is well developed from both a statistical and empirical point of view, methods and evidence about volatility jumps are lagging behind. Empirical evidence about volatility jumps is usually based on methods for price jumps applied to an observable volatility measure like the index of implied volatility of S&P 500 index options (VIX), see Bloom (2009) and Tauchen and Todorov (2011). Such modeling strategies inevitably restrict the number of target variables and the overall scope of empirical insights. The asset pricing model of Pastor and Veronesi (2012) illustrates the economic forces behind contemporaneous price and volatility jumps. In their model, agents learn about the profitability of a firm in a changing political environment. A change in government policy does not only affect the expected profitability of a firm (price jump) but also triggers a simultaneous volatility jump induced by the impact uncertainty of the new policy. Since price jumps have often been associated with macro announcements or firm specific news, a natural empirical question arises, if prices and their volatilities jump at common times stimulated by the same events, or not. Such common jumps of price and volatility are often excluded in the statistics literature to avoid technical difficulties. Beyond the question if one should include simultaneous jump times in price and volatility in a model, testing locally for volatility jumps opens up new ways to study effects of information processing and volatility persistence. This is also reflected in an increasing interest to separate the leverage effect in a continuous and a jump part in the current literature, see Aït-Sahalia et al. (2016) and Kalnina and Xiu (2016).
This article presents a statistical test to decide whether intra-day log-prices exhibit common price and volatility jumps. For this purpose, we introduce a new spot volatility estimator for noisy observations. While we are the first who address the testing problem under noise, a number of spot volatility estimators are available, see Zu and Boswijk (2014), Yu et al. (2013), Mancini et al. (2015) and Kanaya and Kristensen (2016). Our main contribution is to complement the pioneering works by Jacod and Todorov (2010) and Bandi and Renò (2016) and to provide an approach for an observation model that accounts for market microstructure in order to efficiently exploit information from high-frequency data. The test generalizes the theory by Jacod and Todorov (2010) for non-noisy observations, and builds upon a new spectral spot volatility estimator. We obtain a statistical test by a neat combination of a stable central limit theorem at optimal rate for the spectral spot volatility estimator and a suitable test function. In analogy to Jacod and Todorov (2010), the new test is self-scaling in the volatility and rate-optimal. Those two properties are crucial to obtain an efficient method. They rely deeply on the features of the new spectral estimator which, to the best of the authors’ knowledge, is the only estimator for which a stable central limit theorem at optimal rate is available, where the asymptotic variance does not have a more complex sum structure. In contrast, with a more complex asymptotic variance structure, as for instance given by a pre-average or realized kernel spot estimator, preserving the self-scaling property does not seem to be feasible. The development of a test that can cope with noise is of high relevance and importance as Jacod and Todorov (2010) already remark in their empirical application: “presence of microstructure noise in the prices is nonnegligible”. We demonstrate in simulations that compared to an application of the method by Jacod and Todorov (2010) based on skip-sampled returns, we can significantly improve the power of the test.
Jumps in prices and the volatility are of very different nature. Large price jumps become visible through large returns. More precisely, in a high-frequency context truncation techniques as suggested by Mancini (2009), Lee and Mykland (2008) and Jacod (2008) can be used to identify returns that involve jumps. Up to some subtle changes due to dilution by microstructure, this remains valid also in the noisy setup, see Aït-Sahalia et al. (2012) and Bibinger and Winkelmann (2015) for an extended theory. We adopt the methods from Bibinger and Winkelmann (2015) to estimate the spot volatility in presence of price jumps. Contrarily, volatility jumps are latent and not as obvious as price jumps due to the fact that we can not observe the volatility path. The key element to determine volatility jumps even so, will be efficient estimates of the instantaneous volatility from observed prices.
Our spectral spot volatility estimator relies on the Fourier method promoted by Reiß (2011) and Bibinger et al. (2014) for estimating quadratic (co-)variation, combined with truncation techniques of Bibinger and Winkelmann (2015) to deal with price jumps. These methods attain lower variance bounds for integrated volatility estimation from noisy observations and are, compared to simple smoothing methods and especially skip-sampling to lower observation frequencies, more efficient. With this estimation approach at hand, we design a test, comparing estimated local volatilities and their left limits at the estimated price jump times. As a special case, this includes a local test for volatility jumps at some fixed time. For instance, one might want to test for a volatility jump at news arrival times. A test with fast convergence rate based on second order asymptotics of the estimator is suggested. While the overarching strategy follows Jacod and Todorov (2010), the specific test function and construction in the noisy observation case are different and profit from the spectral estimation methodology. In contrast to previous estimation techniques to smooth noise, the asymptotic variance structure of the spectral volatility estimates in Theorem 1 admits a simple form and is not separated in different summands. This facilitates a test statistic which is self-scaling in the local volatility and thus furnishes an asymptotic test with the best possible rate and whose asymptotic law is free from any unknown parameters. The Monte Carlo study demonstrates the high precision of the methods in finite samples. Our data study shows that price volatility co-jumps occur and are practically relevant.
The paper is organized as follows. Section 2 introduces the model and the statistical problem. We discuss the main ideas for the construction of the test including a short review of the approach for non-noisy data. Section 2.2 describes the spectral spot volatility estimation. We state and discuss the assumptions imposed on the model for the asymptotic theory in Section 3.1 before presenting the main results in Section 3.2. Practical guidance for the implementation and a Monte Carlo study are given in Section 4. In Section 5 the methods are used to analyze price and volatility jumps in NASDAQ high-frequency intra-day trading data, reconstructed from the order book. Section 6 concludes. All proofs are gathered in Section 7.

2 Model, testing problem and statistical approach

Let be a filtered probability space satisfying the usual conditions. The latent log-price process follows an Itô semimartingale

(1)

with an -adapted standard Brownian motion, a Poisson random measure on with and an intensity measure (predictable compensator of ) for a given -finite measure . We consider discrete observation times , on the time span . The prevalent model, capturing market microstructure effects which interfere the evolution of an underlying semimartingale log-price process at high frequencies, is an indirect observation model with noise:

(2)

Denote with and . Regularity conditions on the characteristics of the efficient price and the noise, under which we establish asymptotic results, are given in Section 3.1. In particular, we work with a general smoothness assumption on the volatility . Similar to Jacod and Todorov (2010), resulting convergence rates of the spot volatility estimator and the asymptotic test hinge on this smoothness. First, readers may think of the typical case that is an Itô semimartingale with a representation as in (1) and with locally bounded characteristics.

2.1 Test for common price and volatility jumps

In the presence of price jumps, we design a statistical test to decide if contemporaneous price and volatility jumps occur on the considered time interval . Let be a sequence of stopping times exhausting the jumps of . We address the hypothesis of no common jump of volatility and price on :

(3)

against the alternative that there is at least one jump in the volatility at a jump time of . We denote the process of left limits of the volatility .
Our test for (3) relies on two main ingredients. First, localization of price jumps using thresholding. Second, a local test for volatility jumps. Suppose we want to test at a specific time , against the alternative that the volatility exhibits a jump . For such a test we require estimates of the squared volatility at time , , and before time , . An intuitive test statistic is the difference . Considering a more general class with a test function facilitates improved asymptotic properties.
If discrete observations of the efficient log-price , were directly available, and could be estimated by local versions of realized volatility:

For an Itô semimartingale , with some constant , yields rate-optimal spot volatility estimators, that is, . Further, on the test hypothesis that , for with and arbitrarily small, a stable central limit theorem can be proved

For stochastic volatility the limit is mixed normal and it is important that the convergence holds stably in law to obtain confidence. This is a stronger mode of weak convergence which is equivalent to joint weak convergence with every -measurable bounded random variable, see Jacod and Protter (2012) for an overview on stable limit theorems. This limit theorem readily supplies an asymptotic test for a volatility jump at time with a rate of convergence . However, the convergence rate is rather slow and not optimal for this testing problem. For the test statistic

one derives instead with a limit distribution and a much faster rate. This improves the (asymptotic) power significantly. A key property is that the test statistic is self-scaling in the volatility. This means that it does not require some estimated asymptotic variance, since the limit does not depend on any unknown parameter. Such a local test is not separately highlighted in Jacod and Todorov (2010), but is contained as one ingredient of their general method. The final test statistic of Jacod and Todorov (2010) is a sum of these local test statistics over all estimated jump times.
It is not obvious how to construct a generalization of the local test for a volatility jump to the indirect observations setup (2). Spot volatility estimators which are robust to noise are available in the literature, see, for instance, Yu et al. (2013) and Kanaya and Kristensen (2016). For an Itô semimartingale and i.i.d. noise with some moment assumption, stable central limit theorems

with optimal , , can be proved. Based on , a test with rate is obtained. Asymptotic variances of such estimators are usually sums of at least three addends: one depending on the noise variance, one including the quarticity and a cross term depending on both. The form of the variances is thus similar to the ones for integrated volatility estimators, see, for instance, Barndorff-Nielsen et al. (2008), Zhang (2006) and Jacod et al. (2009). Due to the additive structure of the asymptotic variance, it appears to have no prospect to look for a test function that facilitates a self-scaling asymptotic test with improved convergence rate.
Apart from attaining asymptotic efficiency, our main motivation to construct a method based on spectral spot volatility estimation is that we will be able to prove a stable central limit theorem

under mild assumptions for semimartingale volatility. Here, is the variance of i.i.d. noise, while we consider more general heteroscedastic and serially correlated noise in Section 3. As for other spot volatility estimators the rate is quite slow, but we have a simpler structure of the asymptotic variance. This enables us to find a suitable test function , such that

(4)

for a test statistic which is self-scaling in the volatility. The self-scaling property and the much faster convergence rate are key features to derive a reliable testing procedure.
To test the hypothesis (3), local tests are performed at the estimated jump times which can be detected almost surely asymptotically. Our asymptotic analysis provides results for the local test at some time as a special case. The method is of potential interest not only to test for contemporaneous price and volatility jumps. To evaluate the impact of news arrivals and information processing, for instance, economists might be interested to study volatility adjustments in response to certain firm specific or macroeconomic news announcements.
The tests for common price and volatility jumps of Jacod and Todorov (2010) for direct observations and our generalization for noisy observations both restrict to finitely many large price adjustments at whose arrival times local tests are performed. Testing for volatility jumps over an interval instead would require a sequence of tests for volatility jumps at infinitely many points and is rather connected to a high-dimensional testing problem. A theory without noise recently has been presented in Bibinger et al. (2016) and a generalization of the techniques, which are quite different to Jacod and Todorov (2010), to the model with noise is a challenging task for future research. It is clear that detecting volatility jumps from noisy observations of the price is especially difficult if we do not specify where to look for potential volatility jumps and the finite-sample performance of a global test is limited, see Section 6 of Bibinger et al. (2016). Restricting to local tests for volatility jumps as in this work facilitates a larger power in finite-sample applications.

2.2 Spectral spot volatility estimators

Consider a sequence of equispaced partitions of the considered time span into bins . For a simple notation suppose , such that on each bin we enclose noisy observations. A main idea of spectral volatility estimation is to perform optimal parametric estimation procedures localized on the bins. Based on these local estimates, one can build estimators for the spot and the integrated squared volatility. The spectral local method of moments of Bibinger et al. (2014) utilizes -orthogonal functions for spectral frequencies in the Fourier domain up to a spectral cut-off . For and we define

(5)

The indicator functions localize the sine functions to the bins. For the local method of moments, local linear combinations of the noisy data are used with local weights obtained by evaluating the functions (5) on the discrete grid of observation times . This strategy corresponds to performing a discrete sine transformation on the observed returns, similarly as proposed in Curci and Corsi (2012), but localized over the bins. We use the notion of empirical scalar products and norms for functions as follows:

(6)

The empirical norms of the sine functions above give for all bins :

(7)

and we have the discrete orthogonality relations

(8)

where is Kronecker’s delta. The latter rely on basic discrete Fourier analysis, a detailed proof is given in Altmeyer and Bibinger (2015). The central building blocks of spectral volatility estimation are the spectral statistics

(9)

in which observed returns , are smoothed by bin-wise linear combinations with weights from the local discrete sine transformations. Since the weight functions are non-zero only on the th bin, the spectral statistics include returns only over the bin under consideration. In absence of price jumps, bin-wise estimates for the squared volatility , are provided by weighted sums of bias-corrected squared spectral statistics:

(10)

For the moment, readers can interpret as time varying variance function of the observation errors in (2). In Section 3.1, this is further generalized. The oracle optimal weights

(11)

with , follow from minimization of the variance under the constraint of unbiasedness. For a fully adaptive approach we apply a two-stage method and obtain adaptive local estimates by plugging in estimated optimal weights in (10). The integrated volatility estimator of Bibinger et al. (2014) is simply the average .

Remark 1.

Spectral statistics are related to pre-averages used by Jacod et al. (2009), but the two estimators can not be transformed into one another, see Remark 5.2 in Jacod and Mykland (2015) for a discussion of their connection. One difference is that for the spectral method we start with a histogram structure and not a rolling kernel and then bin-wise noisy observations are smoothed in the Fourier domain. The statistics (9) de-correlate the data for different frequencies and form their local principal components. This is key to the asymptotic efficiency attained by the spectral estimators as shown in Reiß (2011) and Bibinger et al. (2014). The latter shows that the estimator’s asymptotic variance coincides with the minimum asymptotic variance among all asymptotically unbiased estimators. We refer to Remark 3.1 of Jacod and Mykland (2015) for a recent discussion about efficient volatility estimation under noise.

The spectral volatility estimation provides local estimates (10) for the squared volatility . In order to derive an estimate at some time , we average the statistics over a local window around of length as , , slowly enough to ensure . In the presence of jumps in (1), truncation disentangles bin-wise statistics (10) which involve jumps from all others. We use the methods from Bibinger and Winkelmann (2015) to cope with price jumps for volatility estimation. If for a threshold sequence , with some constant , the statistic is too large to be driven by the continuous part and is evoked by a jump of . In order to estimate the volatility, we thus truncate for these . For estimating the squared volatility and its left limit at a certain time , we use two disjoint windows after and before , respectively.
When the optimal weights (11) are known, an oracle spot volatility estimator for is:

(12a)
and the estimator for :
(12b)

Close to the boundaries, , we shrink one window length accordingly. Since the optimal weights (11) hinge on the unknown squared volatility and the noise level , we proceed with a two-step estimation approach. First, select a pilot spectral cut-off , and build pilot estimators for the squared volatility

(13)

and analogously. The pilot estimators are hence averages of squared spectral statistics, bias-corrected with the estimated noise level , over bins and spectral frequencies. Estimation of is addressed in Corollary 3.1. In the second step, these pilot estimators are plugged in (11) to determine adaptive weights for the final estimators. We write

(14)

The spectral estimators of the squared spot volatility at time and its left limit are:

(15a)
(15b)

Estimates (15a) and (15b) are local averages of the statistics (14). Thus, this nonparametric spot volatility estimation is closely related to the usual nonparametric kernel estimation when the statistics (14) take the role of de-noised observations which are smoothed over local windows. This illuminates the relation to the nonparametric volatility estimator by Kristensen (2010) for the setup without microstructure noise. Our approach entails several tuning parameters whose practical choice is discussed in Section 4.2.

3 Asymptotic theory

3.1 Assumptions with discussion

We start with the assumptions on the characteristics of in (1) which are similar to the ones in Jacod and Todorov (2010).

Assumption 1.

For the adapted and locally bounded drift process , we require a minimal smoothness condition that for , some constant and some :

(16)

The volatility process is càdlàg and neither nor vanish.

Assumption (H-r).

We assume that is locally bounded for a non-negative deterministic function satisfying .

We index the assumption in to highlight the role of the jump activity index . The larger , the more general jump components are included in our model. In particular for we consider jumps of finite activity. Imposing instead allows for infinite activity jumps which are summable. For the volatility process, our target of inference, we work with the following general smoothness condition determined by a smoothness parameter .

Assumption (-).

The process satisfies with some function , continuously differentiable in both coordinates, and two -adapted processes , where

  • is an Itô semimartingale

    (17)

    with an -Brownian motion independent of , locally bounded characteristics and a random variable . satisfies Assumptions 1 and (H-2) for . For , the continuous martingale part of vanishes and satisfies Assumptions 1 and (H-).

  • lies in a Hölder ball of order almost surely, i.e. , for all and a random variable for which at least fourth moments exist.

The smaller , the less restrictive is Assumption ($\sigma$-$\alpha$). It is natural to develop results for general to cover a broad framework and preserve some freedom in the model. This is particularly important, since the precision of nonparametrically estimating a process (or function) foremost hinges on its smoothness . Therefore, convergence rates in Section 3.2 hinge on . The composition of the volatility in Assumption ($\sigma$-$\alpha$) allows to incorporate recent volatility models and to realistically describe spot volatility dynamics. For instance, can contain a non-Lipschitz seasonality component (Lipschitz continuous seasonalities can as well be modeled by the drift of ). As pointed out by Jacod and Todorov (2010), can also be a long-memory volatility component as the prominent exponential fractional Ornstein-Uhlenbeck model by Comte and Renault (1998).
While an i.i.d. assumption on the noise is standard in most works, empirical findings, for instance by Hansen and Lunde (2006), motivate to allow for serial correlation in the noise. We develop our theory under the following general assumption.

Assumption (-).

The noise process is independent of , , and has finite -th moments, for all . The long-run variance process converges

(18)

for uniformly on compacts in probability and we have the mixing behavior

(19)

for some , which is specified for some of our results later. The process is locally bounded and satisfies for all the mild smoothness condition:

(20)

with some . Furthermore, the noise does not vanish, for all . When is stochastic, for notational convenience, we augment the probability space such that is -adapted.

The case that for all and constant for all is tantamount to the classical setup with i.i.d. noise. In general the noise is serially correlated and heteroscedastic. If we knew the process , Assumption ($\eta$-$p$) with a mild lower bound for would be sufficient for our asymptotic results. For an adaptive method, however, we need to estimate the process . Consistent estimation of the noise long-run variance (18) requires stronger structural assumptions. For a -dependent noise process, that is, for and some given , and if in (18) is time-invariant, consistent estimation with -convergence rate of has been established by Hautsch and Podolskij (2013). Bibinger et al. (2016) show how can be found adaptively if it is unknown. Jacod and Mykland (2015) discuss consistent estimation of the noise variance process under heteroscedasticity, but without serial autocorrelations. For in Assumption ($\sigma$-$\alpha$), we impose a stronger smoothness of in (20), such that roughness of the long-run noise variance process can not manipulate the resulting convergence rates. We formulate (20) as general as possible while from an applied point of view a smoother long-run noise variance process appears realistic. For the fully adaptive method, we tighten the assumptions on the noise as follows.

Assumption 2.

Assumption ($\eta$-$p$) holds. Moreover,

for some .

Corollary 3.1.

Under Assumption 2, for , for all , the locally constant approximated noise long-run variance process can be estimated with accuracy

(21)

When we apply the global method of Bibinger et al. (2016) localized to bins , such that is estimated in the same way as in Equations (19a)-(19c) of Bibinger et al. (2016), but using observations only instead of all observations , the regularity (20) renders for under Assumption 2 such estimators. For fixed , and locally constant , the proof from Bibinger et al. (2016) can be adopted just using observations on instead of all observations. This results in the slower rate instead of . Regularity (20) ensures that the approximation error of setting locally constant on is asymptotically negligible. This readily gives Corollary 3.1.
The assumptions on the noise are more general than in other works on spectral volatility estimation as in Altmeyer and Bibinger (2015) and in Bibinger et al. (2016). In particular, to the best of our knowledge, we consider for the first time heteroscedastic and serially correlated noise.

3.2 Asymptotic results

Our first main result is on the spot squared volatility estimator and its asymptotic distribution.

Theorem 1.

Suppose Assumptions 1, 2 and (H-r) with some and smoothness Assumption ($\sigma$-$\alpha$), . Fix some time , at which we want to estimate and with (15a) and (15b), respectively. Set and with constants and , , as . Then, as and if

(22)

and when moments of the noise exist, with the truncation exponent in the sequence in (13), (15a) and (15b), the estimators satisfy the -stable central limit theorems:

(23a)
(23b)

For the oracle estimators (12a) and (12b) the same limit theorems apply under the less restrictive Assumption ($\eta$-$p$) with , , and if . In fact, we can get arbitrarily close to the optimal rate for estimation which is known to be in this case, see Munk and Schmidt-Hieber (2010). Balancing the squared bias and the variance guarantees that the estimators (15a) and (15b) attain the optimal rate. For a central limit theorem we avoid an asymptotic bias by slightly undersmoothing. Most interesting is the case when and the volatility is a semimartingale. Then the convergence rate is . In case that , we obtain faster convergence rates. In case that , for any in Assumption (H-r), we can choose for any , if all moments of the noise process exist. Under the standard assumption that we only have Assumption ($\eta$-$p$) with , the condition results in . Hence, restricting to the condition that up to 8th moments of the noise exist leads only to a slightly less general condition on the jump activity. We point out that the restriction on the jump activity, to come close to the optimal convergence rate, is less restrictive than the one obtained for integrated squared volatility estimation, in Bibinger and Winkelmann (2015). The reason is that for spot volatility estimation we can only obtain slower convergence rates by local smoothing compared to integrated volatility estimation. This, however, works also under more active jumps.
The limit variables in (23a) and (23b) are mixed normal which we denote by and defined on a product space of the original probability space (on which is defined) and an orthogonal space independent of . The convergence is -stable in law, marked . Stability of weak convergence then allows for a so-called feasible version of the limit theorem (23a) that facilitates confidence sets.

Corollary 3.2.

Under the conditions of Theorem 1, and also for any fixed as :

(24)

with the estimate of , as defined in the weights (11), obtained by inserting the pilot estimates. This works analogously for (23b) for which we self-normalize with instead.

The results proved for the spot volatility estimator provide a main building block for our asymptotic test, but are moreover of interest in their own right. They show that the spectral method renders effective spot squared volatility estimators under noise and in the presence of jumps.
In the sequel, let be a sequence of stopping times exhausting the jumps of . We address hypothesis (3) that no common jumps of volatility and price occur on . Under the alternative, there is at least one contemporaneous jump in volatility and price.
Analogously to Jacod and Todorov (2010), we specify test hypotheses more precisely by focusing on jumps of with absolute values for and write . The reason for this is that a suitable test statistic and associated limit theory for with works under a much more general setup with jumps of infinite variation while testing requires Assumption (H-0) to hold. In both cases, we concentrate on a finite number of (large) price jumps in the hypothesis. From an applied point of view this is reasonable, since we are interested in volatility movements at finitely many relevant price adjustments on a fixed time interval.
Denote by a test function with for all . Let us now state the general form of our test statistics:

(25)

Under mild regularity assumptions on in terms of differentiability in both coordinates, limit theorems for (25) can be proved. For testing , we consider two specific test functions in the following. Adjustments of the test (3) for sub-intervals of are readily obtained by ignoring all jumps elsewhere.

Theorem 2.

Let be a finite collection of jump times of on with for all . Consider , if either and we impose the condition that the Lévy measure of does not have an atom in , or assume . On all assumptions of Theorem 1, when inserting estimates (15a) and (15b) with , , , , in (25) with the test function

(26)

the following asymptotic distribution of the test statistic applies under :

(27)

Under the alternative almost surely. Therefore, we obtain an asymptotic test by the asymptotic -distribution with degrees of freedom which is free from any unknown parameters. The test with critical regions

(28)

where denotes the -quantile of the -distribution, has asymptotic level and asymptotic power 1.

In fact, (28) contains the estimated number of price jumps . Since , (27) applies with also. A naive approach based on the asymptotic normality result (24) with test function yields as well an asymptotic test.

(29)

on the hypothesis . Apparently, the rate , close to for , is slower and thus the test in Theorem 2 is preferable.

4 Implementation and numerical study

4.1 Setup of Monte Carlo simulation study

The simulation study examines the finite-sample performance of the proposed methods. We implement a model where observed log-prices are given by