Modeling Price Clustering in High-Frequency Prices
MModeling Price Clustering in High-Frequency Prices
Vladimír Holý
Prague University of Economics and BusinessWinston Churchill Square 4, 130 67 Prague 3, Czech [email protected] Author
Petra Tomanová
Prague University of Economics and BusinessWinston Churchill Square 4, 130 67 Prague 3, Czech [email protected]
February 25, 2021
Abstract:
The price clustering phenomenon, i.e. an increased occurence of specific prices, is widelyobserved and well-documented for various financial instruments in various financial markets. In theliterature, however, it is rarely incorporated into price models. We consider that there are severaltypes of agents trading only in specific multiples of the tick size resulting in an increased occurrenceof these multiples in prices. For example, stocks on the NYSE and NASDAQ exchanges are tradedwith precision to one cent but multiples of five cents and ten cents occur much more often in prices.To capture this behaviour, we propose a discrete price model based on a mixture of double Poissondistributions with dynamic volatility and dynamic proportions of agent types. The model is esti-mated by the maximum likelihood method. In an empirical study of DJIA stocks, we find that higherinstantaneous volatility leads to weaker price clustering at the ultra-high frequency. This is in sharpcontrast with results at low frequencies which show that daily realized volatility has positive impacton price clustering.
Keywords:
High-Frequency Data, Price Clustering, Generalized Autoregressive Score Model, Dou-ble Poisson Distribution.
JEL Codes:
C22, C46, C58.
Over the last two decades, there has been a growing interest in modeling prices at the highest possiblefrequency which reaches fractions of a second for the most traded assets. The so-called ultra-high-frequency data possess many unique characteristics which need to be accounted for by econometricians.Notably, the prices are irregularly spaced with discrete values . Other empirical properties of high-frequency prices which can be incorporated into models include intraday seasonality , jumps in prices , price reversal and the market microstructure noise . For related models, see e.g. Russell and Engle(2005), Robert and Rosenbaum (2011), Barndorff-Nielsen et al. (2012), Shephard and Yang (2017),Koopman et al. (2017), Koopman et al. (2018) and Buccheri et al. (2020).We focus on one particular empirical phenomenon observed in high-frequency prices – price clus-tering . In general, price clustering refers to an increased occurence of some values in prices. A notabletype of price clustering is an increased occurence of specific multiples of the tick size , i.e. the minimumprice change. For example, on the NYSE and NASDAQ exchanges, stocks are traded with precisionto one cent but multiples of five cents (nickels) and ten cents (dimes) tend to occur much more oftenin prices. In other words, while one would expect the distribution of the second digit to be uniform,the probability of 0 and 5 is actually higher than 0.1. This behavior can be captured by some agentstrading only in multiples of five cents and some only in multiples of ten cents. It is well documentedin the literature that this type of price clustering is present in stock markets (see e.g. Ikenberry andWeston, 2008), commodity markets (see e.g. Narayan et al., 2011), foreign exchange markets (see e.g.1 a r X i v : . [ q -f i n . S T ] F e b opranzetti and Datar, 2002) and cryptocurrency markets (see e.g. Urquhart, 2017). Moreover, priceclustering does not appear only in spot prices but in futures (see e.g. Schwartz et al., 2004), options(see e.g. ap Gwilym and Verousis, 2013) and swaps (see e.g. Liu and Witte, 2013) as well. From amethodological point of view, almost all papers on price clustering deal only with basic descriptivestatistics of the phenomenon. The only paper, to our knowledge, that incorporates price clusteringinto a price model is the recent theoretical study of Song et al. (2020) which introduced the stickydouble exponential jump diffusion process to assess the impact of price clustering on the probabilityof default.Our goal is to propose a discrete dynamic model relating price clustering to the distribution ofprices and to study the high-frequency behavior of price clustering. We take a fundamentally verydifferent approach than Song et al. (2020) and incorporate the mechanism of an increased occurenceof specific multiples of the tick size directly into the model. This allows us to treat the price clusteringphenomenon as dynamic and driven by specified factors rather than given. We also operate withinthe time series framework rather than the theory of continuous-time stochastic processes. In contrastto the existing literature on price modeling, we do not model log returns or price differences butrather prices themselves. Prices are naturally discrete and positive. When represented as integers,they also exhibit underdispersion , i.e. the variance lower than the mean. To accommodate for suchfeatures, we utilize the double Poisson distribution of Efron (1986). It is a less known distribution asnoted by Sellers and Morris (2017) but was utilized in the context of time series by Heinen (2003),Xu et al. (2012) and Bourguignon et al. (2019). Modeling prices directly enables us to incorporateprice clustering in the model. Specifically, we consider that prices follow a mixture of several doublePoisson distributions with specific supports corresponding to agents trading in different multiplesof the tick size. This mixture distribution has a location parameter, a dispersion parameter andparameters determing portions of trader types. In our model, we introduce time variation to all theseparameters. We consider the location parameter to be equal to the last observed price resulting in zeroexpected returns. For the dispersion parameter, we employ dynamics in the fashion of the generalizedautoregressive conditional heteroskedasticity (GARCH) model of Bollerslev (1986). Specifically, weutilize the class of generalized autoregressive score (GAS) models of Creal et al. (2013) and Harvey(2013) which allows to base dynamic models on any underlying distribution. In the high-frequencyliterature, the GAS framework was utilized by Koopman et al. (2018) for discrete price changes andBuccheri et al. (2020) for log prices. To account for irregularly spaced observations, we include thelast trade duration as an explenatory variable similarly to Engle (2000). Finally, we relate the traderportion parameters to the volatility process and potentially to exogeneous variables. The resultingobservation-driven model is estimated by the maximum likelihood method .In the empirical study, we analyze 30 Dow Jones Industrial Average (DJIA) stocks in the firsthalf of 2020. We first focus on price clustering from a daily perspective which is a common approachin the price clustering literature. Using a panel regression with fixed effects, we find a positive effectof daily volatility measured by realized kernels of Barndorff-Nielsen et al. (2008) on price clustering.This finding is in line with the results of ap Gwilym et al. (1998b); Davis et al. (2014); Box andGriffith (2016); Hu et al. (2017); Blau (2019); Lien et al. (2019). Next, we estimate the proposedhigh-frequency price model and arrive at different conclusion – the instantaneous volatility obtainedby the model has a negative effect on price clustering. Note that high instantaneous volatility isstrongly associated with low trading intensity as observed e.g. by Engle (2000). The main messageof the empirical study is therefore that the degree of aggregation plays a pivotal role in the relationbetween price clustering and volatility. While high daily realized volatility correlates with high priceclustering, high instantaneous volatility has the opposite effect.The rest of the paper is structured as follows. In Section 2, we review the literature dealingwith high-frequency price models and price clustering. In Section 3, we propose the dynamic modelaccommodating for price clustering based on the double Poisson distribution. In Section 4, we usethis model to study determinants of price clustering in high-frequency stock prices. We conclude thepaper in Section 5. 2
Literature Review
In the literature, several models addressing specifics of ultra-high-frequency data have been proposed.One of the key issues are irregularly spaced transactions and discreteness of prices. The seminal studyof Engle and Russell (1998) proposed the autoregressive conditional duration (ACD) model to capturethe autocorrelation structure of trade durations, i.e. times between consecutive trades. Engle (2000)combined the ACD model with the GARCH model and jointly modeled prices with trade durations.Russell and Engle (2005) again modeled prices jointly with trade durations but addressed discretenessof prices and utilized the multinomial distribution for price changes.Another approach is to model the price process in continuous time. Robert and Rosenbaum (2011)considered that the latent efficient price is a continuous Itô semimartingale but is observed at thedicrete grid through the mechanism of uncertainty zones. Barndorff-Nielsen et al. (2012) consideredthe price process to be discrete outright and developed a continuous-time integer-valued Lévy processsuitable for ultra-high-frequency data. Shephard and Yang (2017) also utilized integer-valued Lévyprocesses and focused on frequent and quick reversal of prices.Transaction data at fixed frequency can also be analyzed as equally spaced time series withmissing observations. In this setting, Koopman et al. (2017) proposed a state space model withdynamic volatility and captured discrete price changes by the Skellam distribution. Koopman et al.(2018) continued with this approach and modeled dependence between discrete stock price changesusing a discrete copula. Buccheri et al. (2020) also dealt with a multivariate analysis and proposed amodel for log prices accommodating for asynchronous trading and the market microstructure noise.The latter two papers utilized the GAS framework.
The first academic paper on the price clustering was written by Osborne (1962), where the authordescribed the price clustering phenomenon as a pronounced tendency for prices to cluster on wholenumbers, halves, quarters, and odd one-eighths in descending preference, like the markings on a ruler.Since then, there have been many studies focusing on this phenomenon – from Niederhoffer (1965)to very recent papers of Li et al. (2020); Song et al. (2020); Das and Kadapakkam (2020) – showingthat price clustering is remarkably persistent across various markets:• stock market, see Niederhoffer (1965, 1966); Harris (1991); Aitken et al. (1996); Kandel et al.(2001); Ahn et al. (2005); Sonnemans (2006); He and Wu (2006); Cai et al. (2007); Alexanderand Peterson (2007); Brown and Mitchell (2008); Chiao and Wang (2009); Narayan and Smyth(2013); Hu et al. (2017); Lien et al. (2019);• commodity market, see Ball et al. (1985); Narayan et al. (2011); Bharati et al. (2012);• foreign exchange market, see Goodhart and Curcio (1991); Sopranzetti and Datar (2002); West-erhoff (2003); Liu (2011); Lallouache and Abergel (2014);• derivatives market, see ap Gwilym et al. (1998a,b); Schwartz et al. (2004); Chung and Chiang(2006); Capelle-Blancard and Chaudhury (2007); ap Gwilym and Verousis (2013); Meng et al.(2013);• other markets, see Kahn et al. (1999); Palmon et al. (2004); Palao and Pardo (2012); Brookset al. (2013).In recent years, great part of the research on price clustering focuses on Bitcoin due to its enormousprice surge in 2017, see Urquhart (2017); Hu et al. (2019); Baig et al. (2019); Mbanga (2019); Li et al.(2020).In spite of the fact that the price clustering phenomenon attracts researchers’ attention fromthe ‘60s, it has been mainly investigated based on the daily data. In the recent study, Hu et al.32017) claim that “[they] are the first to investigate price clustering using intraday data rather thandaily closing prices” . From then, several high-frequency analysis arises, e.g. Li et al. (2020) analyzedclustering for Bitcoin prices of various time frames with the highest frequency equal to 1 minute.However, despite the frequency of the data, the majority of studies are based on rather descriptivestatistics and basic linear regression analysis.Song et al. (2020) pointed out that all studies are entirely focused on empirically examining theclustering in different financial markets. Except purely theoretical study Song et al. (2020), that onthe other hand lacks the practical applications of their model, the literature uses well-known basicmodels for the empirical examination if any. The common approach is to use a linear regression modelestimated by ordinary least squares method (see, Ball et al., 1985; Harris, 1991; Hameed and Terry,1998; ap Gwilym et al., 1998b; Kandel et al., 2001; Chung et al., 2002; Cooney et al., 2003; Schwartzet al., 2004; Christie et al., 1994; Ahn et al., 2005; Chung and Chiang, 2006; Ikenberry and Weston,2008; Chiao and Wang, 2009; Palao and Pardo, 2012; ap Gwilym and Verousis, 2013; Brooks et al.,2013; Davis et al., 2014; Urquhart, 2017; Hu et al., 2017, 2019; Baig et al., 2019; Li et al., 2020),or its modifications such as two-stage least squares (see, Mishra and Tripathy, 2018), three-stageleast squares (see, Chung et al., 2004, 2005; Meng et al., 2013), or robust regressions that eliminategross outliers (e.g. Mbanga (2019)). Another studies analyzed the panel data by pooled regression(see, Chung et al., 2005; Liu and Witte, 2013; Blau and Griffith, 2016), models with fixed effects(see, Box and Griffith, 2016; Blau, 2019; Das and Kadapakkam, 2020), or random effects (see, Ohta,2006). Part of the literature models price clustering as a binary variable using logit (see, Christie andSchultz, 1994; Aitken et al., 1996; Brown and Mitchell, 2008; Bharati et al., 2012), or probit models(see, Kahn et al., 1999; Sopranzetti and Datar, 2002; Palmon et al., 2004; Ohta, 2006; Alexander andPeterson, 2007; Capelle-Blancard and Chaudhury, 2007; Liu, 2011; Narayan and Smyth, 2013; Lienet al., 2019). Blau (2019) estimated a vector autoregressive process and examined the impulses ofprice clustering in response to an exogenous shock to investor sentiment. Harris (1991) and Hameedand Terry (1998) analyzed a cross-sectional data by static discrete price model.To the best of our knowledge, the literature still lacks dynamic a discrete dynamic model to studythe high-frequency behavior of the price clustering. Thus, in the next section, we propose a newmodel which model high-frequency prices directly at the highest possible frequency and allows us tostudy the main drivers of price clustering such as volatility and trading frequency in the form of tradedurations.
Let us start with the static version of our model for prices. In the first step, we transform the observedprices to have integer values. For example, on the NYSE and NASDAQ exchanges, the prices arerecorded with precision to two decimal places and we therefore multiply them by 100 to obtain integervalues. The minimum possible change in the transformed prices is 1. Empirically, the transformedprices exhibit strong underdispersion , i.e. the variance lower than the mean. In our application,the transformed prices are in the order of thousands and tens of thousands while the price changesare in the order of units and tens. We therefore need to base our model on a count distributionallowing for underdispersion. For a review of such distributions, we refer to Sellers and Morris (2017).Although not without its limitations, the double Poisson distribution is the best candidate for our caseas the alternative distributions have too many shortcomings. For example, the condensed Poissondistribution is based on only one parameter, the generalized Poisson distribution can handle onlylimited underdispersion and the gamma count distribution as well as the Conway–Maxwell–Poissondistribution do not have the moments available in a closed form.The double Poisson distribution was proposed in Efron (1986) and has a location parameter µ > and a dispersion parameter α . We adopt a slightly different parametrization than Efron (1986) anduse the logarithmic transformation for the dispersion parameter making α unrestricted. For α = 0 ,the distribution reduces to the Poisson distribution. Values α > result in underdispersion while4alues α < result in overdispersion. Let Y be a random variable and y an observed value. Theprobability mass function is given by P[ Y = y | µ, α ] = 1 C ( µ, α ) y y y ! (cid:18) µy (cid:19) e α y e α + e α y − e α µ − y , (1)where C ( µ, α ) is the normalizing constant given by C ( µ, α ) = ∞ (cid:88) y =0 y y y ! (cid:18) µy (cid:19) e α y e α + e α y − e α µ − y . (2)The log likelihood for observation y is then given by (cid:96) ( y ; µ, α ) = − ln ( C ( µ, α )) + y ln( y ) − ln( y !) + e α y ln (cid:18) µy (cid:19) + α e α y − e α µ − y. (3)Unfortunately, the normalizing constant is not available in a closed form. However, as Efron (1986)shows, it is very close to 1 (at least for some combinations of µ and α ) and can be approximated by C ( µ, α ) (cid:39) − e α e α µ (cid:18) e α µ (cid:19) . (4)Zou et al. (2013) notes that approximation (4) is not very accurate for low values of the mean andsuggest to approximate the normalizing constant alternatively by cutting off the infinite sum, i.e. C ( µ, α ) (cid:39) m (cid:88) y =0 y y y ! (cid:18) µy (cid:19) e α y e α + e α y − e α µ − y , (5)where m should be at least twice as large as the sample mean. In our case of high mean, approximation(4) is sufficient while approximation (5) would be computationally very demanding and we thereforeresort to the former one. The expected value and variance can be approximated by E[ Y ] (cid:39) µ, var[ Y ] (cid:39) µe − α . (6)The score can be approximated by ∇ ( y ; µ, α ) (cid:39) (cid:18) e α µ ( y − µ ) , e α ( y ln( µ ) − µ − y ln( y ) + y ) + 12 (cid:19) (cid:48) . (7)The Fisher information can be approximated by I ( µ, α ) (cid:39) (cid:18) e α µ (cid:19) . (8) Next, we propose a mixture of several double Poisson distributions corresponding to trading in differ-ent multiples of tick sizes accommodating for price clustring. We consider that there are three typesof traders – one who can trade in cents, one who can trade only in multiples of 5 cents and one whocan trade only in multiples of 10 cents. In Appendix A, we treat a more general case with any numberof trader types and tick size multiples. The distribution of prices corresponding to each trader typeis based on the double Poisson distribution modified to have support consisting only of multiples of k ∈ { , , } while keeping the expected value E[ Y ] (cid:39) µ and the variance var[ Y ] (cid:39) µe − α regardlessof k . For detailed derivation of the distribution, see Appendix A. The distribution of prices for tradertype k ∈ { , , } is given by P (cid:104) Y [ k ] = y (cid:12)(cid:12)(cid:12) µ, α (cid:105) = I { k | y } P (cid:104) Z [ k ] = yk (cid:12)(cid:12)(cid:12) µ, α (cid:105) , Z [ k ] ∼ DP (cid:16) µk , α + ln( k ) (cid:17) , (9)5 .000.050.100.15 100.00 100.05 100.10 100.15 100.20 100.25 100.30 Price P r obab ili t y Price Distribution with Mean 100.13
Price P r obab ili t y Price Distribution with Mean 100.05
Trader
Figure 1: Illustration of the probability mass function for the mixture double Poisson distributionwith parameters µ = 10 013 (left plot), µ = 10 005 (right plot), α = 7 , ϕ = 0 . , ϕ = 0 . and ϕ = 0 . . The prices are reported in the original form with two decimal places.where DP denotes the double Poisson distribution and I { k | y } is equal to 1 if y is divisible by k and 0 otherwise. Note that for k = 1 , it is the standard double Poisson distribution. Finally, thedistribution of all prices is the mixture P [ Y = y | µ, α, ϕ , ϕ , ϕ ] = (cid:88) k ∈{ , , } ϕ k P (cid:104) Y [ k ] = y (cid:12)(cid:12)(cid:12) µ, α (cid:105) , (10)where the parameter space is restricted by µ > , ϕ ≥ , ϕ ≥ , ϕ ≥ and ϕ + ϕ + ϕ = 1 .Parameters ϕ k , k ∈ { , , } are the portions of trader types and parameters µ with α have the sameinterpretation as in the double Poisson distribution. The log likelihood for observation y is given by (cid:96) ( y ; µ, α, ϕ , ϕ , ϕ ) = e α y ln (cid:18) µy (cid:19) + α e α y − e α µ + ln (cid:88) k ∈{ , , } ϕ k I { k | y } √ kC (cid:0) µk , kα (cid:1) (cid:0) yk (cid:1) yk (cid:0) yk (cid:1) ! e − yk . (11)Note that the last logarithm in (11) is not dependent on parameters µ and α besides the normalizingconstant making the approximation of the score quite simple. Additionaly, parameters ϕ , ϕ and ϕ appear only in the last logarithm in (11) making the approximation of the score for parameters µ and α independent of parameters ϕ , ϕ and ϕ . The approximations of the expected value and thevariance as well as the score and the Fisher information for the parameters µ and α of the mixturedistribution are therefore the same as for the regular double Poisson distribution presented in (6),(7) and (8) respectively when assuming C ( µ/k, kα ) = 1 . Figure 1 illustrates the probability massfunction of the mixture distribution. Finally, we introduce time variation into parameters µ, α, ϕ , ϕ , ϕ . We denote the random pricesas Y t ∈ N , t = 1 , . . . , n and the observed values as y t ∈ N , t = 1 , . . . , n . We also utilize the observedvalues of trade durations z t ∈ R + , t = 1 , . . . , n . We assume that Y t follow the mixture double Poissondistribution proposed in Section 3.2 with time-varying parameters µ t , α t , ϕ ,t , ϕ ,t and ϕ ,t . The6ynamics of the location parameter µ t is given by µ t +1 = y t . (12)This means that the expected value of the price is (approximately) equal to the last observed price,i.e. the expected value of the return is zero. This is a common assumption for high-frequency returns(see e.g. Koopman et al., 2017).For the dynamics of the dispersion parameter α t , we utilize the generalized autoregressive score(GAS) model of Creal et al. (2013), also known as the dynamic conditional score (DCS) model by Harvey (2013). The GAS model is an observation-driven model providing a general frameworkfor modeling of time-varying parameters for any underlying probability distribution. It capturesdynamics of time-varying parameters by the autoregressive term and the score of the conditionaldensity function with scaling based on the Fisher information. As the Fisher information for theparameter α t is constant in our case, the score is already normalized and we therefore omit thescaling. Using (7) and (12), we let the dispersion parameter α t follow the recursion α t +1 = c + bα t + a (cid:18) e α t (cid:18) y t ln (cid:18) y t − y t (cid:19) − y t − + y t (cid:19) + 12 (cid:19) + d ln( z t +1 ) , (13)where c is the constant parameter, b is the autoregressive parameter, a is the score parameter and d is the duration parameter. This volatility dynamics corresponds to the generalized autoregressiveconditional heteroskedasticity (GARCH) model of Bollerslev (1986). Similarly to Engle (2000), wealso include the preceding trade duration z t +1 as an explanatory variable to account for irregularlyspaced observations. To prevent extreme values of durations, we use the logarithmic transformation.The portions of trader types are driven by process η t +1 = f η t + g (ln( µ t +1 ) − α t +1 ) , (14)where f is the autoregressive parameter and g is the parameter for the logarithm of the variance ofthe price process µ t +1 e − α t +1 . The portions of trader types are then standardized as ϕ ,t +1 = e η t +1 e η t +1 + h + h , ϕ ,t +1 = h e η t +1 + h + h , ϕ ,t +1 = h e η t +1 + h + h , (15)where h ≥ and h ≥ are parameters capturing representation of 5 and 10 trader types. Themodel can be straightforwardly extended to include additional explanatory variables in (13) and (14). The proposed model based on the mixture distribution for price clustering (10) with dynamics givenby (12), (13) and (15) can be straightforwardly estimated by the conditional maximum likelihoodmethod. Let θ = ( c, b, a, d, f, g, h , h ) (cid:48) denote the static vector of all parameters. The parametervector θ is then estimated by the conditional maximum likelihood ˆ θ ∈ arg max θ n (cid:88) t =1 (cid:96) ( y ; µ, α, ϕ , ϕ , ϕ ) , (16)where (cid:96) ( y ; µ, α, ϕ , ϕ , ϕ ) is given by (11). For the numerical optimization in the empirical study,we utilize the PRincipal AXIS algorithm of Brent (1972).
The empirical study is conducted on transaction data extracted from the NYSE TAQ database whichcontains intraday data for all securities listed on the New York Stock Exchange (NYSE), American7tock Exchange (AMEX), and Nasdaq Stock Market (NASDAQ). We analyze 30 stocks forming theDow Jones Industrial Average (DJIA) index in June 2020. The extracted data span over six monthsfrom January 2 to June 30, 2020, except for Raytheon Technologies (RTX) for which the data areavailable from April 3, 2020.We follow the standard cleaning procedure for the NYSE TAQ dataset described in Barndorff-Nielsen et al. (2009) since data cleaning is an important step of high-frequency data analysis (Hansenand Lunde, 2006). Before the standard data pre-processing is conducted, we delete entries that areidentified as preferred or warrants (trades with the non-empty suffix indicator). Then we follow acommon data cleaning steps and discard (i) entries outside the main opening hours (9:30 – 16:00), (ii)entries with the transaction price equal to zero, (iii) entries occurring on a different exchange thanit is primarily listed, (iv) entries with corrected trades, (v) entries with abnormal sale condition, (vi)entries for which the price deviated by more than 10 mean absolute deviations from a rolling centeredmedian of 50 observations, and (vi) duplicate entries in terms of the time stamp. In the last step, weremain the entry with mode price instead of the originally suggested median price due to avoidingdistortion of the last decimal digit of prices.The first and last step has a negligible impact on our data and steps ii, iv, and vi have noimpact at all. However, the third step causes a large deletion of the data which is, however, in linewith Barndorff-Nielsen et al. (2009). The basic descriptive statistics after data pre-processing areshown in Appendix B. Number of observations ranges from
216 618 (TRV) to (MSFT).Price clustering in terms of the excess occurrence of multiples of five cents and ten cents in pricesranges from 1.45 % (KO) to 11.52 % (BA). First, we investigate the price clustering using a commonapproach of linear regression in Section 4.2 to investigate whether the results for our dataset are inline with the existing literature. Then, we estimate the proposed price clustering model in Section4.3.
In this section, we investigate the main determinants of price clustering for which pervasive evidence isdocumented in the literature: (i) volatility and (ii) trading frequency in the form of trade durations.We use a panel regression with fixed effects to take into account the unobserved heterogeneity ofstocks.Let us define price clustering p i,t as the excess relative frequency of multiples of five cents and tencents in prices of stock i at day t . Then we model p i,t as p i,t = δ i + β + β ln( σ i,t ) + β ln( u it ) + ε i,t , where β , β , and β are parameters to be estimated, δ i accounts for unobserved heterogeneity ofstock i , and ε i,t is the error term. Daily duration u i,t is calculated as the average duration for stock i at day t . Volatility σ i,t is estimated by realized kernel estimator of Barndorff-Nielsen et al. (2008) σ i,t = γ + 2 H (cid:88) h =1 K (cid:16) h − H (cid:17) γ i,t,h , where K ( x ) is the kernel function and γ i,t,h = n (cid:88) j = h r i,t,j r i,t,j − h is the realized autocovariance of returns r i,t,j , j = 1 , . . . , n, for stock i at day t . We use the Parzenkernel given by K ( x ) = (cid:26) − x + 6 x for ≤ x < − x ) for ≤ x < The RTX company results from the merge of the United Technologies Corporation and the Raytheon Company onApril 3, 2020. β is the volatility parameter, β is the duration parameter, and ** denotes the significanceof a coefficient at the 0.01 significance level.I II IIIcoef. s.e. coef. s.e. coef. s.e. β . *** .
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AAPLBAMSFTV
Panel Regression with Fixed Effects
Figure 2: Daily data of four selected stocks and fitted lines from panel regression of price clusteringon volatility where price clustering is measured as the excess relative frequency of multiples of fivecents and ten cents in percentages and volatility is measured by realized kernels.as suggested by Barndorff-Nielsen et al. (2009) and discussed by Holý and Tomanová (2020) in theircomprehensive overview of quadratic covariation estimators.Table 1 reports estimated coefficients of three fixed variants of the effects models. The first variantmodels price clustering on both volatility and the average daily duration, the second model considersthe volatility only, and the third drives the price clustering only by average daily duration. We testthe significance of the estimated coefficients using robust standard errors. The results show thatonly volatility is a significant driver of the price clustering in the full model with both volatility andduration. However, once the volatility is dropped from the model, the average daily duration becomeshighly significant. Figure 2 shows fitted lines from panel regression of price clustering on volatilityfor two stocks traded on NASDAQ – Apple Inc. (AAPL) and Microsoft (MSFT) – and two stockstraded on NYSE – Boeing (BA) and Visa Inc. (V). The positive dependence between price clusteringand volatility is in line with the existing literature (see, ap Gwilym et al., 1998b; Davis et al., 2014;Box and Griffith, 2016; Hu et al., 2017; Blau, 2019; Lien et al., 2019).
Let us analyze the price clutering phenomenon at the highest possible frequency. First, we take abrief look on the relation between trade durations and price clustering. The left plot of Figure 3shows the average duration preceding the trade broken down by the second decimal of the price forthe BA stock. We focus on the BA stock as its price clustering is the most pronounced. We can9learly see that for prices ending with 0 and 5, the average duration is much lower than for the otherdigits. Regarding the 0 digit, the same holds for all other stocks. Regarding the 5 digit, most stockshave lower duration than expected but some do not deviate. Succeeding durations show very similarbehavior. This means that price clustering tends to occur when trading is more intense.Next, we estimate the proposed price clustering model for each of the 30 stocks. Table 2 reportsthe estimated coefficients. For all stocks, the coefficients have the same signs and fairly similar valuesdemonstrating robustness of the model. Parameter d is negative, which means that with longerdurations, dispersion parameter α t is lower and the instantaneous variance µ t exp( − α t ) is higher.This is expected behavior consistent with Engle (2000). The most interesting parameter is g whichlinks price clustering to the instantaneous variance. It is positive and therefore the portion of onecent traders is higher with higher variance. This means that price clustering tends to occur whenprices are less volatile. We further visualize this relation in the right plot of Figure 3.Parameters h and h controlling strength of price clustering are not very informative for readers.It is far better to look at the average values of trader portions ¯ ϕ , ¯ ϕ and ¯ ϕ reported in Table 3. Tencent traders are present for each stock with their average representation ranging from 0.62 percentfor the TRV stock to 9.68 percent for the BA stock. Five cent traders are virtually missing for theJNJ, MCD and MMM stocks while the highest average portion is 2.07 percent for the PG stock. Anexample of the progression of trader type ratios is shown in Figure 4 for the BA stock on the firsttrading day of 2020.Finally, we investigate other specifications of the model. We consider the model without any priceclustering (labeled as II), the model with static price clustering (labeled as III), the model with priceclustering driven by the variance and additionaly by the preceding duration (labeled as IV) and themodel with price clustering driven only by the preceding duration (labeled as V). Table 4 reportsthe Akaike information criterion (AIC) of the proposed model as well as differences in the AIC ofthe alternative specifications. We can see in columns II-I and III-I that the dynamic price clusteringcomponent distinctly increases fit of the model. Less clear issue is whether to let price clustering bedriven by the variance, the preceding duration or both. As we can see in column IV-I, the additionof the preceding duration to the proposed model does not increase fit very much; it is not even worththe additional parameter for 6 stocks with a positive difference in the AIC. The model with priceclustering driven only by the preceding durations, however, performs rather similarly to the proposedmodel. According to the AIC, it is better to have the variance for 19 stocks and the preciding durationfor 11 stocks. We should also note that the sign of the coefficient of the preceding duration differsin models. When only the preceding duration is included, the coefficient is positive. When both thevariance and the preceding duration is included, the coefficent of the preceding duration is half tohalf positive and negative while the coefficient of the variance remains positive. For these reasons,we consider the variance to be more reliable factor and the proposed model to be the most robust. In the paper, we propose a dynamic price model to capture agents trading in different multiplesof the tick size. In the literature, this empirical phenomenon known as price clustering is mostlyapproached only by basic descriptive statistics rather than a proper price model. By analyzing 30DJIA stocks from both daily and high-frequency perspective, we reveal a dissension between the twotime scales. While daily realized volatility has a positive effect on price clustering, instantaneousvolatility obtained by the proposed model has a negative effect.We believe the model to be sufficient for its purpose – capturing price clustering and allowing toexplain it. For the model to be able to compete with other high-frequency price models, however,it would have to be improved. The main limitation lies in the underlying distribution. We haveyet to study how well the double Poisson distribution, which we use, captures the observed prices.However, due to our specific problem, we require the distribution to be defined on positive integersand allow for underdispersion. The range of possible alternatives is therefore severely limited as it isnot a typical situation in count data analysis. Furthermore, the specification of the dynamics couldbe enhanced. We could include a separate model for durations, e.g. in the fashion of Engle (2000),10 .00.61.21.8 0 1 2 3 4 5 6 7 8 9
Second Decimal P r e c ed i ng T r ade D u r a t i on Average Preceding Trade Duration for BA
Second Decimal S t anda r d D e v i a t i on Average Standard Deviation for BA
Figure 3: The average preceding trade duration (left) and the average standard deviation (right)broken down by the second decimal digit of the BA stock prices.
Time R a t i o o f T r ade r T y pe Trader
Composition of Trader Types for the BA Stock on January 2, 2020
Figure 4: The time-varying portions of trader types obtained from the proposed price clusteringmodel for the BA stock on January 2, 2020. 11able 2: Estimated coefficients of the proposed dynamic price clustering model.Stock c b a d f g h h AAPL 6.1399 0.0753 0.2717 -0.3899 0.6853 0.4799 0.0094 0.2927AXP 5.8436 0.0942 0.3350 -0.2785 0.5518 0.3162 0.0493 0.1053BA 5.1272 0.0939 0.2944 -0.2885 0.4059 0.3134 0.0845 0.4425CAT 6.1370 0.0347 0.2611 -0.2720 0.6086 0.2976 0.0620 0.1151CSCO 6.1992 0.1694 0.0834 -0.3548 0.8436 0.2660 0.0000 0.0001CVX 6.0957 0.1101 0.3376 -0.2512 0.5459 0.3453 0.0315 0.0556DIS 6.2872 0.1147 0.3183 -0.2523 0.4614 0.4929 0.0491 0.0870DOW 6.0318 0.1456 0.3183 -0.2165 0.6058 0.3478 0.0124 0.0155GS 5.5290 0.0514 0.2679 -0.3093 0.4174 0.2596 0.0102 0.1307HD 5.4926 0.0772 0.2932 -0.2798 0.4509 0.3238 0.0745 0.1579IBM 6.2749 0.0780 0.3093 -0.2806 0.5351 0.4283 0.0265 0.0820INTC 6.4197 0.1376 0.1312 -0.3543 0.7657 0.2407 0.0009 0.0022JNJ 6.0965 0.1220 0.3612 -0.2400 0.7390 0.3377 0.0000 0.0540JPM 5.9726 0.1707 0.4287 -0.2597 0.5262 0.4836 0.0458 0.0417KO 6.1264 0.2541 0.3182 -0.1757 0.6579 0.5630 0.0019 0.0028MCD 5.6443 0.0770 0.3394 -0.2679 0.5688 0.2998 0.0000 0.2035MMM 5.9815 0.0636 0.2828 -0.2661 0.5437 0.5060 0.0000 0.1592MRK 6.0559 0.2035 0.3617 -0.2184 0.6982 0.2753 0.0165 0.0123MSFT 6.3525 0.1069 0.2602 -0.3711 0.7466 0.4611 0.0005 0.0209NKE 6.3739 0.1000 0.3253 -0.2405 0.3780 0.3882 0.0306 0.0287PFE 5.5838 0.3362 0.2504 -0.1548 0.6953 0.4227 0.0017 0.0011PG 6.1264 0.1093 0.4061 -0.2325 0.7508 0.2379 0.0908 0.0525RTX 6.3903 0.1439 0.2818 -0.2326 0.5423 0.4099 0.0273 0.0264TRV 5.8425 0.0493 0.2926 -0.2941 0.9465 0.1158 0.7529 0.9310UNH 5.0302 0.0656 0.2816 -0.2726 0.7593 0.1717 0.1494 0.3659V 6.2057 0.0599 0.3220 -0.2715 0.3837 0.4104 0.0074 0.1272VZ 5.9740 0.2652 0.3843 -0.1912 0.4941 0.5202 0.0059 0.0051WBA 6.3345 0.1146 0.1765 -0.3348 0.7466 0.3187 0.0006 0.0035WMT 6.4763 0.1159 0.3756 -0.2346 0.4198 0.5391 0.0287 0.0455XOM 6.1758 0.2333 0.2787 -0.1768 0.5437 0.7203 0.0043 0.003612able 3: The average values of the time-varying parameters of the proposed dynamic price clusteringmodel. Stock ¯ µ ¯ α ¯ η ¯ ϕ ¯ ϕ ¯ ϕ AAPL
29 180
17 580
11 791
11 250
19 314
21 544
12 405
14 037
10 365
18 386
14 992
16 929
11 641
11 157
27 172
18 013
11 865
11 896 423 .
38 256 255 .
53 141 466 . − .
33 6 419 . AXP .
39 6 615 .
44 2 185 . − .
36 163 . BA
10 581 225 .
71 179 112 .
36 25 848 . − . − . CAT .
58 4 128 .
02 1 294 . − .
64 28 . CSCO .
77 9 485 .
03 5 465 . − .
98 637 . CVX .
02 9 872 .
48 3 241 .
18 1 .
60 160 . DIS .
58 28 944 .
94 11 321 . − .
77 658 . DOW .
11 2 367 .
96 821 . − .
48 13 . GS .
25 3 893 .
23 769 .
70 1 .
74 38 . HD .
22 5 645 .
84 1 484 . − . − . IBM .
70 3 825 .
06 1 810 . − .
44 92 . INTC .
92 8 913 .
72 4 097 . − .
47 582 . JNJ .
99 659 .
80 611 . − . − . JPM .
44 19 764 .
63 9 849 . − .
65 1 351 . KO .
66 4 639 .
55 2 777 . − .
41 120 . MCD .
09 4 549 .
58 1 172 . − . − . MMM .
36 1 678 .
32 1 042 . − .
97 65 . MRK .
43 3 400 .
43 1 329 .
01 1 .
48 117 . MSFT
10 303 343 .
36 142 555 .
15 97 123 . − .
55 5 727 . NKE .
30 3 733 .
89 1 080 . − . − . PFE .
99 4 357 .
94 2 464 .
60 1 .
98 194 . PG .
39 6 159 .
59 2 024 . − .
40 356 . RTX .
41 4 570 .
86 1 327 . − . − . TRV .
63 547 .
00 409 . − . − . UNH .
08 5 158 .
37 896 . − . − . V .
69 11 050 .
59 3 272 . − . − . VZ .
59 4 056 .
20 1 889 .
09 0 .
09 70 . WBA .
83 4 706 .
12 2 456 .
78 1 .
78 119 . WMT .
90 6 824 .
28 2 840 . − . − . XOM .
42 18 424 .
70 11 978 . − . − . Acknowledgements
Computational resources were supplied by the project "e-Infrastruktura CZ" (e-INFRA LM2018140)provided within the program Projects of Large Research, Development and Innovations Infrastruc-tures.
Funding
The work on this paper was supported by the Czech Science Foundation under project 19-02773S, theInternal Grant Agency of the University of Economics, Prague under project F4/53/2019, and theInstitutional Support Funds for the long-term conceptual development of the Faculty of Informatics,Prague University of Economics and Business.
A Derivation of Distribution for Specific Trader Types
Let there be m types of traders that can trade only in k , . . . , k m multiples of the tick size respectively.For trader type k ∈ { k , . . . , k m } , we derive the distribution of prices P (cid:2) Y [ k ] = y (cid:12)(cid:12) µ, α (cid:3) . We requirethe distribution to be based on the double Poisson distribution, to have the support consisting ofmultiples of k , to have the expected value E (cid:2) Y [ k ] (cid:3) (cid:39) µ and to have the variance var (cid:2) Y [ k ] (cid:3) (cid:39) µe − α .We can modify any integer distribution P (cid:2) Z [ k ] = y (cid:12)(cid:12) µ, α (cid:3) to have support consisting only of multiplesof k as P (cid:104) Y [ k ] = y (cid:12)(cid:12)(cid:12) µ, α (cid:105) = I { k | y } P (cid:104) Z [ k ] = yk (cid:12)(cid:12)(cid:12) µ, α (cid:105) , (17)where I { k | y } is equal to 1 if y is divisible by k and 0 otherwise. We assume that Z [ k ] follows thedouble Poisson distribution with parameters µ [ k ] and α [ k ] , i.e. Z [ k ] ∼ DP (cid:0) µ [ k ] , α [ k ] (cid:1) . The expectedvalue of Y [ k ] is E (cid:104) Y [ k ] (cid:105) = ∞ (cid:88) y =0 y P (cid:104) Y [ k ] = y (cid:12)(cid:12)(cid:12) µ, α (cid:105) = ∞ (cid:88) y =0 y I { k | y } P (cid:104) Z [ k ] = yk (cid:12)(cid:12)(cid:12) µ, α (cid:105) = ∞ (cid:88) y =0 ky P (cid:104) Z [ k ] = y (cid:12)(cid:12)(cid:12) µ, α (cid:105) = k E (cid:104) Z [ k ] (cid:105) (cid:39) kµ [ k ] . (18)15he variance of Y [ k ] is var (cid:104) Y [ k ] (cid:105) = ∞ (cid:88) y =0 (cid:16) y − E (cid:104) Y [ k ] (cid:105)(cid:17) P (cid:104) Y [ k ] = y (cid:12)(cid:12)(cid:12) µ, α (cid:105) = ∞ (cid:88) y =0 (cid:16) y − E (cid:104) Y [ k ] (cid:105)(cid:17) I { k | y } P (cid:104) Z [ k ] = yk (cid:12)(cid:12)(cid:12) µ, α (cid:105) = ∞ (cid:88) y =0 (cid:16) ky − k E (cid:104) Z [ k ] (cid:105)(cid:17) P (cid:104) Z [ k ] = y (cid:12)(cid:12)(cid:12) µ, α (cid:105) = k var (cid:104) Z [ k ] (cid:105) (cid:39) k µ [ k ] e − α [ k ] . (19)Our last requirements E (cid:2) Y [ k ] (cid:3) (cid:39) µ with var (cid:2) Y [ k ] (cid:3) (cid:39) µe − α lead to the system of equations µ = kµ [ k ] µe − α = k µ [ k ] e − α [ k ] (20)with the solution µ [ k ] = µk , α [ k ] = α + ln ( k ) . (21)Everything together gives us the distribution P (cid:104) Y [ k ] = y (cid:12)(cid:12)(cid:12) µ, α (cid:105) = I { k | y } P (cid:104) Z [ k ] = yk (cid:12)(cid:12)(cid:12) µ, α (cid:105) , Z [ k ] ∼ DP (cid:16) µk , α + ln( k ) (cid:17) . (22)Note that the mixture distribution of all prices P [ Y = y | µ, α, ϕ k , . . . , ϕ k m ] = (cid:88) k ∈{ k ,...,k m } ϕ k P (cid:104) Y [ k ] = y (cid:12)(cid:12)(cid:12) µ, α (cid:105) (23)has approximately the same expected value and variance as the distribution of Y [ k ] . This is based onthe identity E [ g ( Y )] = ∞ (cid:88) y =0 g ( y )P [ Y = y | µ, α, ϕ k , . . . , ϕ k m ]= ∞ (cid:88) y =0 g ( y ) (cid:88) k ∈{ k ,...,k m } ϕ k P (cid:104) Y [ k ] = y (cid:12)(cid:12)(cid:12) µ, α (cid:105) = (cid:88) k ∈{ k ,...,k m } ϕ k ∞ (cid:88) y =0 g ( y )P (cid:104) Y [ k ] = y (cid:12)(cid:12)(cid:12) µ, α (cid:105) = (cid:88) k ∈{ k ,...,k m } ϕ k E (cid:104) g (cid:16) Y [ k ] (cid:17)(cid:105) = E (cid:104) g (cid:16) Y [ k ] (cid:17)(cid:105) , (24)where g ( · ) is any function satisfying that E (cid:2) g (cid:0) Y [ k ] (cid:1)(cid:3) are the same for all k .16 Descriptive Statistics of Cleaned Data
Table 5: The table reports a number of observations ( .
095 7 .
324 291 .
707 1 201 .
902 8 . AXP
467 623 6 .
242 176 .
988 98 .
589 318 .
158 4 . BA .
55 25 .
126 176 .
063 3 648 .
516 11 . CAT
413 148 7 .
078 232 .
47 118 .
071 204 .
997 3 . CSCO .
708 30 .
674 42 .
263 17 .
129 2 . CVX
964 508 3 .
025 41 .
271 88 .
337 251 .
894 3 . DIS . .
832 112 .
557 268 .
407 4 . DOW
606 116 4 .
823 109 .
041 36 .
531 69 .
433 2 . GS
376 058 7 .
775 285 .
132 193 .
369 978 .
305 3 . HD
503 522 5 .
799 146 .
819 215 .
652 822 .
868 3 . IBM
563 345 5 .
188 112 .
415 124 .
115 234 .
23 3 . INTC .
416 17 .
841 57 .
839 30 .
544 2 . JNJ
846 988 3 .
451 51 .
012 140 .
419 97 .
392 2 . JPM .
018 18 .
493 103 .
69 301 .
927 3 . KO .
564 44 .
013 48 .
346 33 .
405 1 . MCD
483 508 6 .
047 156 .
726 184 .
021 431 .
146 3 . MMM
445 633 6 .
555 195 .
767 150 .
044 196 .
279 3 . MRK .
612 30 .
813 79 . .
181 1 . MSFT .
943 5 .
779 169 .
252 264 .
697 5 . NKE
687 619 4 .
251 77 .
029 88 .
657 127 .
826 2 . PFE .
029 32 .
856 34 .
476 9 .
348 1 . PG
855 186 3 .
41 50 .
942 116 .
44 46 .
983 2 . RTX
470 061 3 .
037 30 .
12 62 .
283 23 .
712 3 . TRV
216 618 13 .
487 855 .
589 111 .
75 291 .
59 1 . UNH
561 256 5 .
209 147 .
823 271 .
931 714 .
106 3 . V
965 718 3 .
026 36 .
838 180 .
196 337 .
034 3 . VZ .
516 37 .
858 55 .
613 6 .
781 1 . WBA
668 123 4 .
378 111 .
012 45 .
777 23 .
725 2 . WMT
886 255 3 .
299 46 .
504 118 .
66 37 .
725 2 . XOM .
419 11 .
752 45 .
844 86 .
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