Microsoft Word - 00_tresc.docx DYNAMIC ECONOMETRIC MODELS Vol. 10 – Nicolaus Copernicus University – Toruń – 2010 Małgorzata Doman Poznań University of Economics Liquidity and Market Microstructure Noise: Evidence from the Pekao Data† A b s t r a c t. The availability of ultra-high frequency data justifies the use of a continuous-time approach in stock prices modeling. However, this data contain, apart from the information about the price process, a microstructure noise causing a bias in the realized volatility. This noise is connected with all the reality of trade. In the paper we separate the microstructure noise from the price process and determine the noise to signal ratio for the estimates of the realized volatility in the case of the shares of the Polish company Pekao S.A. The results are used to discover the optimal sampling frequency for the realized volatility calculation. Moreover, we check the lin- kages between the noise and some liquidity measures. K e y w o r d s: market microstructure, volatility, realized variance, liquidity, stock market, trad- ing volume, high frequency data. 1. Introduction Continuous-time econometric models are becoming now a standard tool for describing the financial market dynamics. They correspond well to the theoreti- cal models of financial mathematics and can be quite easy estimated due to the availability of ultra-high frequency data. It seems natural that tick-by-tick data are the most useful in the context of continuous-time models. However, it is not the all truth. This kind of data contains apart from the useful information about the price process a noise which, for instance, causes a bias in the daily realized volatility estimates. The sources of the noise are connected with the reality of trade. Dealing with continuous-time models, we make many assumptions that are not satisfied in the real market. They concern time, price process, and mar- ket mechanism. The departures of the observed process from these assumptions † This work was financed from the Polish science budget resources in the years 2007-2010 as the research project NN 111 1256 33. The author would like to thank an anonymous referee for useful comments and suggestions. Małgorzata Doman 6 are very often connected with the so-called market microstructure effects. The most known factors of market microstructure are liquidity, nonsynchronous trade, bid-ask spread, discrete-valued price, irregular time intervals between trades, and existence of diurnal pattern (Tsay, 2000). Usually these effects de- pend on legal regulations, market electronic systems, and traders knowledge and behavior. Volatility is one of the most important parameter in risk management, de- rivative pricing and portfolio allocation. Nowadays, one of the most popular and promising estimator of daily volatility is the daily realized variance (Andersen, Bollerslev, 1998; Barndorff-Nielsen, Shephard, 2002). It is calculated as a sum of squared intraday returns and so it depends on the chosen frequency of obser- vations. The frequency of intraday data should be high enough to capture as much as possible of available information and small enough to avoid including a noise into the realized variance estimates. It seems rather obvious that the problem of separating the noise from the „true price” process is of great impor- tance for quality of the daily volatility estimates. In this connection, the most significant microstructure phenomenon is liquidity. The microstructure noise is usually weaker for very liquid shares. The presented analysis applies the Aït-Sahalia and Yu (2009) approach to separate the microstructure noise from the price process in the case of shares of the Polish company Pekao S.A. Basing on the noise estimates, we determine the noise to signal ratio where the signal is the realized volatility. As a result of the analysis we obtain the optimal sampling frequency for the realized volatility calculation. Since liquidity is considered to be the crucial factor determining the noise level, we try to determine the dependencies between the noise to signal ratio and chosen liquidity measures. Moreover, we apply the signal to noise ratio to compare the strength of market microstructure effects observed in the analyzed Pekao data with that reported from more developed stock markets. 2. Realized Volatility and Market Microstructure Noise We consider a daily log-price process ))(ln()( tPtY = where t is measured in days. Then the logarithmic returns are given by formula )()(),( htYtYhtr −−= , and the daily realized variance (volatility) (Andersen, Bollerslev, 1998; Barndorff-Nielsen, Shephard, 2002) is defined as ,)),1(()( /1 1 2∑ = +−= h j t hjhtrhRV (1) where h denotes time between two consecutive observations. The daily volatility 2tσ of a financial instrument is defined as the conditional variance of its daily return given the set of information 1−Ωt available on day 1−t , i.e. Liquidity and Market Microstructure Noise: Evidence from the Pekao Data 7 ).|))|((( 1 2 1 2 −− ΩΩ−= ttttt REREσ (2) Thus the volatility is an unobservable variable. The realized variance (1) is a possible estimator of it. In the following discussion we assume that )(tY is described by the follow- ing stochastic differential equation ).()()()( tdWtdtttdY σμ += (3) Here )(tW denotes a Brownian motion, )(tσ is an instantaneous volatility and )(tμ is a drift function. In such a framework an ideal ex post measure of the daily volatility 2tσ is the integrated variance .)()( 1 2∫ − = t t duutIV σ (4) From the quadratic variation theory it follows that .0 if ,)()( 1 2 →→ ∫ − hduuhRV t t t σ (5) It means that in absence of market microstructure noise the realized variance is a consistent estimator of the integrated variance. Following Aït-Sahalia and Yu (2009), we assume that the observed price tX is a sum of the “true price” tY and the microstructure noise tε : ,ttt YX ε+= (6) and we are interested in determining the daily volatility 2tσ of the tY basing on discrete observations obtained in moments Tn =ΔΔ ,,,0 … . The model given by (6) is deep-rooted in the market microstructure theory. Many authors consider the noise tε as a result of bid-ask spread (Roll, 1984; Huang Stoll, 1996), transaction costs (Huang, Stoll, 1996; Chan, Lakonishok, 1997), discrete price changes (Gottlieb, Kalay, 1985). Manganelli (2005) and Aït-Sahalia and Yu (2009) associate the noise with the low liquidity level. The framework of the presented investigation is based on the Hasbrouck (1993) model according to which the standard deviation of tε is a total measure of the market quality. In the following empirical analysis our main goal is to separate the micro- structure noise from the fundamental price and evaluate the share of noise in observed values of the daily realized variance. We can use this result to deter- mine the frequency for intraday returns allowing to minimize bias in the rea- lized variance estimates. Moreover, we try to discover the dependencies be- Małgorzata Doman 8 tween liquidity and microstructure noise by modeling dependence of the later on a variety of liquidity measures. From now on we assume that the conditional mean of the return process is equal to 0. It means that (3) reduces to ( ) ( ) ( ).σ=dY t t dW t (7) Aït-Sahalia, Mykland and Zhang (2005) showed that in the parametric case this model is equivalent to that with constant σ . If 0=tε , i.e. if no microstruc- ture noise is present, the observed log returns 1− −= ii XXri ττ are i.i.d. ).,0( 2ΔσN The daily realized volatility is then the maximum likelihood estima- tor for 2σ and ).2,0())(( 42 Δ⎯⎯ →⎯−Δ ∞→ σσ NRVT n (8) In such a case the best estimates of volatility are obtained for the smallest poss- ible Δ (Aït-Sahalia, Yu, 2009; Aït-Sahalia, Mykland, Zhang, 2005). The situation changes in presence of the microstructure noise. Assume now that the noise tε is i.i.d. with mean 0 and variance a. Thus the observed log- returns process is MA(1) ,)( 111 iii uuWWYYr iiiiii ηεεσ ττττττ +≡−+−=−= −−− (9) with ),0.(i.i.d~ 2γiu , 22 2)var( ari +Δ=σ and 2)cov( ari −= . The above dependencies form a theoretical framework for the empirical analysis presented in Section 4. 3. The Data We consider the Polish bank Pekao S.A. stock returns. The period under scrutiny is from August 8, 2006 to February 13, 2009. The tick-by-tick data are provided by Stooq.pl. Table 1. Number of observations in the considered frequencies Type of observations Number of observations transactions 361 314 tick by tick 325 177 5 minute 53 520 10 minute 27 160 daily 629 The analysis was performed for 10, 5, 2 and 1 minute observations and for the duration returns which are calculated from transaction data. The time be- tween the as equal t For th turns. Tab considere sented in Figure 1. D 4. Empi The s volatility structure noise in noise to s NS Noise what degr structure Yu, 2009 friction. The n crostructu types of r ta Liquidity and M e closing of t to 0. he sake of pl ble 1 contain ed frequencie Figure 1. Daily returns o rical Resu steps of the of the funda noise tε fo the daily re signal ratio (N var(sig var(no SR = e to signal ra ree the obse theory it is 9) because in next part of t ure noise on regressions. T 0 1 1−= + +tc c x Market Microstr the stock ex lace, we show ns the inform es. The plot of Pekao S.A. lts presented a amental price or each cons ealized volat NSR) from th . )gnal oise) atio is a mea rved signal h s often used n some sense the investiga liquidity. T The first one ,ν+ t ructure Noise: change and w here only mation about showing the . Period: Augu analysis are e process tX idered day t ility estimat he following asure commo has been cor d as a mark e it allows to ation deals w o determine e is of the for Evidence from its opening n the results fo the number e dynamics o ust 8, 2006 to as follows. and the var t. To evalua tes we calcu g formula only used in rrupted by no ket quality m o evaluate th with depende the possible rm the Pekao Data next day is c or 5 and 10 m of observati of daily retur February 13, First we est riance ta of t ate the partiti ulate for eac science to q oise. In mark measure (Aï he level of t ence of the m e linkages w a 9 considered minute re- ions in the rns is pre- 2009 timate the the micro- ion of the ch day the (10) quantify to ket micro- ït-Sahalia, the market market mi- we run two (11) 10 and captu variance NS allows us quidity. T (V), the d during a d The e presented contains m noise and 5 minute result is s 2009). Table 2. M ta tσ NSR Figure 2. ures the imp ta . The seco 0 1SR = +t c c x s to establish The consider daily mean tr day (DTN). estimates of t d in Figure 2 mean values d noise to si returns. The similar to th Mean and stand The realized (black line) r M act of liquid ond one, 1 ,ν− +t tx h the connec red liquidity ransaction vo the realized v . Figure 3 sh and standar ignal ratio. e mean level hat observed dard deviation Frequency mean standard devia mean standard devia mean standard devia d volatility esti returns Małgorzata Dom dity (measure ctions betwee measures ar olume (DMT volatility bas hows the plo rd deviations The lowest l of noise to in develope n of noise and y ation ation ation imates based man ed by a vari en the noise re logarithm TV), and the n sed on 5 and ot of corresp of the daily t values of n signal ratio ed stock mar d realized vola 5 min 0.1123 0.0953 2.4584 1.4047 0.3381 0.2784 on 10 minute iable tx ) on to signal rat ms of the dail number of tra d 10 minute r ponding nois y volatility, v noise are ob is about 3/1 rkets (Aït-Sa atility 1 3 0 3 0 4 2 7 1 1 0 4 0 (grey line) an n the noise (12) tio and li- ly volume ansactions returns are e. Table 2 variance of btained for 3 and this ahalia, Yu, 10 min 0.1468 0.1408 2.2690 1.3096 0.3314 0.2977 nd 5 minute Liquidity and Market Microstructure Noise: Evidence from the Pekao Data 11 Figure 3. The market microstructure noise estimates for the realized volatility esti- mates based on 10 minute (grey line) and 5 minute (black line) returns The results of analysis on the connections between the microstructure noise and liquidity are presented in Table 3. Table 3. Parameter estimates for regressions (11) and (12) ta Explanatory variable 5 min 10 min 1c 2R 1c 2R Log(DNT) 0.012 (0.008) 0.01 0.042 (0.011) 0.03 Log(MDTV) -0.005 (0.009) 0.004 0.004 (0.0129) 0.001 Log(V) 0.006 (0.006) 0.002 0.0278 (0.008) 0.02 NSR Explanatory variable 5 min 10 min 1c 2R 1c 2R Log(DNT) -0.095 (0.019) 0.1 -0.066 (0.021) 0.02 Log(MVTD) -0.037 (0.028) 0.03 0.017 (0.030) 0.0005 Log(V) -0.072 (0.015) 0.03 -0.037 (0.016) 0.01 Surprisingly, the obtained estimates show rather weak connections between the both measures of the noise level and the considered liquidity measures. In the case of 10 minute returns there exists a positive and significant, though not very strong, dependence of the strength of noise and the number of transac- tions during a day, and the transaction volume. The expectations were that these dependencies should be negative (the higher liquidity, the lower noise). As con- 0,0 0,2 0,4 0,6 0,8 1,0 1,2 1 61 121 181 241 301 361 421 481 541 601 Małgorzata Doman 12 cerns the noise to signal ratio, a significant negative dependence on the daily number of transaction is in agreement with our early conjecture, but the results concerning the remaining liquidity measures are unexpected. It seems that in the case of Pekao S.A. the measures based on trading volume are not good liquidi- ty measures. Some explanation of this fact can be derived from the plots in Fig- ures 4–5, which show a typical dynamics of returns in days with high and low level of the noise. During the days with high noise to signal ratio the tick-by- tick returns exhibit a very regular pattern caused probably by market makers activity. The high values of volume are presumably connected with this spu- rious trade. On the other hand, during the days with the noise to signal ratio close to zero the dynamics of the returns is irregular and strong, which is cha- racteristic for the days with high activity of uninformed traders. So, the conclu- sion is that in the case of analyzed equities the microstructure noise is to a large extent connected with the market makers activity. Figure 4. Tick-by-tick returns with the noise to signal ratio equal to 0.91 observed on May 13, 2008 Figure 5. Tick-by-tick returns with the noise to signal ratio equal to 0.03 observed on September 7, 2006 -0,5 -0,3 -0,1 0,1 0,3 0,5 1 41 81 121 161 201 241 281 321 361 401 441 481 521 561 -0,5 -0,3 -0,1 0,1 0,3 0,5 1 41 81 121 161 201 241 281 Liquidity and Market Microstructure Noise: Evidence from the Pekao Data 13 5. Conclusions Due to the availability of ultra-high frequency data, a continuous-time ap- proach to modeling the stock markets dynamics is still becoming more popular. In fact, many of fruitful research areas in financial econometrics are based on this methodology and use the realized variance as an estimator of true volatility. The daily realized variance is calculated as a sum of squared intraday returns. However, the estimates of volatility obtained in such a way are usually biased due to the presence of the market microstructure noise in the observed data. The market microstructure effects include all the phenomena connected with the reality of the trade that usually contradict the continuous-time model assump- tions. In the paper we considered the quotations of the Polish stock company Pe- kao S.A. and attempted to separate the market microstructure noise from the observed daily realized variance process. Our main findings are as follows. The best volatility estimates are obtained for 5-minute returns. The market micro- structure noise is to a large extent connected with market makers activity. The analyzed liquidity measures (volume, mean volume of transaction, number of transactions during a day) poorly explain the market microstructure noise. The mean level of the noise to signal ratio in the case of the Pekao data is com- parable to that observed in developed markets. This result seems to support the opinion about good quality of market regulations and procedures on the Warsaw Stock Exchange. References Aït-Sahalia, Y., Yu, J. (2009), High Frequency Market Microstructure Noise Estimates and Liquidity Measures, Annals of Applied Statistics, 3, 422–457. Aït-Sahalia, Y., Mykland, P. A., Zhang, L. (2005), How Often to Sample a Continuous-Time Process in the Presence of Market Microstructure Noise, Review of Financial Studies 18(2), 351–416. Andersen, T. G., Bollerslev, T. (1998), Answering the Skeptics: Yes, Standard Volatility Models Do Provide Accurate Forecasts, International Economic Review, 39, 885–905. Andersen, T. G., Bollerslev, T., Diebold, F. X., Ebens, H. (2001), The Distribution of Realized Stock Return Volatility, Journal of Financial Economics, 61, 43–76. Barndorff-Nielsen, O. E., Shephard, N. (2002), Econometric Analysis of Realised Volatility and Its Use in Estimating Stochastic Volatility Models, Journal of the Royal Statistical Society, 64, Series B, 253–280. Chan, L., Lakonishok, J. (1997), Institutional Equity Trading Costs: NYSE Versus Nasdaq, Jour- nal of Finance, 52, 713–735. Gottlieb, G., Kalay, A. (1985), Implications of the Discreteness of Observed Stock Prices, Jour- nal of Finance, 40, 135–153. Hasbrouck, J. (1993), Assessing the Quality of a Security Market: A New Approach to Transac- tion Cost Measurement, Review of Financial Studies, 6, 191–212. Huang, R., Stoll, H. (1996), Dealer Versus Auction Markets: A Paired Comparison of Execution Costs on NASDAQ and the NYSE, Journal of Financial Economics,41 (3), 313–357. Manganelli, S. (2005), Duration, Volume and Volatility Impact of Trades, Journal of Financial Markets, 8, 377–399. Małgorzata Doman 14 Roll, R. (1984), A Simple Model of the Implicit Bid–Ask Spread in an Efficient Market, Journal of Finance, 39, 1127–1139. Tsay, R. S. (2002), Analysis of Financial Time Series, Wiley Series in Probability and Statistics, John Wiley& Sons, New York. Płynność a szum mikrostruktury rynku na przykładzie notowań spółki Pekao Z a r y s t r e ś c i. Dostępność danych giełdowych o bardzo wysokiej częstotliwości stanowi argument za stosowaniem do opisu dynamiki cen akcji modeli z czasem ciągłym. Jednak dane takie zawierają oprócz informacji na temat procesu ceny także szum mikrostruktury rynku, które- go obecność powoduje obciążenie oszacowań zmienności. Szum ten jest związany z rzeczywi- stymi warunkami, w jakich odbywa się handel. W pracy dokonano oszacowania szumu mikro- struktury rynku w zmienności zrealizowanej cen akcji spółki Pekao SA oraz wyliczono stosunek sygnału do szumu. Wyniki badań wskazują, że optymalna częstotliwość wyliczania stóp zwrotu przy wyznaczaniu zmienności zrealizowanej to częstotliwość pięciominutowa, a obserwowany stosunek sygnału do szumu jest na poziomie zbliżonym do obserwowanego na rozwiniętych rynkach giełdowych. Ponadto, przeprowadzona została analiza powiązań pomiędzy wybranymi miarami płynności a poziomem szumu mikrostruktury rynku. S ł o w a k l u c z o w e: mikrostruktura rynku, zmienność, wariancja zrealizowana, płynność, rynek giełdowy, wolumen obrotu, dane wysokiej częstotliwości.