A Simple Way to Estimate Bid-Ask Spreads from Daily High and Low Prices

Transcription

1 A Simple Way o Esimae Bid-Ask Spreads from Daily High and Low Prices Shane A. Corwin and Paul Schulz * January 010 * Boh auhors are from he Mendoza College of Business a he Universiy of Nore Dame. We are graeful o seminar paricipans a he Universiy of Nore Dame and he Naional Bureau of Economic Research Marke Microsrucure meeing, and o Shmuel Baruch, Rober Baalio, Hank Bessembinder, Ryan Davies, Larry Harris, Joel Hasbrouck and Asani Sarkar for useful commens.

2 A Simple Way o Esimae Bid-Ask Spreads from Daily High and Low Prices Absrac We develop a new mehod for esimaing bid-ask spreads from daily high and low prices. Daily high (low) prices are almos always buy (sell) orders. Hence he high-low price raio reflecs boh he sock s variance and is bid-ask spread. Furher, he variance componen of he high-low raio is proporional o he reurn inerval, while he bid-ask spread componen is no. This allows us o derive a spread esimaor as a funcion of high-low raios over one-day and wo-day inervals. Through simulaions and comparisons o TAQ daa, we show ha he esimaor is accurae and dominaes oher spread esimaes from daily daa.

3 In his paper, we derive a simple way o esimae bid-ask spreads from daily high and low prices. The esimaor is based on wo unconroversial ideas. Firs, daily high prices are almos always buy orders and daily low prices are almos always sell orders. Hence he raio of high-o-low prices for a day reflecs boh he fundamenal volailiy of he sock and is bid-ask spread. Second, he componen of he high-olow price raio ha is due o volailiy increases proporionaely wih he lengh of he rading inerval, while he componen due o bid-ask spreads is consan over differen rading inervals. This implies ha he sum of he price ranges over wo consecuive single days reflecs wo days volailiy and wice he spread, while he price range over one wo-day period reflecs wo days volailiy and one spread. This allows us o derive an esimae of a sock s bid-ask spread as a funcion of he high-o-low price raio for a single wo-day period and he high-o-low raios for wo consecuive single days. Simulaions reveal ha under realisic condiions, he correlaion beween high-low spread esimaes and rue spreads is abou 0.9 and he sandard deviaion of high-low spread esimaes is one-fourh o one-half as large as he sandard deviaion of esimaes from he popular Roll (1984) covariance spread esimaor. Our spread esimaor should prove useful o researchers in a wide variey of applicaions. Even wih inraday daa now widely available, researchers make frequen use of he covariance esimaor of Roll (1984) or is exensions in applicaions ranging from asse pricing, o corporae finance, o ess of efficien markes. In some cases, his is because he researcher is sudying a ime period ha predaes inraday daa or inernaional markes wihou inraday daa. 1 In oher cases, hese measures are used when inraday quoes and rades canno be reliably mached. Oher low-frequency spread measures based on he frequency of zero reurns are developed in Lesmond, Ogden, and Trzcinka (1999) and have been 1 See Bessembinder and Kalcheva (008), Bharah, Pasquariello, and Wu (008), Gehrig and Fohlin (006), Kim, Lin, Singh, and Yu (007, Lesmond, Schill and Zhou (004), or Lipson and Moral (007) for esimaes of spreads for periods before he availabiliy of inraday daa. See Amihud, Lauerbach, and Mendelson (00), Chakrabari, Huang, Jayaraman, and Lee (005) or Griffin, Kelly, and Nardari (007) for examples of he applicaion of Roll spread esimaors o inernaional daa. See Anunovich and Sarkar (006), Fink, Fink, and Weson (006), or Schulz (000). 1

4 applied in a number of recen papers. 3 The high-low spread esimaor derived here has a number of advanages over he daily esimaors used in prior research. Firs, using TAQ daa from , we show ha i is much more accurae han he sill popular Roll (1984) covariance esimaor or he LOT esimaor of Lesmond, Ogden, and Trzcinka (1999). Anoher advanage is ha i is easy o use. We provide a closed-form soluion for he spread which can be easily programmed, unlike measures ha require an ieraive process (Hasbrouck (006)) or maximum likelihood esimaion (LOT). Third, unlike Hasbrouck s (006) Gibbs esimaor or Holden (009) measure, he high-low spread esimaor is no compuer-ime inensive, making i ideal for large samples. Finally, he high-low spread esimaor is derived under very general condiions. I is no ad-hoc and does no depend on insiuional quirks of a paricular marke for is accuracy. We es he accuracy of high-low spread esimaes by comparing hem wih monhly effecive spreads from TAQ from 1993 hrough 006. For comparison purposes, we also esimae spreads from daily daa using he covariance spread esimaor of Roll (1984), he effecive ick esimaor of Holden (009), and he LOT measure of Lesmond e al. (1999). Because researchers ackling differen problems may care abou differen characerisics of he spread esimaor, we provide several differen ess of accuracy. We firs examine he performance of he various spread esimaors in he pooled sample of ime-series and cross-secional observaions. The resuls sugges ha he high-low spread esimaor is very accurae and dominaes he alernaive spread esimaors. Across all sock-monhs, he correlaion beween TAQ effecive spreads and high-low spreads is The comparable correlaions for he Roll spread, he effecive ick spread, and he LOT measure are 0.707, 0.734, and 0.699, respecively. We nex calculae cross-secional correlaions beween spread esimaes and TAQ effecive spreads on a monh-by-monh basis from 1993 hrough 006. Examining cross-secional correlaions 3 See Bekaer, Harvey, and Lundblad (007), Lesmond, Schill, and Zhou (004), Mei, Scheinkman, and Xiong (005) for applicaions of he LOT measure. Amihud (00) and Pásor and Sambaugh (003) provide low frequency measures ha aemp o capure liquidiy more generally. These measures end o be highly correlaed wih low frequency spread esimaes bu incorporae boh spreads and he price impac of rades.

5 serves wo purposes. Firs, in many cases, researchers care abou he abiliy of he spread esimaor o capure he cross-secional disribuion of spreads. Second, looking a cross-secional correlaions on a monh-by-monh basis allows us o examine he performance of he esimaors in differen rading environmens. The hree subperiods ha we examine, , , and , correspond closely o periods when he minimum ick size in U.S. markes was one-eighh, one sixeenh, and one penny, respecively. In all subperiods, cross-secional correlaions beween high-low spreads and TAQ effecive spreads are higher han cross-secional correlaions beween TAQ effecive spreads and any of he oher esimaors. As addiional evidence, we examine cross-secional correlaions in monhly spread changes. For he enire period, he high-low spread esimaor dominaes wih an average cross-secional correlaion of 0.47, compared o 0.49 for he Roll spread, for he effecive ick spread, and for he LOT measure. Thus, he high-low spread esimaor ouperforms he alernaive measures in capuring he cross-secions of boh spread levels and monh-o-monh changes in spreads. Noably, he high-low spread esimaor performs paricularly well during he subperiod when he ick size was one-eighh, suggesing ha i should perform well during earlier ime periods when inraday daa were no available. Finally, we calculae sock-by-sock ime-series correlaions beween each of he spread esimaors and TAQ effecive spreads. This analysis serves wo purposes. Firs, for some applicaions, researchers may be paricularly ineresed in he abiliy of he esimaor o capure he ime-series of spreads. Second, his allows us o see how well he esimaors perform for differen ypes of socks. For all size quiniles and all exchange lisings, we find ha high-low spreads have much higher average imeseries correlaions wih TAQ effecive spreads han do Roll spreads or spreads esimaed from he LOT measure. For he grea majoriy of socks, high-low spreads also have higher ime-series correlaions wih TAQ effecive spreads han do effecive ick spreads. Effecive ick spread esimaes have higher correlaions for he larges socks. 3

6 The high-low spread esimaor is derived under very general condiions. I is simple, accurae, and easy o use. Boh simulaions and comparisons o TAQ daa sugges ha, for mos applicaions, he high-low spread esimaor is more accurae han alernaive low-frequency spread esimaors. I clearly dominaes oher low-frequency measures in capuring he cross-secions of bid-ask spreads and monh-omonh changes in spreads. For he vas majoriy of socks, i also dominaes oher measures in capuring he ime-series variaion in individual sock spreads. I is imporan o noe ha he high-low spread esimaor capures liquidiy more broadly han jus he bid-ask spread. Price pressure from large orders will ofen lead o execuion a daily high or low prices. Likewise, a succession of buy or sell orders in a shallow marke may resul in execuions a daily high or low prices. The high-low spread esimaor capures hese ransiory price effecs in addiion o he bid-ask spread. The remainder of he paper is organized as follows. The high-low spread esimaor is derived in Secion 1. Secion discusses pracical issues in esimaing spreads using high and low prices. Secion 3 discusses exising spread esimaors ha use daily daa and reviews empirical ess of hese esimaors. We presen simulaion resuls for he high-low spread esimaor in Secion 4. In Secion 5, spread esimaes from he high-low spread esimaor are compared wih TAQ effecive spreads and wih esimaes based on he Roll spread, he effecive ick spread, and he LOT measure. Secion 6 concludes. 1. A New Class of Spread Esimaor: The High-Low Price Esimaor The high-low spread esimaor is based on a simple insigh. The high-low price raio reflecs boh he rue variance of he sock price and he bid-ask spread. While he variance componen grows proporionaely wih he ime period, he spread componen does no. This allows us o solve for boh he spread and variance by deriving wo equaions: he firs a funcion of he high-low raios on wo consecuive single days and he second a funcion of he high-low raio from a single wo-day period. 4,5 4 Blume and Sambaugh (1983) show ha bid-ask bounce in closing prices leads o an upward bias in esimaes of mean reurns. The bias increases in he size of he spread. Using buy-and-hold reurns eliminaes he problem. 5 We could also esimae spreads by comparing one single-day high-low raio wih one wo-day high-low raio. However, using he sum of wo single days produces a more powerful esimae. 4

7 We assume ha he rue or acual value of he sock price follows a diffusion process. We also assume ha here is a spread of S%, which is consan over he wo-day esimaion period. Because of he spread, observed prices for buy orders are higher han he acual values by (S/)%, while observed prices for sell orders are lower han he acual value by (S/)%. We assume furher ha he high price of he day is a buy order and is herefore grossed up by half of he spread, while he low price of he day is a sell order and is discouned by one half of he spread. Hence he observed high-low price range conains boh he range of he acual prices and he bid-ask spread. Wih H A (L A ) as he acual high (low) sock price on day and H O ( L O ) as he observed high (low) sock price for day, we can wrie (1) ln( / ) ln ( / ) ( / ). H L H S L S O O A A 1 1 Rearranging (1) gives () ln / ln ln ln ln. H L H L H L S S S S O O A A A A This equaion can be simplified by noing ha he naural log of he raio of high o low prices ha appears as he firs erm in () is proporional o he sock s variance. Specifically, under he assumpions ha sock prices follow he usual geomeric Brownian moion and he price is observed coninuously, Parkinson (1980) and Garman and Klass (1980) show ha (3) E T H L k T HL ln, where H (L ) is he high (low) on day and k 1 = 4ln(). 6 Similarly, Parkinson (1980) shows ha (4) E T H L k where k T HL ln,. 6 Using a sample of 08 socks over 9 quarers from January 1973 hrough March 1980, Beckers (1983) demonsraes ha he high-low variance esimaor is more accurae han he radiional variance esimaor based on closing prices. See Gallan, Hsu, and Tauchen (1999) for an applicaion of high-low volailiy esimaors o he esimaion of sochasic volailiy. 5

8 Taking expecaions of () and subsiuing from (3) and (4) yields (5) E H L k k S S S S o O HL HL ln ln ln. 1 The expecaion of he sum of (5) over wo single days is (6) E H L k k S S S S j o j O j HL HL ln ln ln To simplify he noaion going forward, we se (7) ln, ln 0 1 S S E H L j O j O j This allows us o rewrie (6) as (8) k k HL HL. Equaion (8) links he high-low price raios on wo consecuive single days wih wo unknowns: α and σ. To solve for hese unknowns, we define a second equaion ha links he high-low raio from he wo-day period and he same wo unknowns. Squaring he log price range over a wo-day period yields (9) ln ln ln ln ln,,,,,,, H L H L H L S S S S O O A A A A where H,+1 is he high price over he wo days and +1 and L,+1 is he low price over days and +1. To furher simplify he noaion we se (10) ln.,, H L O O 1 1 Using his noaion and aking expecaions in (9) yields (11) 0 1 k k HL HL. This leaves wo equaions, (8) and (11), and wo unknowns, σ and α. Because he spread is posiive, α mus also be posiive. Hence we choose he posiive roo for α. 6

9 k k k HL HL 1. Subsiuing from (1) ino (11) and rearranging yields k k1 k HL k k1 HL 0. (1) (13) Equaion (13) can be easily solved numerically for σ and he resul insered ino (1) o obain a value for α. A simple ransformaion of α in (7) hen provides he high-low spread esimae: S e 1 1 e. (14) A furher simplifying assumpion allows us o obain closed form soluions for σ and α. If we ignore Jensen s inequaliy in (4) and assume ha T T 1 H 1 H E ln E ln k1 HL k1 HL, T 1 L T 1 L (15) hen k k1 and (1) and (13) become k 3 HL k k 0. (1') (13') Rearranging yields HL HL k k 3 ( 3 ) 0. (16) Solving for σ and using he posiive roo o insure a posiive esimae yields HL k k 3 k 3. (17) Insering he sandard deviaion from (17) ino (1') provides an esimae of α: 7

10 3 3. (18) This closed form soluion for α can be insered ino (14) as before o yield our simple high-low spread esimaor. The spread esimaor given in (14) is easy o compue and does no require he researcher o ierae hrough successive esimaes of he spread o ge he correc value. 7 Insead, he procedure we ouline above produces an esimae of he spread and an esimae of he daily sandard deviaion using only he high and low prices from wo consecuive days. To ge spreads for longer periods like a monh, we average he spread esimaes from all overlapping wo-day periods wihin he monh. One noe of cauion is needed here. In esimaing spreads and variances, we use he observed raio of high o low prices, while he esimaor is derived using he expeced raio. Because he variance and he spread are non-linear funcions of he high-lo price raio, an average of spread esimaes is no an unbiased esimae of he spread. However, boh our simulaion resuls and empirical analysis sugges ha his is no a problem in pracice. 8. Using he High-Low Spread Esimaor in Pracice There are a number of implici assumpions underlying he high-low spread esimaor. One is ha he sock rades coninuously. Anoher is ha sock values do no change while he marke is closed. These assumpions are no rue, of course, raising some issues for he esimaion of high-low spreads in pracice..1 Adjusmen for Overnigh Price Changes Because markes are closed overnigh, he raio of high o low prices for he wo-day period reflecs boh he range of prices during each day and he overnigh reurn. On he oher hand, he wo 7 We have also esed spread esimaors derived from variance esimaors in Garman and Klass (1980) ha incorporae high, low, and closing prices. These esimaors are more complex bu fail o produce beer spread esimaes. 8 To address he imporance of his problem, we reesimaed he resuls based on an alernaive mehod for aggregaing daa a he monhly level. Specifically, we average he high-low raio parameers over he monh raher han averaging he spread esimaes over he monh. We find in boh our simulaions and empirical ess ha his mehod does no produce more accurae monhly spread esimaes. We are graeful o an anonymous referee for his suggesion. 8

11 single-day high-low raios reflec only he range of prices during rading hours. Though sock prices are more volaile during he rading day han a oher imes, sock prices ofen move significanly over nonrading periods (see French and Roll (1986) and Harris (1986)). This causes he high-low price raio (and hence variance) esimaed using one wo-day period o be inflaed relaive o he variance esimaed using wo one-day periods. Wihou an adjusmen for overnigh reurns, he spread porion of he high-low price raio will herefore be underesimaed. To correc for overnigh reurns, we deermine wheher he close on day is ouside he range of prices for day +1 for every pair of consecuive rading days. If he day +1 low is above he day close, we assume he price rose overnigh from he close o he day +1 low and decrease boh he high and low for day +1 by he amoun of he overnigh change when calculaing spreads. Similarly, if he day +1 high is below he day close, we assume he price fell overnigh from he close o he day +1 high and increase he day +1 high and low prices by he amoun of his overnigh decrease. As an alernaive, we could adjus for overnigh reurns using he difference beween he day close price and he day +1 open price. There are hree reasons why we do no use his adjusmen. Firs, we wan o adjus only hose cases where he rue value changes overnigh. For many socks, he change from close o open is more likely o occur as a resul of bid-ask bounce han from an overnigh reurn. Second, a primary use of his esimaor is o esimae hisoric rading coss during periods when daa on open prices may no be available. For example, open prices are missing on CRSP from July, 196 hrough June, 199. Finally, we found a small number of cases where he open price was ouside he high-low price range repored by CRSP, suggesing ha open price daa may be unreliable... True High and Low Prices are no Observed for Infrequenly Traded Socks High and low prices are observed rade prices. Garman and Klass (1980) noe ha if a sock rades infrequenly, he observed high price will be lower han he rue high price for he day and he observed low price will be greaer han he rue low price for he day. In pracice, i seems likely ha he probabiliy 9

12 of a rade will be especially high when prices are near heir high and low values for he day. Infrequen rading is clearly a problem if a sock rades only once during a day or, more generally, if all rades occur a he same price. In such cases, if he rade price is wihin he previous days price range, we assume he same high and low prices as he previous day. In hose less common cases where he high and low are equal, bu a a price ouside he previous day s range, we use he same dollar range as he previous day assuming he high and low are increased or decreased by he amoun he price lies ouside he previous day s high-low price range. When a sock does no rade a all during a day, CRSP liss closing bid and ask prices in place of high and low prices. In pracice, a researcher may benefi from using his informaion when esimaing spreads. However, o provide a fair comparison wih oher esimaors, we eliminae he bid and ask prices provided by CRSP in hese cases and replace hem wih he mos recen high and low rade prices available from a prior rading day..3 High-Low Spread Esimaes May Be Negaive The high-low spread esimaor assumes ha he expecaion of a sock s rue variance over a woday period is wice as large as he expecaion of he variance over a single day. Even if his is rue in expecaion however, he observed wo-day variance may be more han wice as large as he single day variance during volaile periods, in cases wih a large overnigh price change, or when he oal reurn over he wo consecuive days is large relaive o he inraday volailiy. If he observed wo-day variance is large enough, he high-low spread esimae will be negaive. For mos of he analysis o follow, we se all negaive wo-day spreads o zero before calculaing monhly averages. As described in more deail below, his produces more accurae monhly spread esimaes han eiher including or deleing negaive wo-day spread observaions. 3. Oher Classes of Spread Esimaors ha Use Daily Daa To our knowledge, his is he firs use of high and low prices o esimae rading coss. Researchers have derived several oher classes of spread esimaors based on daily daa. We describe several of hese 10

13 alernaive esimaors below. 3.1 Spread Esimaors Derived from Reurn Covariances Roll (1984) assumes ha he rue value of a sock follows a random walk and ha P, he observed closing price on day, is equal o he sock s rue value plus or minus half of he effecive spread. Under hese condiions, he auocorrelaion of reurns from observed prices will be negaive and Roll derives he following simple esimaor for he spread: S Cov( P, P 1) (19) Roll s measure is simple, inuiive, and easy o compue. I provides accurae spread esimaes wih inraday daa if a researcher has rade prices bu no quoes (Schulz (000)). Even wih a long ime-series of daily daa hough, he covariance of price changes is frequenly posiive, forcing he researcher o arbirarily conver an imaginary number ino a spread esimae. In fac, Roll (1984) finds ha crosssecional average covariances are posiive for some enire years. In hese cases, researchers usually do one of hree hings: 1) rea he observaion as missing, ) se he Roll spread esimae o zero, or 3) muliply he covariance by negaive one, esimae he spread, and muliply he spread by negaive one o produce a negaive spread esimae. Harris (1990) examines he small-sample properies of he Roll esimaor. He demonsraes ha he esimaor is noisy even in relaively large samples and shows ha he large number of posiive auocovariance esimaes is no surprising given he level of noise. He also shows ha as a resul of Jensen s inequaliy, spread esimaes are significanly downward biased. Researchers have proposed and esed a number of refinemens o he Roll esimaor. George, Kaul, and Nimalendran (1991) noe ha he Roll esimaor is downward biased if expeced reurns are ime-varying and hence posiively auocorrelaed. They propose using a covariance esimaor ha is based on he residual of he regression of a sock s reurn on a measure of is expeced reurn. Holden (009) noes ha when a sock does no rade for a day, CRSP records he midpoin of is bid-ask range as is 11

14 closing price. He proposes a revised version of he Roll esimaor in which he covariance of price changes is divided by he percenage of days wih rading. Hasbrouck (004, 006) uses a Gibbs sampler and Bayesian esimaion o improve he simple Roll esimaor. As in Roll (1984), price changes are assumed o occur as a resul of new, serially uncorrelaed informaion, and as a resul of shifs beween bid and ask prices. The Gibbs esimaor hen uses informaion in he series of prices o assign a poserior probabiliy ha each specific rade is a buy or sell order. Hasbrouck (006) noes ha Gibbs esimaes of annual effecive spreads are more accurae han spreads esimaed wih he basic Roll esimaor, bu ha he procedure is compuaionally inensive. 3.. Spread Esimaors Derived from Transacion Price Tick Size The effecive ick esimaor, developed in Holden (009) and Goyenko e al. (009), is based on he idea ha wider spreads are associaed wih larger effecive ick sizes. For example, heir model assumes ha when boh he ick size and he bid-ask spread are one eighh, all possible prices are used, bu when he ick size is one eighh and he spread is one quarer, only prices ending on even-eighhs, or quarers, are used. Chrisie and Schulz (1994) documen a very srong relaion beween effecive ick size and bid-ask spreads for Nasdaq socks in he early 1990's, bu he relaion is much weaker for NYSE socks. Goyenko e al. (009) show ha heir assumed relaion beween spreads and he effecive ick size allows researchers o use price clusering o infer spreads. Suppose ha here are four possible bid-ask spreads for a sock: $1/8, $1/4, $1/ and $1. The number of quoes wih odd-eighh price fracions, associaed only wih $1/8 spreads is given by N 1. The number of quoes wih odd-quarer fracions, which occur wih spreads of eiher $1/8 or $1/4, is N. The number of quoes wih odd-half fracions, which can be due o spreads of $1/8, $1/4, or $1/, is N 3. Finally, he number of whole-dollar quoes, which can occur wih any spread widh, is given by N 4. To calculae an effecive spread, he proporion of prices observed a each price fracion is calculaed as 1

15 F j N j N for j J J 1,...,. 1 j j (0) The unconsrained probabiliy of he j h spread (which corresponds o he j h price fracion), U j, occurring is given by F j 1 j U F F j,..., J j j j1 F F j J. j j1 (1) The effecive ick measure is a probabiliy-weighed average of all possible spreads. However, using unconsrained probabiliies can be problemaic. When he number of observed prices on finer incremens is high, he effecive ick esimaor s unconsrained probabiliy of a narrow spread can exceed one and he unconsrained probabiliy of a wider spread may be negaive. In he example above, if en prices were observed and six had odd-eighh price fracions, he unconsrained probabiliy of a one-eighh spread would be 1.. If one of he en prices had an odd-quarer fracion, he probabiliy of a one-quarer spread would be. -.6 = -.4. Holden (009) and Goyenko e al. (009) consrain he probabiliies of spreads esimaed by he effecive ick mehod o be non-negaive and consrain he probabiliy of an effecive spread o be no more han one minus he probabiliy of a finer spread, a pracice we also adop in our examinaion of he effecive ick esimaor. 9, Spread Esimaors Derived from he Frequency of Zero Reurns Lesmond, e al. (1999) develop an effecive spread esimaor (he LOT esimaor) based on he idea ha a sock s rue reurn is given by he marke model, bu he observed reurn is only differen from zero if he rue reurn exceeds he coss of rading. Wih α 1 < 0 as he cos of selling and α > 0 as he cos O of buying, he observed day sock reurn R is: 9 During decimal pricing, we assume he effecive ick can be 1, 5, 10, 5, 50, or $ Holden (009) derives a spread esimaor ha ness boh he Roll covariance spread esimaor and he effecive ick esimaor as special cases. The esimaor performs well, bu is compuaionally inensive. 13

16 O A R R if R m 1 1 O A R 0 if R 1 O A R R if R. m () Using his relaion beween rading coss and observed reurns, Lesmond e al. (1999) esimae rading coss by maximizing he likelihood funcion for a year of daily sock reurns wih respec o α 1, α, β and σ. The LOT esimae of he effecive spread is hen defined as α - α Simulaion Resuls for he High-Low Spread Esimaor To see how well he high-low spread esimaor works under differen condiions, we simulae 10,000 monhs of sock daa. Each monh conains 1 days and each day has 390 minues. A he beginning of each monh, he sock price is arbirarily se o $100. Then for each minue of each day, m, he rue value of he sock price, P m, is simulaed as P P e m where σ is he sock sandard deviaion per minue and x is a random drawing from a uni normal m1 x, (3) disribuion. The bid price for each minue is obained by muliplying P m by one minus half he bid-ask spread and he ask price is obained by muliplying P m by one plus half he bid-ask spread. We assume a 50% chance ha he observed price a minue m is a bid, and a 50% chance i is an ask. The high and low for he day are he highes and lowes observed prices, respecively, wheher a bid or ask. The closing price for he day is he observed price for minue The Disribuion of Simulaed Spread Esimaes We firs examine he performance of he high-low spread esimaor under he near ideal condiions of no overnigh reurn and prices observed every minue. For comparison, we also presen simulaed resuls for he Roll spread esimaor. For hese simulaions, we assume ha he rue value for he firs minue of he day is equal o he rue value for he las minue of he previous day. We assume he daily sandard deviaion of reurns is 3% and repea he simulaions for spreads of 0.5%, 1%, 3%, 5%, and 8%. Resuls are shown in Panel A of Table I. 14

17 Column 1 repors simulaion resuls for he simple closed-form high-low spread esimaor defined in equaions (18) and (14). To esimae monhly spreads, we esimae spreads separaely for each wo-day period and calculae he average across all overlapping wo-day periods in he monh. For his spread esimaor, he mean esimae across 10,000 simulaions is very close o he assumed spread regardless of he spread widh. For example, he mean esimae from he simple high-low spread esimaor is 7.84% when he rue spread is 8% and is.9% when he rue spread is 3%. Regardless of he size of he rue spread, he sandard deviaion of he high-low spread esimaes is around 0.6%. For spreads of 3% or more, none of he monhly high-low spread esimaes are negaive. However, for spreads of 0.5%, almos 0% of monhly high-low spread esimaes are negaive. The nex wo columns repor simulaion resuls when ad-hoc adjusmens are made for negaive spread esimaes. The firs adjusmen, shown in he second column, ses all negaive wo-day esimaes o zero before aking he monhly average. Under hese near ideal condiions, wih no overnigh reurns and almos coninuous observaion of prices, his adjusmen produces spread esimaes ha are oo large for spreads of 0.5% o 3%. For wider spreads, seing negaive wo-day spreads o zero before aking monhly averages leads o a sligh improvemen in he average spread esimaes. The second adjusmen, shown in he following column, includes negaive wo-day spread esimaes in he monhly average, bu ses negaive monhly spread esimaes o zero. Using his alernaive adjusmen produces spread esimaes ha are comparable o hose from he simple unadjused high-low spread esimaor. Thus, under hese near ideal condiions, i is no worhwhile o adjus for negaive esimaes when examining means. The fourh column of Panel A provides he performance of he high-low spread esimaor wih he adjusmen for Jensen s inequaliy in equaions (1), (13), and (14). The performance of his alernaive version of he esimaor is very similar o, bu slighly worse han he performance of he simple high-low spread esimaor. Under hese near ideal condiions, here is lile benefi from incorporaing he adjusmen for Jensen s inequaliy. 15

18 As noed above, he spread esimae is a non-linear funcion of he β and γ parameers. As a resul, averaging wo-day spread esimaes can produce a biased esimae of he spread. To address he imporance of his Jensen s inequaliy problem, he esimaors in he fifh and sixh columns ake a slighly differen approach o aggregaing daily high and low prices ino a monhly esimae. Raher han averaging spread esimaes across wo-day periods wihin he monh, we average he β and γ parameers across wo-day periods wihin he monh. The fifh column repors resuls when hese monhly parameer averages are plugged ino he simple high-low spread esimaor, while column six repors resuls from he more complicaed esimaor ha incorporaes an adjusmen for he oher Jensen s inequaliy complicaion - ha he square roo of he high-low variance esimae is no an unbiased esimae of he sandard deviaion. As he able shows, averaging he β and γ parameers over he monh produces less accurae spread esimaes and a larger proporion of non-posiive spreads han when wo-day spread esimaes are averaged over he monh. The las column provides simulaion resuls for he Roll spread esimaor. As described in Secion 3, we se he Roll spread o zero in cases where he serial covariance is posiive. The simulaion resuls show ha he Roll esimaor performs far worse han he high-low spread esimaor, especially when spreads are narrow. When he rue spread is 8%, he mean high-low spread esimae is 7.84% and he mean Roll esimae is 7.54%. More imporanly, he sandard deviaion of spread esimaes is for he simple high-low spread esimaor, compared o for he Roll esimaor. When he rue spread is 0.5%, he mean high-low spread esimae is 0.59% and he mean Roll esimae is 1.18%. Here again, he Roll esimaes have a much larger sandard deviaion (0.0137) han he high-low esimaes (0.006). 11 Under he near ideal condiions of hese simulaions, he simple version of he high-low spread esimaor appears o work bes, wihou adjusmens for negaive spreads or Jensen s inequaliy. Monhly spread esimaes are also more accurae when wo-day spread esimaes are averaged wihin he monh, as 11 Alhough no shown, we also esimae Roll spreads assuming spreads are negaive when he serial correlaion is posiive. This reduces he upward bias in he Roll spread, bu significanly increases he sandard deviaion of Roll spread esimaes. 16

19 opposed o using average parameer esimaes wihin he monh. The resuls also sugges ha he highlow spread esimaor performs significanly beer han he Roll covariance esimaor. There are several ways in which he above simulaion assumpions depar from marke realiies. We nex examine wo imporan complicaions ha may affec he performance of he spread esimaor. Firs, overnigh reurns affec he high-low raio for a wo-day period, bu do no affec he high-low raios for eiher of he single days. This leads o an underesimae of spreads. Second, wih infrequenly observed prices, he observed high and low price may no reflec he rue high and low, leading o a misesimaion of he rue spread. We simulae infrequen observaion of prices by assuming here is a 10% chance of observing a price a any given minue. This corresponds o an average of 39 rades per day - a realisic assumpion for mos Nasdaq and smaller NYSE socks. We simulae overnigh reurns ha are normally disribued wih zero mean and sandard deviaion equal o 0.5 imes he open-o-close sandard deviaion of reurns. 1 We hen adjus for he effecs of overnigh reurns based on he mehod described in Secion. Simulaion resuls incorporaing boh overnigh reurns and infrequenly observed prices are described in Panel B of Table I. Even afer incorporaing an adjusmen for overnigh reurns, mean spread esimaes decline for all versions of he high-low spread esimaor. The simple high-low spread esimaor provides mean spread esimaes of 6.65% when he rue spread is 8%, 3.69% when he rue spread is 5%, 0.05% when he rue spread is 1%, and -0.4% when he rue spread is 0.50%. In conras o Panel A, when prices are observed infrequenly and here are overnigh reurns, ad-hoc adjusmens for negaive spread esimaes make he esimaes more accurae. For spreads of 1% or greaer, seing negaive wo-day spreads o zero before aking monhly averages produces mean esimaes ha are much closer o he assumed spreads han he simple high-low spread esimaor wih no adjusmen for negaive spreads. Seing negaive wo-day spreads o zero before aking monhly averages also resuls in a much 1 Lockwood and Linn (1990) (Table I) esimae he average o be for he Dow Jones Indusrials over Oldfield and Rogalski (1980) (Table I) provide daa ha allows esimaion of he raio for five large individual socks for Ocober, December, The average raio across he five sock is

20 smaller sandard deviaion of spread esimaes han does he unadjused esimaor. As in Panel A, adjusing for negaive spreads a he daily level appears o work beer han adjusing for negaive monhly spreads, and here appears o be lile benefi o eiher incorporaing an adjusmen for Jensen s inequaliy or aking average parameer esimaes raher han average wo-day spreads wihin he monh. The las column of he able shows ha overnigh reurns and infrequenly observed prices have lile impac on he Roll spread esimaor. In Panel B, mean spread esimaes from he Roll esimaor are slighly closer o rue spreads han are high-low spread esimaes for spreads of 3% or greaer. However, he high-low spread esimaor wih negaive wo-day spreads se o zero provides beer mean esimaes han he Roll esimaor for spreads of 0.5% or 1%. More imporanly, he sandard deviaion of spread esimaes from he high-low spread esimaor wih negaive wo-day spreads se o zero is only one-half o one-fourh as large as he sandard deviaion of Roll spread esimaes. Even under hese unfavorable condiions, he high-low spread esimaor appears o ouperform he Roll esimaor. 4. The Cross-Secional Correlaion of Simulaed Spread Esimaes The simulaions described in Table I illusrae he accuracy of he high-low spread esimaor. Nex, we examine how well he differen versions of he esimaor capure he cross-secion of spreads under alernaive assumpions abou prices and spreads. Our simulaions again consis of 10,000 sockmonhs wih 1 days in each monh and 390 minues in each day, wih sock prices simulaed as in equaion (7). However, for each of he 10,000 sock monhs, we randomly assign a spread from a uniform disribuion wih a range from 0% o 6%. As in Table I, here is a 50% chance ha an observed price is a bid and a 50% chance ha i is an ask. Table II repors he correlaions of spread esimaes wih simulaed rue spreads across he 10,000 sock monhs, where he sandard deviaion of daily reurns is se o eiher 3% or 5%. The firs wo rows repor simulaions under he near-ideal condiions of no overnigh reurn and prices observed each minue. Under hese condiions, correlaions beween high-low spread esimaes and simulaed spreads are high for all versions of he high-low spread esimaor, ranging from 0.96 o when he daily reurn 18

21 sandard deviaion is 3%. When he sandard deviaion of daily reurns is 5%, he correlaion beween high-low spread esimaes and simulaed spreads ranges from 0.8 when β and γ parameers are averaged over he monh o when spreads are calculaed for wo-day periods and averaged over he monh wih negaive wo-day spreads se o zero. The correlaions beween Roll spread esimaes and simulaed spreads are much lower. When Roll spreads are se o zero for posiive serial correlaions, he correlaion is for a daily reurn sandard deviaion of 3% and for a sandard deviaion of 5%. However, while seing Roll spreads o zero is he common ad hoc adjusmen when he auocovariance is posiive, i is no clear a priori wheher his adjusmen increases or decreases he correlaion beween Roll spreads and simulaed spreads. As an alernaive, he las column repors correlaions beween Roll spreads and simulaed spreads when posiive serial correlaions are reaed as negaive spreads. Here, he correlaion beween Roll spreads and simulaed spreads drops o 0.54 when he daily sandard deviaion is 3% and o 0.97 when he sandard deviaion is 5%. The nex wo rows of he able repor correlaions from simulaions ha incorporae overnigh reurns and he overnigh reurn adjusmen described in Table I. As in Table I, he sandard deviaion of close-o-open reurns is assumed o be 0.5 imes he open-o-close reurn sandard deviaion. Wih overnigh reurns, correlaions decline slighly for all version of he high-low spread esimaor. However, correlaions remain highes for he simple version of he esimaor in which negaive wo-day spreads are se o zero before aking he monhly average. For his version of he esimaor, he correlaion falls from o 0.95 for a sandard deviaion of 3% and from o for a sandard deviaion of 5%. The middle wo rows of Table II provide correlaions beween spread esimaes and simulaed spreads incorporaing boh overnigh reurns and infrequenly observed prices. As in Table I, we assume here is a 10% chance of observing a price a any specific minue. Incorporaing infrequen observaion of prices reduces correlaions slighly for all versions of he high-low spread esimaor. As in Table I, however, infrequen observaion of prices has lile impac on he Roll spread esimaor. Again, he 19

22 correlaions sugges ha all versions of he high-low spread esimaor dominae he Roll spread esimaor, and he simple version of he high-low esimaor in which negaive wo-day esimaes are se o zero ouperforms oher versions of he high-low spread esimaor. This version of he high-low spread esimaor produces a correlaion of 0.9 for a daily reurn sandard deviaion is 3% and 0.88 for a sandard deviaion is 5%. The nex wo rows of he able repor correlaions when daily reurns are posiively auocorrelaed. Specifically, we assume ha he innovaion o he expeced reurn is normally disribued wih a sandard deviaion of 1% per day. The daily expeced reurn is hen defined as he sum of he innovaion plus 0.5 imes he previous day s expeced reurn and he expeced reurn for each one minue ime inerval is he daily expeced reurn divided by 390. When reurns are posiively auocorrelaed, he correlaion wih simulaed spreads declines for every version of he high-low spread esimaor, bu remains high. Again, he version of he high-low spread esimaor in which negaive wo-day spread esimaes are se o zero produces he highes correlaion and significanly ouperforms he Roll esimaor. For his version of he high-low spread esimaor, he correlaion beween high-low spread esimaes and simulaed spreads is for a daily sandard deviaion of 3% and 0.81 for a sandard deviaion of 5%. Finally, he las wo rows of Table II repor correlaions under he assumpion ha spreads hemselves change randomly. In hese simulaions, we coninue o assume ha sock prices change overnigh, ha here is only a 10% chance ha a price is observed a a paricular minue, and ha daily reurns are posiively auocorrelaed. For each sock, an iniial spread is drawn randomly from a uniform disribuion over 0% o 6%. Then, each day s spread is obained by muliplying he previous day s spread by e δ, where δ is normally disribued wih zero mean and sandard deviaion equal o 0.1. The simulaed monhly spread is hen defined as he average simulaed spread across he 1 days wihin he monh. Ineresingly, he correlaions beween spread esimaes and mean simulaed spreads are higher for all esimaors han he correlaions when spreads are assumed o be consan wihin he monh. This may reflec ha he range of mean spreads in hese simulaions is wider han he original 0% o 6%. More 0

23 imporan hough is ha he relaive performance of he spread esimaors is unaffeced by allowing spreads o vary over ime. The simulaion resuls presened in Tables I and II provide several key findings. Firs, he simple high-low spread esimaor in which wo-day spread esimaes are averaged over he monh works well and is far more accurae han Roll spread. Second, adjusing for Jensen s inequaliy complicaes he esimaion bu doesn improve he accuracy of he esimaes. Third, esimaing spreads over wo-day periods and averaging he spreads over a monh works beer ha averaging he parameers and calculaing a single spread for he monh. Finally, seing negaive wo-day spread esimaes o zero before calculaing monhly spreads improves esimaes, paricularly when reurns are generaed overnigh and socks do no rade coninuously. Throughou he remaining empirical analysis, we presen resuls based on he simple high-low spread esimaor, where monhly spreads are based on an average of wo-day spread esimaes afer seing negaive wo-day esimaes o zero. 5. A Comparison of Spread Esimaes from Daily Daa wih TAQ Spreads In his secion, we compare he accuracy of monhly high-low spread esimaes o esimaes generaed by hree common alernaive spread esimaors: he Roll spread esimaor, he effecive ick esimaor, and LOT esimaor. These alernaives provide esimaes based on he auocovariance of reurns, he price fracion of rade prices, and he frequency of zero reurns, respecively. We focus on hese specific esimaors because hey have been used as building blocks for oher, more complex mehods of esimaing spreads (see, for example, Holden (009)). Goyenko e al. (009) provide a comprehensive sudy of he properies of hese esimaors and esimaors derived from hem. Given is simpliciy and accuracy, we believe ha he high-low spread esimaor may also serve as a foundaion for more complex esimaion echniques. Monhly Roll, effecive ick, LOT, and high-low spread esimaes are calculaed using daily daa from CRSP for he period from January 1993 hrough December 006. For each spread esimaor, we 1

24 require a leas 1 daily observaions o calculae a monhly spread esimae. The CRSP sample includes all NYSE, Amex, and Nasdaq socks wih CRSP share codes of 10 or 11 (i.e., U.S. common shares). To assess he accuracy of hese monhly spread measures, we compare hem o rade-weighed effecive spreads and ime-weighed quoed spreads esimaed for each securiy each monh using he NYSE s TAQ daa. 13 For each securiy and each rading day, we firs deermine he highes bid and lowes ask across all quoing venues a every poin during he day. 14 A any ime, le Bid equal he inside bid, Ask equal he inside ask, and Midpoin equal (Bid + Ask )/. The percenage quoed spread a ime is hen defined as (Ask -Bid )/Midpoin. For each securiy, he average quoed spread for he day is defined as a weighed average across all spreads during he day, where each spread is weighed by he number of seconds i is in place. The monhly Quoed Spread for each securiy is obained by averaging he wo-day esimaes across all rading days wihin he monh. To esimae effecive spreads, we compare each rade price during he day o he inside bid and ask posed one second prior o he rade. For each rade I, le Price i equal rade price and Midpoin i equal he bid-ask midpoin ousanding a he ime of rade I. The percenage effecive spread for rade I is hen defined as * P I Midpoin I /Midpoin I. The average effecive spread for each day is a rade-weighed average across all rades during he day. The monhly Effecive Spread for each securiy is hen defined as he average across all rading days wihin he monh. 13 To mach securiies in he CRSP daa o securiies in he TAQ daa, we follow a muli-sep maching procedure. We firs idenify all unique cusip-icker combinaions in boh he TAQ and CRSP daases from 1993 hrough 006. We use eigh-digi cusip numbers, where cusip numbers for TAQ securiies are aken from he monhly TAQ Maser Files. We hen merge he TAQ and CRSP samples by cusip and icker, assigning a CRSP perm number o each TAQ securiy. For hose securiies ha canno be mached in he firs sep, we hen mach based solely on he eigh-digi cusip number. Finally, we aemp o mach any remaining securiies by eiher icker symbol or six-digi cusip number. All securiies mached solely by icker or cusip are hen hand verified for accuracy and correcions are made, where needed. As a final sep, we hand verify any CRSP-TAQ maches where he number of daily observaions in he wo daases differs by more han 10 days. 14 For Nasdaq securiies, we firs esablish he bes bid and ask across all Nasdaq marke makers. These inside quoes are hen compared o he quoes on oher venues. We apply several sandard filers o he rade and quoe daa. We include only regular NBBO-eligible quoes wih posiive prices and posiive deph. We also exclude quoes if he ask is less han or equal o he bid or if eiher he bid or ask differs by more han 5% from he previous quoe. We use only rades ha occur during regular rading hours, have a posiive price and quaniy raded, have normal condiion codes, and have rade correcion codes less han wo. We also exclude he firs rade each day and rades for which he price differs by more han 5% from he preceding price. Finally, we exclude observaions for which eiher he effecive or quoed spread exceeds $1 wih a midpoin of $5 or less, $5 wih a midpoin of $100 or less, or $10 wih a midpoin greaer han $100.

25 Any comparison of alernaive spread esimaors mus be qualified, because he esimaors may capure differen componens of liquidiy or ransiory volailiy. As noed earlier, he high-low esimaor capures ransiory volailiy over wo rading days, which may include emporary price pressure from large orders in addiion o bid-ask spreads. Noneheless, i is informaive o analyze how well he alernaive esimaors capure bid-ask spreads as measured by inraday TAQ daa. 5.1 Pooled Time-Series and Cross-Secional Esimaes Table III provides summary saisics for spread esimaes using he pooled sample of all socks and all monhs from 1993 hrough 006 for which all four spread esimaors could be calculaed. For comparison purposes, daa on effecive spreads and quoed spreads from TAQ are presened firs. The simple average effecive spread from TAQ across all sock monhs is.38%, while he average quoed spread from TAQ is 3.05%. Roll spread esimaes are repored nex. For he full sample of socks over , posiive monhly serial correlaion esimaes occur for 38.0% of he sock monhs. We adop he common ad-hoc adjusmen of seing Roll spreads o zero in hese cases. This yields a mean Roll spread of.4%, which is very close o he mean TAQ effecive spread. If he posiive correlaions are insead omied, more han a hird of he observaions are los and he mean Roll spread is 3.90%, much greaer han eiher he mean quoed or effecive spread from TAQ. In he analysis o follow, we use he version of he Roll spread esimaor in which posiive auocorrelaions are defined as zero spreads. Spread esimaes obained from he effecive ick esimaor and he LOT esimaor are presened nex. By consrucion, effecive ick esimaes are always posiive. The mean effecive ick spread is 1.67% and he median is 0.7%, boh of which are less han he comparable effecive spread esimae from TAQ. The mean and median LOT esimaes are.15% and 0.86%, wih 4.4% of monhly LOT esimaes being non-posiive. Resuls for hree versions of he high-low spread esimaor are repored nex. The firs high-low spread esimaor ses all negaive wo-day spread esimaes o zero before calculaing he monhly 3

26 average. This high-low spread esimaor produces a mean spread of.10%, compared o he mean TAQ effecive spread of.60%. The median spread esimae from his version of he high-low spread esimaor is 1.3%, which is very close o he median TAQ effecive spread of 1.9%. When negaive spreads are included, he mean high-low spread esimae equals 1.6%, wih 4.0% of monhly spread esimaes being non-posiive. When negaive wo-day spreads are omied, he mean high-low spread increases o.76%. These findings are consisen wih he simulaion resuls and sugges ha he high-low spread esimaor performs bes when negaive wo-day esimaes are se o zero before aking he monhly average. The resuls hroughou he remainder of he paper are herefore based on he simple version of he high-low spread esimaor in which negaive wo-day spreads are se o zero. 15 Table IV repors correlaions beween he various spread esimaes and he effecive and quoed spreads from TAQ. As in Table III, he pooled sample includes observaions from all socks-monhs for which each of he four spread esimaors could be calculaed. The resuls demonsrae how well he highlow spread esimaor works. The correlaion beween he high-low spread esimaes and he TAQ effecive spread is The comparable correlaions for he Roll spread, he effecive ick spread, and he LOT measure are 0.707, 0.734, and 0.699, respecively. The second column of IV repors correlaions beween spread esimaors and he TAQ quoed spread. Given he high correlaion beween effecive and quoed spreads from TAQ (0.979), i is no surprising ha he correlaion resuls for quoed spreads are similar o hose for effecive spreads. Again, he high-low spread esimaor dominaes, wih a correlaion of The comparable correlaions for he Roll spread, he effecive ick spread, and he LOT measure are 0.695, 0.74, and 0.676, respecively. In he remainder of he paper, we repor ess based on effecive spreads only, bu resuls are similar when quoed spreads are used. 15 Alhough no shown, we also examined he mean absolue error for each spread esimaor relaive o he TAQ effecive spread. The mean absolue error is highes for he Roll spread a and lowes for he high-low spread esimaor a The comparable values for he effecive ick and LOT measures are and respecively. 4