QUANTITATIE METHODS IN ECONOMICS ol. XI, No. 1, 2015, pp. 137 146 TRADE DURATION AND MARKET IMPACT Marek Andrzej Kocńsk Department of Appled Mathematcs Warsaw Unversty of Lfe Scences SGGW e-mal: marek_kocnsk@sggw.pl Abstract: In ths artcle the problem of the algorthm of the transacton executon as the factor n market mpact modellng s studed. The current state of research n ths area s presented and dscussed. The paper adds new arguments to the dscusson on ths topc. Moreover, the soluton to the problem of the trade executon s duraton n practcal applcaton of [Almgren et al. 2005] market mpact model s proposed. Keywords: market mpact, square root mpact law, trade duraton, transacton cost INTRODUCTION Market mpact (also called prce mpact) can be defned as some sort of a change n the asset prce wth respect to adequate reference prce, caused by tradng. Ths change, f occurs, s aganst the trade ntator, that s the prce grows when buyng and drops when sellng, and thus the prce mpact s a source of transacton costs. It n ntutve and a standard n economc theory that a demand ncrease should result n growth of the prce and a supply ncrease should result n the prce drop. The concept of market mpact s closely related to the noton of bdask spread, whch s the dfference between the best avalable n the market bd and ask prces (called just bd and ask prces, respectvely) and often s expressed as a fracton of a so called md-prce whch s defned as the average between best bd and ask prces and represents the market value of an asset. If, as some authors do, the md-prce s the reference prce, then bd-ask spread s a part of market mpact. However, t s often assumed that the reference prce s a bd prce n case of a seller-ntated trade and an ask prce n case of a buyer-ntated trade. In such approach the spread and prce mpact are treated separately.
138 Marek A. Kocńsk Market mpact s the man source of lqudty rsk. It s the reason of not beng able to execute a transacton at the current quoted prce because executon moves the prce n an unfavourable manner. Spectacular examples showng how mportant s market mpact are: the fasco of Metallgesellschaft n 1993, the LTCM crss n 1998 and the cancellng of the portfolo of Jérôme Kervel by Soceté Générale n 2008 [Sched and Slynko 2011]. Snce market mpact moves adversely the prces at whch transactons are made, t can, sometmes sgnfcantly, reduce profts and turn theoretcally proftable strategy nto a fnancal falure. Therefore, t s no surprse that modellng, estmaton and analyss of market mpact nterests many asset managers and scholars. In fact, research on prce mpact has become one of the most popular actvtes n quanttatve fnance snce the md-1990s [Tóth et al. 2011]. The am of ths artcle s to analyse the problem of mportance of the algorthm of the trade executon as the varable n modellng of market mpact. The paper contrbutes to the lterature on market mpact by addng new arguments confrmng that the executon speed s of mnor sgnfcance n prce mpact modellng. Next contrbuton s a new vew on the model of [Almgren et al. 2005] whch allows for better practcal use of ths model. Ths artcle contans also the example of the calculaton of market mpact n Warsaw Stock Exchange, wth use of order book. MARKET IMPACT MODELS AND TRADE DURATION Market mpact modellng and estmaton has been very mportant to scholars nterested n market mcrostructure and practtoners. A well-calbrated prce mpact model s an mportant part of quanttatve nvestment management. It s a useful tool n predctng transacton costs and prce changes due to tradng actvty. Such expectatons allow to forecast the consequences of mplementng portfolo strateges. Today, any decent pre-trade analytc software takes nto account the prce mpact of a proposed transactons as a functon of trade-based parameters and characterstcs of the traded securty [Gatheral 2010]. The smple and popular approach to modellng prce mpact suggested n the lterature s to consder t as one of the components of transacton costs. Then the formula for market mpact as a relatve fracton of the prce of the traded securty at the begnnng of the trade, s gven as follows: trade MI c (1) where s the daly volatlty, trade s the volume of the executed trade, s the average daly volume, c s the numercal constant of order unty that can be estmated from the representatve sample of transactons and the exponent does not exceed 1 and ts estmaton has often the range between 0.4 and 0.7, however
Trade duraton and market mpact 139 an mportant practcally and theoretcally case s lnear functon of market mpact wth 1. A partcular varant of the formula (1) s the so-called square root mpact law whch s wdely used n academa and fnancal servce ndustry: trade MI c (2) Equaton (2) s strongly supported by the emprcal data, reasonable arguments gven n [Grnold and Kahn 2000] and t s consstent wth a tradng rule of thumb accordng to whch the transacton cost of the volume equal to the average one day s volume, costs roughly one day s volatlty of the prce. [Grnold and Kahn 2000], [Gatheral 2010]. Formula (1) suggests that the only trade-based varable whch s necessary to calculate the market mpact s the transacton sze, t does not take nto account the executon algorthm used by the trader. In ths context t should be notced that there s a great varety of executon strateges apart from statc (determned n advance of tradng) there are also dynamc whch are condtoned on movement of the securty prce durng executon of transacton. The trade executon s roughly characterzed by duraton whch descrbes how long the executng lasts. The duraton s determned by the tradng rate (the speed of executon) and the transacton volume. Low mportance of the executon characterstcs s more emphassed by some authors [Tóth et al. 2011], [Zarnell et al. 2014] by usng, for the volume of the executed trade, n the market mpact equaton the name metaorder, whch denotes the sequence of tradng decsons. A metaorder s usually fragmented and traded ncrementally by sngle orders whch are, n ths context, called chld orders. Such approach s however contrary to the wdespread opnon that market mpact can be reduced by dvdng ntended transacton nto smaller orders and placng them n separate tme ntervals. In short, the popular vew s that slower trade executon lowers prce mpact. It s also emprcally confrmed that the tradng rate can, n some crcumstances, sgnfcantly affect the market mpact. There s an extensve theoretcal research and practcal solutons on the problem of optmal counteractng market mpact whle executng transacton. The queston arses, therefore, about the explanaton of ths conceptual contradcton. In order to answer ths queston, t s worth pontng out that the observatons where the duraton was mportant pertan to the cases of very large tradng rates were tradng szes were large relatve to the volume of trade offers n the order book. For reasonable tradng rates (about 1% to 25% of average daly volume per day), t seems that the market mpact s roughly ndependent of trade duraton [Gatheral 2010]. It s even presented n [Gatheral and Sched 2013] as the emprcal rule of thumb that market mpact s roughly proportonal to the sze of the transacton and not very dependent on the tradng rate. There s also a heurstc argument that trade
140 Marek A. Kocńsk duraton can or maybe even should be omtted as varable n formula (1). Namely, [Grnold and Kahn 2000] clamed that n a framework of nventory rsk model, for a proposed trade of sze trade, the estmated tme before a suffcent number of opposng trades appears n the market to execute the transacton (tme to clear the transacton) s gven by the formula trade clear (3) Formula (3) establshes a strctly lnear relatonshp between the sze of the transacton and the tme of executon. Thus, snce t s natural that the duratons can be measured by tme clear, t s, accordng to (3), fully characterzed by the tradng volume. Consequently, snce the sze of the transacton s a varable n (1), duraton does not have to appear there. Weak dependence of market mpact on trade duraton s also confrmed by the emprcal data n [Engle et al. 2008]. In order to provde new arguments for dscusson on the meanng of executon algorthm n market mpact modellng I would lke to notce that t s not uncommon to consder the problem of the optmal portfolo selecton n multperod settng where nether the total transacton volume nor the nvestment horzon has to bounded n advance. Then, t s approprate to ask how long lasts the market mpact effect of tradng n one perod, on the asset s prce dynamcs. It s clear that the value of market mpact n next tme perod strongly depends on the answer to ths queston. Most practtoners n executon models use the decomposton of the market mpact nto permanent and temporary market mpact [Guéant 2014]. Temporary prce mpact affects a sngle transacton and may be consdered as the cost of provdng enough lqudty to absorb the trade. The permanent prce mpact component s an nformaton-based effect and measures the change of the market value before and after trade. Ths s due to the fact that there s no easy, method to dstngush not nformed traders from nformed traders and therefore each transacton s consdered as a source of nformaton on the market value of the traded asset. Thus, a buyer-ntated transacton tells the market partcpants that an asset may be underpced and a seller-ntated transacton s a sgnal that an asset s overvalued. As a result, the transacton causes the change n the theoretcal value of the asset whch s unfavourable to the ntator of the trade. It seems that the speed executon has dfferent effects on the levels of the consdered components of market mpact. The hgher tradng rate results n larger temporary mpact and lower permanent mpact, n case of lower tradng rate t s the other way round. Therefore the coexstence of the two components of market mpact whch dfferently react on the speed of tradng I fnd as one of the arguments for low sgnfcance of the executon style n modellng market mpact.
Trade duraton and market mpact 141 In case of nformed traders there s also another factor that counteracts the effect of reducng market mpact of the strategy of slower executon. It s opportunty cost. Ths noton assumes that, f the motvaton of the trade s nformaton on the future value of the traded assets, then quck executon s necessary because such nformaton can be used only for a lmted tme. Rapd executon enables to beneft from the underprcng n case of buyng and from overprcng n case of sellng. Physcal tme s not the only method of measurng the duraton of the executon. The duraton s sometmes quantfed n so called volume tme [Almgren et al. 2005], [Zarnell et al. 2014] whch s calculated for tme perods shorter than tradng day, as the fracton of an average daly volume that has been executed up to physcal tme t. Speakng formally let t be the total volume traded n the market from the tradng day s open up to physcal tme t. olume tme (also called t t volume duraton) s defned as v t c or v, where t c s the market close. It s easly seen that ndependently of the total daly volume, the volume tme defned that way equals 0 at the market openng and 1 at market closng tme [Zarnell et al. 2014]. The duraton measured n volume tme s an nput varable n the elaborately worked out and seemngly ready for use model used by [Almgren et al. 2005]. In ts estmaton [Almgren et al. 2005] used the data set of almost 700,000 trade orders from the US market, executed by Ctgroup equty tradng desk from December 2001 to June 2003, n whch a drecton of the trade (buyer or seller ntated) s known. The market mpact defned as the executon cost n the model of [Almgren et al. 2005], assumng that prce mpact s postve for buy as well as for sell orders (n the orgnal verson of ths model the executon cost can be negatve), s gven by the formula: 1 4 1 trade trade MI (4) 2 T where s the total number of shares outstandng, T s volume duraton of actve tradng,, are the constants. The estmated values of and were calculated by lnear regresson [Almgren et al. 2005] and they calculated that 0.314 0. 041 and 0.142 0.0062. An example of the applcaton of ths model s presented n [Kocńsk 2014] where the duraton was assumed to be an arbtrary value. The varable T I fnd the most problematc n the model gven n [Almgren et al. 2005]. It seems that the trader s rather not able to control the duraton of executon to the extent whch s necessary to produce relable estmator of the volume duraton. However, by reasonable assumptons, applyng of low-frequency estmator whch uses only the 3 5
142 Marek A. Kocńsk daly volumes n estmatng the tme to clear the trade and applcaton of lnear approxmaton, t s possble to elmnate T from the formula (3). Namely, to justfy the use of low frequency estmaton, t seems sound to assume that average tme to fulfl an order does not depend on whether t s a buy or sell order. By ths assumpton, the formula (3) and the fact that the coeffcent of proportonalty n (3) whch s estmated by usng daly volumes, equals 1, t s possble to wrte the followng formula for physcal tme of transacton executon: trade clear (5) t The volume tme gven n [Almgren et al. 2005] s defned as v. From the assumpton that t s a lnear functon of t, t follows that physcal tme s dentcal wth volume tme and consequently from (6) t follows that the value of T s gven by the formula: trade T (6) The equatons (5) and (6) mply the followng, smplfed expresson for the market mpact: 1 trade MI (7) 2 Accordng to (7) the prce mpact s a lnear functon of the traded volume, what s an nterestng n vew of the popularty of the square root model. However the assumpton that market mpact s lnear n the traded volume one can meet n the lterature (see for example [DeMguel et al. 2014]). Such approach can be partly justfed n market mcrostructure theory by the Kyle model [Kyle 1985]. The model gven n [Almgren et al. 2005] treats the bd-ask spread as a part of the market mpact and ths allows to nterpret the second component of the sum n (7) as the bd-ask spread, and n ths approach t s a lnear functon of volatlty. Moreover, an nterestng observaton s that n vew of (7) the average volume s not a sgnfcant determnant of the spread. The neglgblty of the market volume n case of the bd-ask spread n the opton market was found n [Cho and Engle 1999]. EMPIRICAL RESEARCH To verfy the conclusons on the role volatlty and volume n determnng bd-ask spread followng from formula (7) the emprcal research was carred out on a random sample of 300 stocks quoted on the Warsaw Stock Exchange (WSE) n 2014. The annual volatlty was computed from the formula: 1 4 P volatlty ln max (8) P mn
Trade duraton and market mpact 143 where P max and P mn denote the maxmal and mnmal prce of the stock n year 2014, respectvely. Usng data 2014 WSE Statstc Bulletn I calculated the Pearson correlaton coeffcents between the average spread and volatlty ( r volatlty ) and the average spread and the average daly volume ( r volume ). Then, I verfed ther sgnfcance by the standard sgnfcance test wth test statstcs n 2 t r where r s the 2 1 r correlaton coeffcent and n s a number of observatons. The followng results were obtaned ( p s the p-value): r volatlty 0,256; p 0,0000068 r volume 0,037, p 0,5256843. The correlaton between volatlty and spread s hghly statstcally sgnfcant, n contrary to the correlaton between the spread and volume. I also estmated the regresson coeffcents of the spread on volatlty: spread 142,48 63,18 * volatlty (9) (2,95*10 25 6 ) (6,94*10 The results concernng correlaton coeffcents presented above are consstent wth equaton (7). However, the non-zero constant n regresson equaton suggests that formula (7) should be supplemented by an addtve constant. PRICE IMPACT IN MARKET MICROSTRUCTURE: AN EXAMPLE FROM WARSAW STOCK EXCHANGE One of the most mportant characterstcs of a market s the type of ts executon system. In ths respect there are three major types of markets: quote drven markets, order drven markets and brokered markets. Warsaw Stock Exchange (WSE) s classfed as order drven market [Doman 2011]. Table 1 shows the frst fve rows of an order book for the stock of the company Stalexport Autostrady S.A. (denoted as STALEXPORT), from the WSE at some pont n tme durng the tradng sesson on 09 September 2015. Table 1. The frst fve rows of the order book for the shares of STALEXPORT at some moment durng the tradng day n WSE on 09 Sep 2015 Bd sze Bd prce (PLN) Ask prce (PLN) Ask sze 2000 3.21 3.24 2700 1600 3.18 3.25 3582 2375 3.17 3.27 1500 1900 3.16 3.28 5821 4433 3.15 3.29 2550 Source: http://bznes.onet.pl/gelda/notowana/gpw-rynek-glowny/akcjewszystke,101,notowana-gpw-cagle-szczegolowe.html )
144 Marek A. Kocńsk The second row of Table 1 represents the hghest avalable bd prce (3.21), the number of stocks to buy at ths prce (2000), the lowest avalable ask prce (3.24) and the number of stocks to sell at ths prce (2700). The theoretcal prce of the stock STALEXPORT s the average of the best bd and ask prces and equals 3.243.21 3.225. The spread s computed as 3. 225 and equals approxmately 0.009. Consder an nvestor who wants to buy some shares and places market buy order. For the sake of transparency of ths example I assume that commsson here s neglgble, snce ths sort of cost s just an addtve constant. On the frctonless market the cost of hs or her transacton would be 3.225 PLN per share. The transacton cost here s calculated as the relatve ncrease n average prce per share wth respect to the theoretcal prce. On real market f the number of shares does not exceed 2700 then the requred by the market prce per share s 3.24 PLN and transacton cost s just half of the spread. However, f the tradng volume ncreases then the average prce per share grows. It s clear that the average prce per share can be computed from the formula: S n 1 n 1 x S where n s the number of prce levels n the order book, and x s the number of shares for whch the ask prce s average cost per share equals S S S 0 0 x (10) S s the -th prce level S, respectvely. Then the where S 0 s the theoretcal prce. Market mpact s here the dfference between transacton cost and the half of the spread. The calculaton of the average costs and market mpact for tradng volumes correspondng to the cumulated values of ask szes from the order book are shown n Table 2. Table 2. The average cost per share for purchase transactons of the stock Tradng sze Average cost Market mpact 2700 0.0047 0.0000 3582 0.0064 0.0018 1500 0.0079 0.0032 5821 0.0118 0.0071 2550 0.0131 0.0085 Source: data from Table 1 and own elaboraton The graph of the transacton costs functon s presented n Fgure 1.
Trade duraton and market mpact 145 Fgure 1. The functon of transacton cost Source: data from Table 2 and own elaboraton The dashed lne corresponds to the case of proportonal transactons costs, where the only source of payments for exchange of shares s the bd-ask spread. The market model wth proportonal cost s much more realstc then the assumpton of frctonless market where there are no costs assocated wth tradng. Fgure 1, however, shows, that n order to precsely estmate the rsk n asset management, one should take the prce mpact nto account. CONCLUSIONS The paper presents the problem of the nfluence of the method of transacton executon on the magntude of market mpact. Neglectng the style of tradng as factor affectng prce mpact s popular among both theoretcans and practtoners. Takng nto account the lterature on ths subject and the arguments presented n ths artcle, It appears reasonable to assume that when the transacton volume s not extremely large and the nvestment horzon s not very long the executon style s not very mportant. The effect of lowerng market mpact wth executon method s usually of mnor order than the value of prce mpact, and therefore the executon algorthm need not be taken nto account when estmatng market mpact. REFERENCES Almgren R., Thum C., Hauptmann E., L H. (2005) Drect Estmaton of Equty Market Impact Rsk 18, pp. 58 62. Cho Y.-H., Engle R. F. (1999) Modelng the Impacts of Market Actvty on Bd-Ask Spreads n the Opton Market, http://www.nber.org/papers/w7331.pdf DeMguel., Martín-Utrera A., Nogales F. J. (2014) Parameter uncertanty n multperod portfolo optmzaton wth transacton costs, http://faculty.london.edu/avmguel/dmn- 2014-01-09.pdf
146 Marek A. Kocńsk Doman M. (2011) Mkrostruktura gełd paperów wartoścowych, Wydawnctwo Unwersytetu Ekonomcznego w Poznanu, Poznań. Elton R. J., Gruber M. J., Brown S. J., Goetzmann W. N. (2010) Modern Portfolo Theory and Investment Analyss, John Wley & Sons, Hoboken. Engle R., Ferstenberg R., Russel J. (2008) Measurng and modelng executon cost and rsk, http://faculty.chcagobooth.edu/jeffrey.russell/research/transcost.pdf Gatheral J., No-Dynamc-Arbtrage and Market Impact (2010) Quanttatve Fnance 10, pp. 749 759. Gatheral J., Sched A. (2013) Dynamc models of market mpact and algorthm for order executon, [n:] Fouque J.-P., Langsam J. A. (eds.) Handbook on Systemc Rsk, Cambrdge Unversty Press, Cambrdge. Grnold R. C., Kahn R. N. (2000) Actve Portfolo Management, McGraw Hll, New York. Guéant O. (2014) Permanent market mpact can be nonlnear, http://arxv.org/pdf/1305.0413v4.pdf Kocńsk M. (2014) Transacton Costs and Market Impact n Investment Management, e- Fnanse, pp. 28 35. Kyle A. S., (1985) Contnuous auctons and nsder tradng, Econometrca 53, pp. 1315-1335. Scherer B. (2010) Portfolo Constructon and Rsk Budgetng, Rsk Books, London. Sched A., Slynko A. (2011) [n:] Blath J., Imkeller P., Roelly S. (eds.) Surveys n Stochastc Processes, European Mathematcal Socety Publshng House, Zürch. Tóth B., Lempérère Y., Deremble C., de Latallade J., Kockelkoren J., Bouchaud J.-P. (2011) Anomalous Prce Impact and the Crtcal Nature of Lqudty n Fnancal Markets, http://journals.aps.org/prx/pdf/10.1103/physrevx.1.021006 Zarnell W., Treccan M., Doyne Farmer J., Llo F. (2014) Beyond the square root: evdence for logarthmc dependence of market mpact on sze and partcpaton rate, http://arxv.org/pdf/1412.2152v1.pdf WSE Statstcs Bulletn (2014) http://www.gpw.pl/statystyk_roczne.