International Review of Financial Analysis

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1 Iteratioal Review of Fiacial Aalysis 23 (2012) Cotets lists available at ScieceDirect Iteratioal Review of Fiacial Aalysis Liquidity cost of market i the Taiwa Stock Market: A study based o a order-drive aget-based artificial stock market Yi-Pig Huag a,, Shu-Heg Che b, Mig-Chi Hug c, Tia Yu d a Fiace Departmet, Chughwa Telecom Co., Ltd., Taipei, Taiwa b Departmet of Ecoomics, Natioal Chegchi Uiversity, Taipei, Taiwa c Departmet of Fiacial Egieerig ad Actuarial Mathematics, Soochow Uiversity, Taipei, Taiwa d Departmet of Computer Sciece, Memorial Uiversity of Newfoudlad, St Joh's, NL, Caada article ifo abstract Available olie 19 July 2011 Keywords: Order-drive Liquidity cost Zero-itelligece traders Aget-based artificial stock market We developed a order-drive aget-based artificial stock market to aalyze the liquidity costs of market i the Taiwa Stock Market (TWSE). The aget-based stock market was based o the DFGIS model proposed by Daiels, Farmer, Gillemot, Iori ad Smith (Daiels et al., 2003). We also improved the DFGIS model by usig two average order size parameters. Whe tested o 10 stocks ad securities i the market, the model-simulated liquidity costs were higher tha those of the TWSE data. We idetified some possible factors that have cotributed to this result: 1) the overestimated effective market order size, which ca be improved by usig two average order size parameters; 2) the radom market order arrival time desiged i the DFGIS model; 3) the zero-itelligece of the artificial agets i our model; ad 4) the price of the effective market order. We cotiued improvig the model so that it could be used to study liquidity costs ad to devise liquidatio strategies for stocks ad securities traded i the Taiwa Stock Market Elsevier Ic. All rights reserved. 1. Itroductio Market liquidity, or the ability of a asset to be sold without causig a sigificat amout of price movemet ad with miimum loss of value, plays a importat role i fiacial ivestmet ad i securities tradig. Oe recet evet that highlighted the impact of asset liquidity o fiacial istitutios was the collapse of Bear Stears. Bear Stears was ivolved i securitizatio ad issued a huge amout of asset-backed securities, mostly mortgage-backed assets. Due to the subprime crisis i 2007, the compay issued subprime hedge fuds that had very low market liquidity ad subsequetly lost most of their value. I March 2008, the Federal Reserve Bak of New York provided a emergecy loa to try to avert a sudde collapse of the compay. However, the compay could ot be saved ad was subsequetly sold to JP Morga Chase i I large ivestmet istitutios, the liquidatio of a large block of assets withi a give time costrait to obtai cash flow arises frequetly. For example, a fiacial istitutio may eed to liquidate part of its portfolio to pay for its immediate cash obligatios. Oe possible liquidatio strategy is to sell the etire block of assets at oce. However, this high-volume tradig ca cause the price of the share to drop betwee the time the trade is decided ad the time the trade is completed. This implicit cost (due to the price declie) is kow as the market impact cost (MIC) or liquidity cost (the umerical defiitio is Correspodig author. addresses: berthuag@cht.com.tw (Y.-P. Huag), che.shuheg@gmail.com (S.-H. Che), hugg@scu.edu.tw (M.-C. Hug), gwoigyu@gmail.com (T. Yu). give i Sectio 3). To miimize such cost, a better strategy is to divide the block of assets ito chuks ad sell them oe chuk at a time. However, i what way should those chuks be sold so that the liquidity cost is miimized? I Algorithmic Tradig, where computer programs are used to perform asset tradig icludig decidig the timig, price, or the volume of a tradig order, this liquidatio problem is characterized as a optimizatio problem. With a smooth ad differetiable utility fuctio, the problem ca be solved mathematically (Almgre & Chriss, 2000) (Kali & Zagst, 2004). However, this mathematical approach to fid a optimal liquidatio strategy has some shortcomigs, such as the imposed assumptio that risk has a liear impact o prices. I this paper, we adopt a differet approach by devisig a aget-based artificial stock market, which has more relaxed assumptios (explaied i Sectio 3). By performig simulatios ad aalyzig liquidity costs iduced uder differet market scearios, we hope to uderstad the dyamics of liquidity costs, ad hece to devise a more realistic optimal liquidatio strategy. The rest of this paper is orgaized as follows. I Sectio 2,weprovide the backgroud ad summarize related works. Sectio 3 explais the aget-based artificial stock market we developed based o the DFGIS model ad the data from the Taiwa Stock Market (TWSE). I Sectio 4, the 10 securities ad stocks that we selected to coduct our study are preseted. Sectio 5 provides the model parameters used to perform our simulatio. We aalyze the simulatio results i Sectio 6 ad preset our discussios i Sectio 7. Fially, Sectio 8 cocludes the paper with a outlie of our future work /$ see frot matter 2011 Elsevier Ic. All rights reserved. doi: /j.irfa

2 Y.-P. Huag et al. / Iteratioal Review of Fiacial Aalysis 23 (2012) Backgroud ad related work This study implemets a aget-based model for a order-drive double auctio market, which is the most commo fiacial market i the world. We shall first provide a brief itroductio of the basic microstructure ad tradig mechaism of a stadard order-drive double-auctio market. Next, the DFGIS model (Daiels, Farmer, Gillemot, Iori, & Smith, 2003), o which our aget-based artificial stock market is based, will be preseted. After that, we will summarize the work of (Guo, 2005) o aget-based models used to study liquidatio strategies at the ed of the sectio Order-drive double-auctio markets I a order-drive double-auctio market, prices are determied by the publicatio of to buy or sell. This is differet from a quote-drive market where prices are determied from quotatios made by market makers or dealers. There are two basic kids of i a order-drive market. Impatiet traders submit market, which are requests to buy or sell a give umber of immediately at the best available price. More patiet traders submit limit, which specify the limit (best acceptable) price for a trasactio. Sice limit ofte fail to result i immediate trasactios, they are stored i a limit order book. As show o the left of Table 1, limit buy are stored i decreasig order of limit prices while limit sell are stored i icreasig order of limit prices. The buy limited are called bids ad the sell limited are called asks. For a ormal double-auctio market, the best (highest) bid price is lower tha the best (lowest) ask price. The differece betwee the two is called the spread of the market. I the example i Table 1 (left), the spread is $0.07. Whe a market order arrives, it is matched agaist limit o the opposite side of the book. For example, whe a market sell order for 30 arrives at the market whose order book is as that o the left of Table 1,itwillfirst be matched agaist the curret best bid (20 at $1.10 per share). Sice the size of the sell order (30 ) is larger tha that of the best bid (20 ), the remaider of the market sell order (10 ) will be matched agaist the ext best bid (25 at $1.09 per share). After the trasactio is completed, the limit order book will chage to the right of Table 1 ad the market spread will wide to $ The DFGIS model I the origial DFGIS model (Daiels et al., 2003), all the order flows (icludig limit ad market ) are modeled as a Poisso process. Market arrive at the market i chuks of σ (where σ is a fixed iteger) at a average rate of μ per uit of time. A market order may either be a buy market order or a sell market order with equal probability. Limit arrive at the market i chuks of σ, at a average of α per uit price per uit of time. A limit order may either be a limit buy order or a limit sell order with equal probability. The limit prices i limit are geerated radomly from a uiform distributio. I particular, the limit buy prices have a rage betwee Table 1 A example of a order book before (left) ad after (right) a trasactio. Limit buy Limit sell Limit buy Limit sell Size Price Size Price Size Price Size Price 20 $ $ $ $ $ $ $ $ $ $ $ $ $ $ $1.25 Table 2 Summary of DFGIS parameters. Parameter Descriptio Dimesio α Avg. limit order rate Share/price time μ Avg. market order rate Share/time δ Avg. limit order decay rate 1/time σ Order size A costat share dp Tick size Price Table 3 A example of a order book before (left) ad after (right) hadlig a limit sell order with price equal to the best bid. Limit buy Limit sell Limit buy (, a(t)), where a(t) is the best (lowest) ask price i the market at time t. Similarly, the limit sell prices have a rage betwee (b(t), ), where b(t) is the best (highest) bid i the market at time t. I additio, the price chages are ot cotiuous, but have discrete quata called ticks (represeted as dp). Tick size is the price icremet/decremet amout allowed i a limit order. 1 DFGIS also allows the limit order to expire or to be caceled after beig placed i the market. Limit are expired ad caceled accordig to a Poisso process, aalogous to radioactive decay, with a fixed-rate δ per uit of time. Table 2 lists the parameters of the DFGIS model. To keep the model simple, the DFGIS does ot explicitly allow limit whose prices cross the best bid price or the best ask price. I other words, the price of a limit buy order must be below the best ask price ad the price of a limit sell order must be above the best bid price. Farmer, Patelli ad Zovko (2005) implemeted the model to explicitly hadle this type of order. They defied effective market as that result i trasactios immediately ad effective limit as that remai o the order book. A limit order with a price that crosses the opposite best price is split ito effective market ad effective limit accordig to the above defiitio. For example, whe a limit sell order of 30 at price of $1.10 arrives at the market (with the order book as that listed i Table 3 (left)), the order will be split ito a effective market order of 20 ad a effective limit sell order of 10. After the executio of the 20 of the effective market order (at price $1.10), the order book is chaged to that o the right of Table The Guo aget-based stock market model Limit sell Size Price Size Price Size Price Size Price 20 $ $ $ $ $ $ $ $ $ $ $ $ $ $1.21 Guo (2005) implemeted a aget-based artificial stock market based o the DFGIS model (he called it the SFGK model) to study timecostraied asset liquidatio strategies through market sell oly. I particular, he compared the performace of two strategies. The first oe uiformly divides the liquidatio X ad the time costrait T ito N chuks. This uiform rhythm strategy istructs a trader to sell X/N every T/N secods, regardless of the market coditio. 1 The rage of price has as the lower limit because i the DFGIS model prices are first coverted to logarithms. As we shall see later i Sectio 3, we do ot use the logarithm trasformatio of price. Prices ad ticks are all i their origial form.

3 74 Y.-P. Huag et al. / Iteratioal Review of Fiacial Aalysis 23 (2012) The secod strategy is the o-uiform rhythm strategy which also divides the liquidatio ad time uiformly. However, withi each time segmet, this strategy requires a trader to cotiuously observe the market spread ad iitiates the sellig of the X/N as soo as the curret market spread, for the first time withi the time segmet, falls below the pre-determied spread threshold. If the market spread ever falls below the spread threshold for a time segmet, the strategy will ivolve sellig the X/N at the ed of the time segmet. Guo devised oe aget (A) with the uiform rhythm strategy ad aother aget (B) with the o-uiform rhythm strategy. He tested each aget idividually by ruig 200 simulatios idepedetly. For each simulatio ru, the umber of liquidatio (X) is 20 ad the time costrait (T) is 5 mi. The total assets are divided ito 10 chuks, each of which cotais 2 that will be sold withi a time segmet of 30 s. The performace of the two agets is evaluated by the average sellig price per share relative to the volume weighted average price i the market. I other words, it is the measure of how much better or worse a aget performs, whe tested compared to the market. His simulatio results idicated that aget B (based o the o-uiform rhythm strategy) outperformed the market while aget A (based o the uiform rhythm strategy) uderperformed the market. Guo did ot explicitly study liquidity costs whe devisig his liquidatio strategy. By cotrast, we are iterested i quatifyig the cost i a real-world stock market. We therefore used TWSE stock order ad trasactio data (see Sectio 4) to costruct the aget-based artificial stock market usig the DFGIS model. We describe this agetbased system i the followig sectio. 3. A aget-based Taiwa Stock Market model The aget-based artificial stock market cosists of zero-itelliget agets who place buy, sell or cacelatio at radom, subject to the costraits imposed by the curret prices. The distributio of the order prices ad quatities ad the distributio of the time itervals of the order submissios i the market follow that of the DFGIS model. Ulike the Algorithmic Tradig model, which imposes urealistic assumptios, the aget-based model is govered by the 5 DFGIS model parameters. The market properties emerge from the stochastic simulatio. We describe our implemetatio of the abstract DFGIS model for the TWSE i the followig subsectios Buy ad sell O the TWSE, a submitted order (either buy or sell) eeds to specify the price, i additio to the quatity, that the trader is willig to accept. However, the price ca cross the disclosed best prices (explaied i Sectio 3). Whe the price of a buy order is greater tha or equal to the disclosed best ask price or the price of a sell order is less tha or equal to the disclosed best bid price, there is a match for the trasactio. We follow that defied by Farmer et al. (2005) (see Sectio 2) ad call the portio of a order that might result i a trasactio i the curret matchig period a effective market order. The possibly o-trasacted portio of a order might be recorded o the order book ad is called the effective limit order Evet-time model We implemeted the artificial stock market as a evet-time model, where the evets i the model are ot coected to the real time. We first partitioed a tradig day ito a fixed umber of time itervals. The evets takig place at each time iterval become the evet-time series describig the market activities of that day. The TWSE opes at 9:00 am ad closes at 1:30 pm. With a time iterval of 0.01 s, the evet-time series of a tradig day is 1,620,000 time itervals log. The TWSE is a call auctio market where the submitted are matched oce every 25 s. Hece, there are 648 ordermatchig evets i a daily evet-time series ad the umber of evets betwee two order-matchig evets is There are five possible evets i a evet-time series: Effective limit order submissio: this has a average rate α ad is deoted by L. Effective market order submissio: this has a average rate μ ad is deoted by M. Order cacelatio submissio: this has a average rate δ ad is deoted by C. Order matchig: the buy ad sell are matched for trasactios ad are deoted by T. No activity: where oe of the above evets occurred i the market is deoted by N. From a simulatio poit of view, the simulatio result based o a evet-time series ad that based o a real-time series are equivalet. For example, the evet-time series LNNNM is equivalet to the real-time series L (real time elapse) M. However, the evet-time model is easier to implemet ad has a shorter simulatio ruig time because there is o eed to hadle the time elapse betwee two evets. We therefore adopted the evet-time model to implemet our artificial stock market Order pricig rules As metioed previously, the TWSE matches submitted oce every 25 s. The five best prices are the disclosed to the public. Durig the followig 25 s of the waitig-for-matchig period, the TWSE does ot disclose ay iformatio about the ewly-submitted. Cosequetly, ivestors do ot have the updated best prices, but rather the best prices from the previous matchig period to make tradig decisios. These disclosed best prices are the used to decide order pricig rages ad to calculate the liquidity costs (explaied i the ext subsectio). The TWSE has a pricig rule whereby the price rage of a order has to be betwee the closig price o the previous tradig day (cp) (1±7%). Thus, the submitted i our simulatio system have the followig price rages: effective limit buy : uiform probability i the rage (cp (1 7%), da(t)), where da(t) is the disclosed best (lowest) ask price i the market at time t, effective limit sell : uiform probability i the rage (db(t), cp (1+7%)), where db(t) is the disclosed best (highest) bid i the market at time t. effective market buy : cp(1+7%). This guaratees a immediate trasactio. effective market sell : cp(1 7%). This guaratees a immediate trasactio. The tick sizes (the price icremet/decremet amout) are as defied by the TWSE Liquidity costs We defie the liquidity cost of a effective market order as the differece betwee the expected trasactio paymet ad the actual trasactio paymet of a effective market order. The expected 2 However, o the TWSE, i the last 5 mi, i.e., after 1:25 pm ad before 1:30 pm, there is o further matchig. The the last matchig happes at 1:30 pm whe the market closes. I our simulatio, we do ot isolate these last 5 mi ad cotiue matchig i this time iterval as we do for the others. 3 See

4 Y.-P. Huag et al. / Iteratioal Review of Fiacial Aalysis 23 (2012) trasactio paymet is calculated by multiplyig the disclosed best price by the umber of of a effective market order. Sice the disclosed best price is from the previous trasactio period, this is the expected amout of paymet to be made durig the curret matchig period. The actual trasactio paymet, however, ca be differet from what was expected. It is calculated as the executed trasactio price (explaied i the ext paragraph) multiplied by the umber of i a trasactio. We ca iterpret liquidity cost as the differece betwee a ivestor's expected trasactio paymet ad the actual trasactio paymet he/she made. The TWSE uses a special rule to decide the executio trasactio price to match the submitted. Istead of the best ask ad the best bid i the curret matchig period, the rule selects the price that gives the maximum trasactio volume as the executio trasactio price. Table 4 gives a example of how the trasactio price is decided. I colum 2, the buy order quatities uder the prices i colum 3 are give. Colum 4 gives the sell order quatities uder the prices i colum 3. As a example, the first row shows that there is a buy order biddig $102.5 for 99 ad there is a sell order askig for the same price for 1 share. Colum 6 gives the umber of trasactio uder the price i colum 3. I this case, $101.5 gives the largest trasactio volume (100 ) ad is selected as the executio trasitio price for this matchig period. With the selected executio trasactio price (TP) ad the disclosed best ask (DBA), the liquidity cost of a effective market buy order with volume V is defied as: LC buy = TP V DBA V DBA V = TP DBA DBA : ð1þ Similarly, with the disclosed best bid (DBB), the liquidity cost of a effective market sell order is defied as: LC sell = DBB V TP V DBB V = DBB TP DBB We have ormalized the liquidity cost such that the value is the ratio of the origial cost to the expected trasactio paymet. Both LC buy ad LC sell ca be positive or egative. This is because uder the TWSE pricig regulatio, the executio trasactio price may be higher or lower tha what a trader has expected. Liquidity cost with a egative value meas that the executio trasitio price is better tha what the trader has expected, while liquidity cost with a positive value meas that the opposite situatio applies. Whe a order is oly partially or ot executed i the curret matchig period, the o-executed portio remais i the order book. These effective limit may later be executed ad become effective market. However, effective limit may lead to a loss of opportuities related to the chagig market prices or a decayig value of the iformatio resposible for the origial tradig decisio. This so-called opportuity cost is difficult to estimate, ad hece is ot cosidered i our liquidity costs calculatio. Table 4 A example of the determiatio of the executio trasactio price. Accumulated buy Buy Price Sell Accumulated sell Trasactio volume () ð2þ Table 5 Average order sizes of effective limit ad effective market rage over all 77 days of data, without weights. Ticker Eff. limit () Eff. market () Test for equality p-value , , , T 24, , T 24, , , , , , Measuremet of model parameters The model parameters are estimated usig real data from the TWSE (see Sectio 4). These parameters iclude a effective market order rate (μ), effective limit order rate (α) cacelatio order rate (δ) ad order size (σ). Note that tick size (dp) has bee discussed previously i Sectio 3. For each parameter, we calculated the mea of the daily value weighted by the umber of daily evets. For example, the parameter p t (μ) is the ratio of the umber of effective market order evets (icludig buy ad sell ) to the total umber of buy, sell, ad cacelatio ad o-active evets (1,619,352) o day t. The weight factor w t is the ratio of the umber of order evets (icludig effective market, effective limit ad cacelatio ) o day t to the total umber of order evets for the etire period: w t = i =1 μt + αt + δt ð3þ μi + αi + δi where μt is the umber of effective market o day t; αt is the umber of effective limit o day t; δt is the umber of cacelatio o day t; ad is the umber of days i the etire period. We measured w t p t (μ) across the whole period (77 tradig days i this case) ad the added them together, which becomes the average daily effective market order evet rate p(μ). Note that its dimesio is orderevet/time, which is slightly differet from μ of (Daiels et al., 2003) whose dimesio is share/time (see Table 2). We applied the same method to calculate p(α) adp(δ). With that, we ca calculate the average daily o-activity evet rate (p()) as 1 p(μ) p(α) p(δ). To calculate the average order size, we first computed the average umber of i the effective market ad effective limit submitted o day t (σ t ) (excludig those submitted before the first best prices were disclosed ad after the market was closed). The summatio of σ t w t for the etire period becomes the average order size σ. Ithe simulatio, ulike the DFGIS model, we used a variable order size, which is geerated rffiffiffiffiffiffiffiffiffiffi radomly from a half-ormal distributio with stadard π 2 deviatio σ (σ is the average order size, ot stadard 2 deviatio) (Weisstei, 2005). 4 From real data ad our simulatio experiece, we fid that the average order sizes of effective limit ad effective market are ot the same. The average effective limit order size, average market order size ad the p-value of the test for equality are show i Table 5. Therefore, we use differet average order sizes for effective limit ad effective market i the secod model (i.e., DFGIS-II). Accordig to the TWSE regulatio, the maximum order size is 499,000. Table 6 summarizes the model parameters implemeted i our 4 Accordig to Daiels et al. (2003), the variable order size gives the same result as that produced from the costat order size σ. However, we are ot sure whether this equality also applies to TWSE, so we still cosider the variable order size.

5 76 Y.-P. Huag et al. / Iteratioal Review of Fiacial Aalysis 23 (2012) Table 6 Model parameters estimatio. Parameter Descriptio Value Dimesio p(μ) Avg. daily effective market t =1 p t (μ) w t Order evet/time order evet rate p(α) Avg. daily effective limit t =1 p t (α) w t Order evet/time order evet rate p(δ) Avg. daily cacelatio t =1 p t (δ) w t Order evet/time order evet rate σ Avg. Order size t =1 σ t w t Share/order σ limit Avg. effective limit t =1 σ limit, t w t Share/order order size σ market Avg. effective market order size t =1 σ market, t w t Share/order system, where stads for the umber of tradig days (77) i the data set. Note that this approach to the estimatio of the model parameters is similar to that of Farmer et al. (2005). It assumes that the daily probability distributios of these parameters are idetical hece it is a importat assumptio i this study Program implemetatio ad system flow The simulatio program was implemeted i the Pytho programmig laguage. Fig. 1 depicts the overall system workflow. Each simulatio is for oe tradig day for the TWSE. Iitially a series of evets o a tradig day is geerated, based o p(μ), p(α), p(δ) ad p(). The umber of evets is 1,620,000. These evets are the executed sequetially, accordig to what types of evets they are. If the evet is a order matchig evet (T), the program matches ad carries out trasactios. If it is a effective market order submissio (M) or a effective limit order submissio evet (L), the program decides the order size based o the half-ormal distributio of σ. Next, the program decides if it is a buy or a sell order with a equal probability (50%). After that, the order price is determied (see Sectio 3) ad the order is submitted. If it is a order cacelatio submissio evet (C), the program decides whether to cacel a buy or a sell order with equal probability (50%). Next, a order o the order book is caceled radomly. If it is a o-activity evet (N), the program cotiues to process the ext evet. Table 7 The 10 selected securities ad their characteristics. Security Ticker Characteristics Taiwa Top ETF with the highest tradig volume Tracker Fud Polaris/P ETF with a low tradig volume Taiwa Divided+ETF Cathay No. 2 Real 01007T REIT with a high tradig volume Estate Ivestmet Trust Gallop No. 1 Real Estate 01008T REIT with a low tradig volume Ivestmet Trust Fud Chia Steel 2002 Blue chip stock i TWSE TSMC 2330 Stock with a high tradig volume ad the largest market capitalizatio MediaTek 2454 Stock with a high uit price HTC 2498 Stock with a high uit price Presidet Chai Store 2912 Stock with a large market capitalizatio but a low tradig volume Iotera 3474 No-blue chip stock o the TWSE (there is a et loss durig the fiscal year) 4. The data set I recet years, the types of securities traded i a stock market have icreased from stocks ad warrats to Exchage Traded Fuds (ETFs) ad Real Estate Ivestmet Trust fuds (REITs). ETFs are baskets of stocks which are vehicles for passive ivestors who are iterested i log-term appreciatio ad limited maiteace. REITs are popular ivestmet optios as they have better liquidity, i theory, tha real estate. We therefore selected a variety of stocks, ETFs ad REITs traded o the TWSE to study liquidity costs. They are selected to cover a wide variety of characteristics (see Table 7). The data provided by the TWSE iclude daily order data, trasactio data ad disclosed price data from February 2008 to May 2008 (77 tradig days). Based o Eqs. 1 ad 2, we calculate the liquidity costs of all effective market order trasactios for the 10 securities. We igore the that arrive before the first best price is disclosed, sice the opposig best prices of these are ot available. The same applies to the that arrive after the market is closed. We are particularly iterested i the liquidity costs of effective market that were traded immediately right after the were submitted, as they were a strog idicatio of the market liquidity of a security. The liquidity costs for, whose trasactios took place Start TMLLCM Geerate evet series Order matchig evet (T) Market order submissio (M) Limit order submissio (L) Order cacelatio submissio (C) No activity (N) Determie the order size Determie the order size Decide to cacel a buy or a sell order matchig Matchig Determie to buy or sell Determie the order price, submit the order Determie to buy or sell Determie the order price, submit the order cacel Cacel a order radomly from the order book Fig. 1. The overall system workflow.

6 Y.-P. Huag et al. / Iteratioal Review of Fiacial Aalysis 23 (2012) Table 8 Descriptive statistics of the liquidity costs of effective market with immediate trasactios, based o TWSE data. Ticker Max Mi Mea Std. dev. Kurtosis Sum sq. dev No. of trasactios % 0.92% 0.02% , % 1.4% 0.01% , T 2.55% 1.58% 0% T 1.12% 1.68% 0.01% % 1.89% 0.05% , % 1.88% 0.06% , % 1.47% 0.04% , % 1.63% 0.04% , % 2.67% 0.07% , % 2.97% 0.05% ,450 after the have bee etered (ad waited for) i the order book, might be iflueced by other factors, such as opportuity cost, ad hece are less iformative about the liquidity of a security. Table 8 gives the liquidity cost statistics of the effective market with immediate trasactios that were executed durig the 77 tradig days. It shows that most securities have egative average liquidity costs, except for the two securities that have the lowest trasactio frequecy (01007T.TW ad 01008T.TW). The maximum amout of liquidity cost for a trasactio is less tha 3%. This idicates that these securities have high market liquidity. We oted that Chia Steel (2002.TW) ad TSMC (2330.TW) have the highest tradig frequecy (472,354 ad 483,565). This might be due to the fact that 2002.TW is a blue chip stock while 2330.TW has a high market capitalizatio. They are attractive to domestic ad foreig ivestors who seek secure ad stable returs. To further aalyze the market liquidity of these 10 securities, we partitioed the liquidity costs ito 6 differet value rages. After that, we computed the ratio of the trasactio volume of the effective market with immediate trasactios to the total trasactio volume of all (which iclude limit, ot immediately executed market ad so o). As show i Table 9, the tradig volume of this type of effective market order is more tha 50% of the total trasactio volume for all 10 securities. Meawhile, the majority (40 50%) of the trasactio volumes for these securities have their liquidity costs betwee 1% ad 0% (see the 6th colum of Table 9). These statistics further support them as high market liquidity securities. 5. Experimetal setup Based o the defiitio i Table 6, wecalculated p(μ), p(α), p(δ), p(), σ, σ limit ad σ market for the 10 securities i Table 10. Although TSMC (2330.TW) is the largest market capitalizatio stock ad is traded frequetly, the average daily order size is 14,000, which is much lower tha that of the Taiwa Top 50 Tracker Fud (0050.TW) (45,000 ). This might be because the Taiwa Top 50 Tracker Fud (0050.TW) is mostly traded by market makers, who ormally trade with a large volume, while TSMC (2330.TW) is traded by may differet kids of ivestors. For each of the securities, we made 10 simulatio rus, each of which simulates oe tradig day for the TWSE. The simulatio results are preseted ad aalyzed i the followig sectio. 6. Simulatio results ad aalysis Usig the 10 days of simulatio data, we calculated the daily average tradig volume ad the daily average umber of trasactios. We the compared them with that calculated from the 77 days of TWSE data. As show i Table 11, the results calculated from the simulatio data are higher tha those calculated from the TWSE data for almost all of the securities. This might be because, i TWSE, a crossig order could be split ito effective market ad effective limit (see Sectio 3). I most cases, the effective market order volume () is smaller tha the volume () of the effective limit order. However, the DFGIS model assumes that the effective market order size is the same as the effective limit order size σ, which is calculated as the Table 9 Ratio of the trasactio volume of the effective market with immediate trasactios to the total trasactio volume of all, based o TWSE data. LC rage ( 3%, 2%] ( 2%, 1%] ( 1%, 0%] (0%, 1%] (1%, 2%] (2%, 3%] Ticker Avg Max Avg Max Avg Max Avg Max Avg Max Avg Max % 65.88% 6.15% 24.51% % 0.71% 40.88% 79.82% 9.22% 52.45% 0.01% 1% 0.05% 0.05% 01007T 0% 0.12% 44.7% 98.79% 4.17% 50.83% 0.01% 2.22% 0.04% 0.04% 01008T 0.05% 7.35% 45.39% 100% 2.59% 71.43% 0.02% 3.45% % 12.74% 50.54% 71.49% 4.45% 26.33% 0.21% 18.4% % 17.1% 47.38% 69.5% 4.07% 33.37% 0.43% 24.09% % 4.56% 51.34% 68.51% 7.75% 21.25% 0.14% 8.45% % 1.51% 48.62% 67.43% 5.59% 18.9% 0.05% 5.81% % 8.9% 0.14% 6.54% 44.18% 77.66% 2.7% 21.82% 0.18% 11.41% 0.01% 0.01% % 13.68% 0.17% 8.91% 45.54% 67.54% 7.03% 50.54% 0.51% 17.13% 0.35% 0.35% Table 10 Computed p(α), p(μ), p(δ), p(), σ, σ limit ad σ market for the 10 studied securities. Ticker p(α) 10 3 p(μ) 10 3 p(δ) 10 3 p() 10 3 σ 10 3 σ limit 10 3 σ market T T

7 78 Y.-P. Huag et al. / Iteratioal Review of Fiacial Aalysis 23 (2012) Table 11 A compariso of daily tradig volumes ad the umber of trasactios. Ticker TWSE data Simulatio data DFGIS Simulatio data DFGIS-II Daily tradig volume (share) Daily o. of trasactios Daily tradig volume (share) Daily o. of trasactios Daily tradig volume (share) ,411, ,289, ,773, ,438, ,334, ,048, T 1,731, ,136, ,995, T 332, , , ,360,139 12,062 64,442,500 11,826 55,480,600 11, ,341,433 12,582 95,335,300 12,765 72,635,300 11, ,749, ,604, ,401, ,475, ,357, ,248, ,228, ,120, ,305, ,124, ,510, ,545, Daily o. of trasactios average size of both types of. I other words, the average order rate for effective market is overestimated. Cosequetly, the effective market order volume simulated based o σ is higher tha that for the real TWSE data. By usig two differet order size parameters, oe for effective market ad oe for effective limit, the DFGIS- II model effectively reduces the tradig volume. We also evaluated the liquidity cost statistics of effective market with immediate trasactios, based o the simulatio data (see Table 12). Compared to Table 8, the liquidity costs geerated by the simulatio data are higher tha those for the TWSE data. This might also be partially due to the overestimated effective market order size σ i our system. With a higher effective market order size, the liquidity costs of effective market with immediate trasactios are likely to be higher. The liquidity costs geerated from the DFGIS-II model are simulatio results after removal of the overestimatio of market order size. They are closer to the TWSE data tha the results of the DFGIS model, although they are still higher tha the TWSE data. We will discuss this issue further i Sectio 7. To rigorously evaluate the similarity betwee the simulated liquidity costs ad the TWSE data, we performed the Ma Whitey Wilcoxo (MWW) test o all 10 securities. The resultig p-values are 0 across all 10 securities, idicatig they are ideed differet from oe aother. Similar to Table 9, we computed the ratio of the trasactio volume of the effective market with immediate trasactios to the total trasactio volume for all uder liquidity cost for 6 differet rages. However, the value rages are partitioed slightly differetly from those of Table 9. This is because the liquidity costs of the simulatio data have a wider spread ( 11.11% 9.65% i DFGIS, 7.51% 7.01% i DFGIS-II) tha those of the TWSE data. Sice we are more iterested i positive liquidity costs, which are idicators of poor market liquidity, we grouped the egative liquidity costs ito oe bi ad added two bis for liquidity costs beyod 3%. The results are give i Table 13. As show, the liquidity cost upper boud of Chia Steel (2002.TW) ad TSMC (2330.TW) is 2%, which is the same for both DFGIS-II ad TWSE data. Meawhile, the two securities have higher liquidity cost trasactios ratios (1% 2%) for simulatio ad TWSE data that are similar to each other (the differece is 1% i DFGIS, 0.2% i DFGIS-II). Similarly, the egative liquidity cost trasactio ratio ( 12% 0%) for the simulatio ad TWSE data of these two securities are ot too far from each other either. However, they have may more trasactios with liquidity costs betwee 0% ad 1% for the simulatio data tha for the TWSE data. For a ivestor, whose mai cocer is to avert high liquidity costs, the DFGIS-II model produces liquidity costs that are cosidered to be similar to those for the TWSE data. Whe devisig liquidatio strategies for these two securities, this model ca be used to simulate liquidity costs uder differet strategies to idetify the optimal oes. What the, has distiguished these two securities from others? We examied the data statistics i Table 8 ad foud that they have a high umber of trasactios. This idicates that these two agetbased systems simulate liquidity costs more accurately for securities with a higher tradig frequecy. 7. Discussios The simulated liquidity costs have a wider spread ad higher values tha those for the TWSE data. This might be due to the followig reasos: 1. The average effective market order size (σ) used to ru the simulatio was overestimated. This ca be improved by usig two differet average effective order size parameters, as show i DFGIS-II. 2. Durig the simulatio, the time iterval betwee two order evets is radom ad idepedet, which is differet from that observed i the real fiacial markets. Frequetly, are clustered Table 12 Descriptive statistics of the liquidity costs of effective market with immediate trasactios, based o simulatio data. Ticker Max Mi Mea Std. dev. Kurtosis Sum sq. dev No. of trasactios DFGIS DFGIS-II DFGIS DFGIS-II DFGIS DFGIS-II DFGIS DFGIS-II DFGIS DFGIS-II DFGIS DFGIS-II DFGIS DFGIS-II % 0.65% 3.05% 0.82% 0.12% 0.06% ,470 16, % 1.25% 4.41% 1.72% 0.18% 0.01% T 9.65% 7.01% 6.42% 7.51% 0.5% 0.51% T 6.39% 3.74% 1.94% 1.49% 0.74% 0.51% % 1.47% 2.22% 1.68% 0.18% 0.16% ,047 80, % 1.41% 2.35% 1.84% 0.21% 0.17% ,490 81, % 2.39% 3.62% 2.8% 0.28% 0.24% ,213 71, % 2.68% 4.65% 3.5% 0.3% 0.27% ,727 38, % 5.45% 11.11% 6.28% 0.11% 0.12% ,320 10, % 2.58% 6.43% 3.23% 0.28% 0.19% ,556 15,594

8 Y.-P. Huag et al. / Iteratioal Review of Fiacial Aalysis 23 (2012) Table 13 Ratio of the trasactio volume of effective market with immediate trasactios to the total trasactio volume of all, based o the simulatio ad TWSE data. LC rage ( 12%, 0%] (0%, 1%] (1%, 2%] Ticker DFGIS DFGIS-II TWSE DFGIS DFGIS-II TWSE DFGIS DFGIS-II TWSE % 42.49% 45.52% 21.39% 9.04% 6.15% 2.69% % 30.09% 40.89% 22.7% 12.97% 9.22% 5.09% 0.21% 0.01% 01007T 13.51% 14.78% 44.7% 17.21% 15.87% 4.17% 7.72% 7.82% 0.01% 01008T 9.25% 13.74% 45.44% 13.14% 10.54% 2.59% 4.42% 7.1% 0.02% % 45.72% 50.68% 20.85% 19.93% 4.45% 1.21% 0.46% 0.21% % 48.06% 47.84% 19.61% 18.19% 4.07% 1.96% 0.29% 0.43% % 48.44% 51.43% 20.48% 20.05% 7.75% 4.54% 2.85% 0.14% % 43.35% 48.64% 19.1% 19.14% 5.59% 4.99% 3.25% 0.05% % 34.13% 44.41% 18.31% 18.26% 2.7% 4.31% 2.02% 0.18% % 38.64% 45.84% 17.09% 16.13% 7.03% 5.41% 1.99% 0.51% LC rage (2%, 3%] (3%, 4%] (4%, 10%] Ticker DFGIS DFGIS-II TWSE DFGIS DFGIS-II TWSE DFGIS DFGIS-II TWSE % % 0.05% 0.28% 0.06% 01007T 3.28% 3.17% 0.04% 1.57% 1.93% 1.26% 1.33% 01008T 3.25% 3.73% 2.09% 2.29% 2.94% % % 0.03% % 0.44% 0.21% % 0.98% 0.01% 1.03% 0.53% 0.74% 0.1% % 0.19% 0.35% 0.96% 0.28% umber of trasactios umber of trasactios Fig. 2. The trasactio volume (share) vs. the umber of trasactios i the TWSE (left) ad simulatio (right) data. together i a certai umber of time periods, ad ot evely distributed throughout a day. This might have cotributed to the higher liquidity costs i the simulatio data. 3. Durig simulatio, the order evets were geerated radomly, based o the model parameters, without cosultig the order book. This is differet from the reality, where a ivestor ormally checks the order book of the opposite side to make sure a profitable matchig is possible before submittig a order. I other words, although the order distributio i the simulatio system is the same as that for the TWSE (we used the TWSE data to estimate the probability of order submissios), the sequece of the order submissios i the simulatio system is ot optimized as is that devised by huma traders. Cosequetly, the simulated liquidity costs are likely to be higher tha those for the TWSE data. 4. I the simulatio system, the price of a effective market order is set to be the highest possible bid (buy order) or the lowest possible ask (sell order) allowed by the TWSE to guaratee a immediate trasactio. This hardly happes i reality. Normally, a trader would seek a price that geerates a trasactio, without goig to the extreme of the highest possible bid/lowest possible ask. As a result, the simulated liquidity costs are likely to be higher tha those for the TWSE data. The secod issue has bee ivestigated by Egle ad Russell (1998). I particular, they devised a Autoregressive Coditioal Duratio (ACD) model to more realistically simulate the order arrival time, price ad volume i a stock market. Huag (2010) 5 showed a simple example of itegratig ACD without diural adjustmet i the DFGIS model, but diural adjustmet is actually eeded to geerate a iverted U shaped daily duratio patter. The aalysis of items 3 ad 4 suggests that traders who employ itelligece (e.g., icorporatig order book iformatio) to make tradig decisios i a real stock market produced trasactios with lower amouts of liquidity costs tha that produced by the zeroitelliget agets i our artificial stock market. To simulate the real market behavior, i terms of the liquidity costs, we eed to istall itelligece (e.g., learig ability) i the artificial agets i our system. We will explore this aveue of research i future work. 5 For the Eglish versio, please cotact the correspodig author.

9 80 Y.-P. Huag et al. / Iteratioal Review of Fiacial Aalysis 23 (2012) Oe itelliget behavior demostrated by the TWSE traders is a more profitable liquidatio strategy. As show i DFGIS-II i Table 13, of the daily total tradig volume of the Taiwa Top 50 Tracker Fud (0050.TW), 51.53% cosists of tradig volume from the effective market with immediate trasactios with 42.49% payig egative liquidity cost, ad 9.04% payig liquidity cost of betwee 0 ad 1%. By cotrast, the TWSE data show that 51.67% of the daily total tradig volume of this security is tradig volume from the effective market with immediate trasactios with 45.52% payig egative liquidity cost ad 6.15% payig liquidity cost of betwee 0 ad 1%. I other words, give the task of liquidatig a large block of securities (aroud 50% of the daily tradig volume i this case), the TWSE traders accomplished the task by payig a lower amout of liquidity cost tha the cost paid by the zero-itelligece artificial traders. What strategy has delivered such savig? We aalyzed the Taiwa Top 50 Tracker Fud (0050.TW) trasactios data from effective market o March 20, Fig. 2 (left) shows that there are may more small-volume trasactios tha largervolume oes. I particular, more tha 500 trasactios are with 5000 or 10,000 tradig. This is very differet from the simulatio data (see the right of Fig. 2), where the umber of small-volume trasactios is ot dramatically differet from that of the large-volume oes (the scale is 20 to 1). This suggests that TWSE traders submitted may smaller-size istead of a large-size order to coduct trasactios. This strategy has led to a lower amout of liquidity costs. We pla to icorporate this itelliget behavior i the artificial agets i our system. 8. Cocludig remarks The market liquidity of a security plays a importat role i fiacial ivestmet decisios ad i the liquidatio strategies of the security. As a alterative to Algorithmic Tradig, this study has developed a aget-based model to examie the liquidity costs of stocks ad securities traded i the Taiwa Stock Market. For the 10 TWSE stocks ad securities that we studied, the modelsimulated liquidity costs are higher tha those for the TWSE data. We idetified four possible factors that cotribute to this result: The overestimated effective market order size, which ca be improved by usig two average order size parameters. The radom market order arrival time desiged i the DFGIS model, which might be improved by icorporatig the ACD model i our system. The zero-itelligece of the artificial agets i our model. The price of the effective market order. We ca cotiue improvig the model by addressig the abovemetioed issues. A model that behaves i a similar way to the TWSE i terms of the liquidity costs ca be used to study liquidity costs ad to devise liquidatio strategies for stocks ad securities traded o the TWSE. Ackowledgmets A early versio of this paper was preseted at the Ecoophysics Colloquium 2010, Academia Siica, Taipei, Taiwa, November 4 7, The authors beefited sigificatly from the discussios with coferece participats. This versio has bee substatially revised i light of two aoymous referees' very paistakig reviews, for which we are most grateful. The NSC grat H MY3 is also gratefully ackowledged. Refereces Almgre, R., & Chriss, N. (2000). Optimal executio of portfolio trasactios. Joural of Risk, 3(22), Daiels, M. G., Farmer, J. D., Gillemot, L., Iori, G., & Smith, E. (2003). Quatitative model of price diffusio ad market frictio based o tradig as a mechaistic radom process. Physical Review Letters, 90(10), Egle, R. F., & Russell, J. R. (1998, September). Autoregressive coditioal duratio: a ew model for irregularly spaced trasactio data. Ecoometrica, 66(5), J.D. Farmer, P. Patelli ad I.I. Zovko. Supplemetary material for the predictive power of zero itelligece i fiacial markets, T.X. Guo. A aget-based simulatio of double-auctio markets. Master's thesis of Graduate Departmet of Computer Sciece, Uiversity of Toroto, Yi-Pig Huag. Liquidity cost of market i the Taiwa Stock Market: A study based o a order-drive aget-based artificial stock market. Master's thesis of Departmet of Fiacial Egieerig ad Actuarial Mathematics, Soochow Uiversity, Kali, D., & Zagst, R. (2004). Portfolio optimizatio uder liquidity costs. Workig paper, Germay: Risklab Germay GmbH. Weisstei, E. W. (2005). Half-ormal distributio. Half-NormalDistributio.html