Is the Extension of Trading Hours Always Beneficial? An Artificial Agent-Based Analysis

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1 Is the Extension of Trading Hours Always Beneficial? An Artificial Agent-Based Analysis KOTARO MIWA Tokio Marine Asset Management Co., Ltd KAZUHIRO UEDA Interfaculty Initiative in Information Studies, The University of Tokyo / CREST, Japan Science and Technology Agency Abstract The extension of trading hours to provide more trading opportunities and improve price efficiency has increasingly been discussed. However, currently, there is limited trading activity during the stock market s extended-hours trading session. Thus, we should examine whether the extension of trading hours is still effective for creating more trading opportunity and price efficiency even if there are few market participants during the extended session. For this study, we build an agent-based market model based on that of Brock and Hommes (1998) and analyze the effect of extending trading hours. We find that although extending trading hours could increase daily trading volume, it could distort price formation and trade opportunity if market participants are limited during the extended-hours session. Specifically, the extension could result in more concentrated trading in the opening session, wider divergence between market prices and the fundamental value of stocks, and higher return volatility (especially at the open). 1. Introduction Recently, the extension of trading hours for stocks has increasingly been discussed. In several markets (e.g., NYSE and NASDAQ), both pre-market and after-hours trading sessions have already been introduced, and the Tokyo Stock Exchange is considering extending trading hours by introducing extended-hours sessions and/or shortening the midday recess. The extension of trading hours is intended to provide more trading opportunities and improve price efficiency (Osaki, 2014). Periodical market closures can cause significant negative effects on price efficiency and trading opportunity. First, periodical market closures might impede stock prices from incorporating public and private information. Kyle (1985), Glosten and Milgrom (1985), Foster and Viswanathan (1990), and Easlay and O Hara (1992) show public and private information accumulates overnight while information asymmetry declines over the course of trading periods. These studies suggest market closures may induce a delay in the incorporation of information into stock prices, which can widen divergence between stock prices and their fundamental values. Second, periodical market closure may cause excessive price fluctuations, especially at the beginning and end of the trading session on 1

2 an intraday basis. Wood et al. (1985) and Harris (1986) find that a standard deviation of returns is especially high at the open and close of trading; this U-shaped pattern of return volatility is also found in non-u.s. markets (Hamao and Hasbrouck, 1995; Abhyankar et al., 1997). Third, periodical market closures can cause skewed trading activity, i.e., trading is concentrated at the beginning and end of trading sessions. Jain and Joh (1988) document a U-shaped intraday pattern of trading volume; trading volume is especially high at the open and close of trading. Finally, obviously, periodical market closures could reduce investors trading opportunities. Therefore, the extension of trading hours is likely to mitigate market inefficiencies caused by market closures, i.e., extending trading hours could lower divergence between market prices and fundamental values, lower stock return volatility (especially at the open and close), increase daily trading volume, and ease concentration of trading activity at the open and close. However, limited investor participation during extended hours is a concern regarding the effect of extending trading hours. In the U.S., where extended trading hours have already been introduced, the trading volume per unit of time during extended hours is less than 5% as that during regular trading hours (Barclay and Hendershott, 2004). Although trading during extended hours allows investors to quickly react to after-market news, market prices are less efficient during extended hours compared to regular hours due to reduced liquidity (Barclay and Hendershott, 2003). It is quite uncertain whether extending trading hours would mitigate the market inefficiencies if there are few market participants during extended-hours sessions. In this study, we uncover the effect of extending trading hours assuming limited market participation during the extended-hours sessions. There are a few studies that empirically analyze the effect. For example, Houstion and Ryngaert (1992) find reductions in NYSE trading hours had little effect on return volatility and trading volume during the week the reductions occurred, but did have an effect on the distribution of return volatility and trading volume during the week. The study of Fan and Lai (2006) reveals significant change in the intraday pattern of return volatility and trading volume could not be observed after extending the trading session of the Taiwan Stock Exchange by 1.5 hours. Although these studies might indicate the market inefficiencies are not easily mitigated by a change in trading hours, the result might also be due to an insufficient change in trading hours. The drawback of empirical analyses on the extension of trading hours is there is no perfect sample with which to compare prices and trading behavior between a market with extended-hour sessions and one without extended hours. There is an obvious limitation with respect to showing the effect of extended hours via empirical analyses. On the other hand, model-based analysis allows for direct comparison of price behavior and trading activity between a market with extended hours and one without extended hours. In addition, we can easily understand underlying reasons the extension of trading hours is effective or ineffective 2

3 when there are limited market participants during extended-hours sessions. Thus, in this study, we perform model-based analysis to analyze the effect. We specifically designed our simulation model based on that of Brock and Hommes (1998) rather than existing models that are built for analyzing the effect of market closures; a few studies built analytical models to analyze the effect of market closure on price and trading volume as well. Brock and Kleidon (1992) expand on Merton s model to analyze the effect of market closures on the trading concentration at the open and close. To analyze the intraday pattern of price and trading behavior, Hong and Wang (2000) develop a competitive market model with periodic closures where investors trade for allocation and informational reasons. The model by Brock and Kleidon (1992) is in a partial equilibrium setting and cannot show the equilibrium dynamics between returns and trading volume. More importantly, since both models are analytically tractable general equilibrium models, in these models it is assumed there are enough market participants; in other words, the models cannot analyze the effect of the number of market participants on market prices and trading activity. Therefore, these models do not allow for analyzing the effect of extended-hours sessions with limited market participants. On the other hand, the model by Brock and Hommes (1998) is a simple simulation-based model with evolutional dynamics; the model s strengths are simplicity and high flexibility (high scalability). Thus, in this study, we expand the model by incorporating extended trading hours with limited investor participation. First, we show the extended model can reproduce the U-shaped intraday pattern of return volatility and volume, gradual incorporation of fundamental information during regular trading hours, and two important stylized facts: fat-tailed returns and clustered volatility. Next, we examine the effect of illiquid extended-hours sessions. Specifically, we compare the following two cases: Case 1 includes investors who can only trade during regular hours and case 2 involves investors who can trade 24 hours per day, but there are limited market participants during extended-hour sessions. Then, we examine differences in the following three factors between the cases: the deviation between stock prices and fundamental values, volatility of stock returns (especially at the open and close), and trading volume (especially at the open and close). If illiquid extended-hours sessions mitigate the negative effect induced by market closure, the deviation between market price and fundamental value, return volatility, and trading concentration at the open and close should be smaller, and daily trading volume should be higher for case 2 than for case 1. The paper is structured as follows. Section 2 derives the market model with limited extended-hours trading. Section 3 presents the simulation results for whether the extension of trading hours is beneficial, even if there are limited market participants during the extended-hours session Section 4 concludes. 3

4 2. Market Model In this section, we explain the agent-based market model used. We built a simple artificial market model based on that of Brock and Hommes (1998), including the extended-hours session with limited investor participants. 2.1 Basic Model Price Determination Process Agents can either invest in a risk free asset or in a risky asset. The risk free asset has perfect supply elasticity, and in the short-term investment horizon, the interest rate and dividend yield are irrelevant; therefore, we suppose the interest rate is zero and the risky asset (e.g. a stock) pays no dividend. Let be the price per share of the risky asset at time t. First, we show a price determinant process when there is no periodic market closure. The process is similar to that of Brock and Hommes (1998) under the condition the assets pay no dividend and the interest rate is zero. Agents are myopic mean-variance maximizers, so the demand per trader i for the risky asset is calculated: E i,t and V i,t denote the beliefs (forecasts) of trader i about conditional expectation and conditional variance of return -1, and a is the risk aversion parameter. Bold face variables denote random variables at date t + 1. The conditional variance V i,t = σ 2 is assumed to be equal and constant for all investors. Let z s denote the supply of outside risky shares per investor, and it is assumed to be constant. When there are N traders, equilibrium of demand and supply yields: (1) Brock and Hommes (1998) focus on the special case of zero supply of outside shares, i.e. z s = 0, for which the Walrasian market clearing price satisfies: (2) The volume of trade TV at time t is given by: 1 TVt zi, t zi, t1 2 i (3) Heterogeneous Beliefs and Evolutionary Selection of Strategies Regarding traders heterogeneous expectations about future prices, we basically follow the 4

5 heterogeneous expectation process of Brock and Hommes (1998). All traders are assumed to be able to derive the fundamental price p t * that would prevail in a perfectly rational world. The fundamental price continuously reflects the upcoming fundamental news; the value is assumed to be varied over time as: p * t1 p * W t e W ~ N(0, ) f (4) Traders believe that in a heterogeneous world prices may deviate from their fundamental value p t *.Following the four investor type model of Brock and Hommes (1998), we assume the model has four investor types: fundamentalists (denoted as X 1 ), trend followers (denoted as X 2 ) who allow price deviation from fundamental value, and two investor types with purely biased beliefs: optimists (denoted as X 3 ) and pessimists (denoted as X 4 ), who expect a constant price above (optimists) or below (pessimists) the fundamental price. * E [ ] i p t 1 p t 1 X1 * * g t 1] pt 1 ( pt / pt X 2 * t 1] pt 1 (1 d) X3 * t 1] pt 1 (1 d) X4 E [ p ) E[ p E[ p where d > 0, 1 i Fundamentalists (5) i Trend followers (6) i Positively biased (optimists) (7) i Negatively biased (pessimists) (8) The evolutionary part of the model, describing how beliefs are updated over time, follows the endogenous selection of forecasting rules introduced by Brock and Hommes (1997); the probability Pr{ i X } is given by: t s P Pr t{ i X s} upd exp( U s P s, t1 exp( U upd ) (1 P ) s, t1 exp( U s s, t1 exp( U ) s, t1 ) upd ) if i X if i X s s at time t 1 (9) at time t 1 Where 0 P upd 1 U s,t is the fitness measure of strategy s evaluated at time t. A natural candidate for evolutionary fitness is realized profits, given by: U s, t E[ pt ]/ pt ( pt / pt 1 1)( 2 a 1 1 ) 2.2 Extension of the Model Price Determination Process during the Extended-Hours Session Now, we consider the case when there is a periodic market closure or the extended-hours 1 The way in which each type of investors forecast future price is slightly different from that of Brock and Hommes (1997), due to keep (and asset price) positive. 5

6 session has imperfect market liquidity. We assume investors can trade a risky asset 24-hours day. A daily trading session consists of a regular-hours session (during which there is successive N regular time steps) and an extended-hours session (successive N extended time steps). We call the first time step of the regular-hours session and the last time step of the regular-hours session the opening session and the closing session, respectively. In addition, we call a step subsequent to the opening session the subsequent session. All the investors trade the asset during the regular-hours session; on the other hand, only a limited number of investors trade outside the regular hour session (during the extended-hours session). X both denotes investors who trade the asset during both the regular-hours and extended-hours sessions, X regular denotes investors who trade the asset only during the regular-hours session, and P e denotes the ratio of investors who trade the asset during the extended-hours session (P e =0 means all the investors trade only during the regular-hours session). During the regular-hours session, except for the last time step of the regular-hours session, the demand z i,t. can be given by (1). Thus, a price of the risky asset is given by (2). On the other hand, the price is determined differently in the closing session and during the extended-hours session. The demand z i,t of investors who trade only during the regular-hours session is set to be constant during the extended-hours session; this setting means there is no trading activity among these investors during the extended-hours session. Thus, the demand z i,t during the extended-hours session is given by: z i, t Ei, t [pt 1 / pt ] 1 i X 2 both a (10) z i X i, t1 regular Thus, equilibrium of demand and supply yields: ix E i, t [pt 1 / pt ] 1 2 a i both X regular z i, t1 0 Thus, market clearing price satisfies: pt N both ix both E 2 a i, t[pt 1] ix z regular i, t 1 (11) Price Determination Process in the Closing Session As discussed in the arguments of Brock and Kleidon (1992) and Hong and Wang (2000), investors who trade only during the regular-hours session assume overnight risk in the closing session. The risk aversion parameter for these investors could be higher than that for investors who also trade during the extended-hours session. To incorporate this possibility, we define the risk 6

7 aversion parameter b i (b i >=a) at the close by: abi i X regular b i (12) a i X both Where B i -1 is assumed to follow exponential distribution: B i 1~ Exp (1/ ) (13) Therefore, the demand z i,t is given by: z i, t E i, t [p t1 2 bi / p ] 1 t (14) Thus, equilibrium of demand and supply yields: i E i, t [pt 1 / pt ] b i The market clearing price satisfies: p t i E i, t[ pt 1 b i ] i 1 b i (15) 3. Simulation Results 3.1 Simulation Settings We define 24 time steps as comprising one trading day (1 time step per hour). We denote each time step as T1, T2 T24. The first 6 time steps each day comprise the regular-hours session and the remaining 18 time steps comprise the extended-hours session. We set the simulation length at 250 days, the number of investors at 100, the risk aversion parameter in Formula 1 ( at 1, the volatility parameter for the fundamental value in Formula 4 ( at 0.01, and the constant estimated return volatility in Formula 1 ( at 1. We choose to let P e (the ratio of investors who trade during the extended-hours sessions) = {0, 0.1, , 1.0}, d (the parameter of biased estimation in Formulas 7 and 8) = {0, 0.1, 0.2, 0.3, 0.4}, g (the parameter of trend chasing in Formula 6) = {0, 0.25, 0.5, 0.75, 1.0, 1.25,1.5}, (the parameter of evolutionary updating of strategies in Formula 9) = {0, 5, 10, 25, 50}, and P upd (the parameter of strategy update speed in Formula 9) = {0, 0.1, , 1.0}. We choose to let (the parameter of overnight risk in Formula 13) = {1, 3, 5, 7, 9} and examine when investors do not concern overnight risk ( ). All statistics and the following figures use an average of 100 simulation runs. 7

8 3.2 The Effect of Periodic Market Closures In this section, we examine whether our model can reproduce a U-shaped intraday pattern of return volatility and trading volume when there is periodic market closure. In addition, we examine whether our model explains two important stylized facts: fat-tailed returns and clustered volatility, which are reported by several prior studies (e.g. Mandelbrot, 1963, 1997; Pagan, 1996; Cont et al., 1997). Several artificial market model studies (e.g. LeBaron, 2006; Chen et al., 2012) examine whether their model can explain these stylized facts to verify their artificial market models. The simulation analyses in Section 3.2 are performed under the condition there is periodic market closure Trading Concentration at the Open and Close We evaluate average trading volume in the opening and closing sessions. In addition, since the effect of market closure on trading activity could last more than one time step after the opening bell, we also calculate average volume in the subsequent session. We calculate the ratios of those to average trading volume in other regular-hours sessions (defined by the time steps during regular-hours sessions except in the opening, subsequent, and closing sessions). To test the trading concentration, we examine whether those ratios are higher than 1. The simulation test reveals the trading concentration can be observed unless P upd is large enough. Figure 1(a) and Table 1(a) show the result when we change the parameter P upd from 0 to 1 at an interval of 0.1, under the condition d, g, and λ are set at 0.2, 1.25, and 5, respectively. As shown in the table and figure, the larger P upd is, the smaller the ratio is. This simulation result reveals the trading volume at the close and open could be lower than that during the other regular-hours session if P upd 0.5. A large value for P upd means the investor strategy is highly influenced by the latest price movement. Thus, if P upd is large, trading volume is mainly determined by a change in investor strategy induced by immediate price movement rather than a change in investor position induced by periodic market closures. Thus, the trading concentration at the open and close cannot be observed when P upd is large. In addition, average volume in the closing session is not significantly higher than that during the other regular-hours session if there is no biased trader (d=0). Figure 1(b) and Table 1(b) show the result when we change the parameter d from 0 to 0.4 at an interval of 0.1, under the condition P upd, g, and are set at 0.2, 1.25, and 5, respectively. As shown in the table and figure, only when d=0 does the ratio of average volume at the close to that during the other regular-hours session almost equal 0, and trading activity is extremely concentrated at the open. If there is no biased trader, prices converge to fundamental values during the regular-hours session. The convergence to fundamental values means expected returns have converged to zero during the regular-hours session. Since expected returns are close to zero at the close, investors positions regarding the risky asset is almost 8

9 zero at the close. Therefore, average volume at the close is lower than that during the other regular-hours session. Finally, the result reveals that average volume in the closing session is not significantly higher than that during the other regular-hours session if investors do not assume overnight risk ( ). Assuming overnight risk causes heterogeneity with respect to their investment policy that increases trading volume. Thus, the result indicates that overnight risk plays a key role for trading concentration at the close. In sum, trading concentration at the open and close can be observed unless d = 0 (there is no biased traders), P upd is large, and (investors assume overnight risk at the end of the regular trading session). [Table 1] [Figure 1] U-shaped Intraday Volatility Pattern We calculate a standard deviation of stock returns in the opening session, subsequent session, closing session, and during the other regular-hours session. To test the U-shaped pattern of return volatility, we examine whether volatility at the open and close is higher than that during the other regular-hours session. The simulation result reveals the volatility-pattern can be observed unless P upd = 0. Especially, return volatility at the close is not higher than that during the other regular-hours session if P upd = 0. Table 2 and Figure 2 show the result when we change the parameter P upd from 0 to 1 at an interval of 0.1 under the condition d, g, and are set at 0.2, 1.25, and 5, respectively. As shown in the table and figure, return volatility at the close is not higher than that during the other regular-hours session only when P upd is set at 0. P upd = 0 means investors do not update their investment strategy (there is no evolutional dynamic in investors strategy). Thus, change in investors expected returns is mainly determined by a change in fundamental value. Thus, while return volatility is high at the open due to a large change in fundamental value, which has been accumulated at the market close, return volatility at the close is the same as that during the other-regular-hours session because the change is not accumulated. Thus, the U-shaped intraday pattern of return volatility cannot be observed when P upd = 0. Finally, higher volatility at the close cannot be observed if investors do not assume overnight risk ( ). Thus, the result indicates that overnight risk plays a key role for high return volatility at the close. [Table 2] [Figure 2] 9

10 3.2.3 Gradual Incorporation of Fundamental Information We evaluate a degree of incorporation of fundamental information by dispersion between market price and fundamental value. To analyze gradual incorporation of fundamental information during the regular-hours session, we examine whether the dispersion is lower during the other regular-hours session than at the open. The result reveals gradual incorporation can be observed regardless of the parameters settings. The model can reproduce the price fluctuation where the gradual incorporation of fundamental information into asset prices is observed during the regular-hours sessions Fat-tail Return Distribution A fat-tail means the kurtosis of stock returns is more than 3. Thus, we evaluate fat-tail return distribution by calculating the kurtosis of single period returns (returns for 1 hour). The simulation results reveal the kurtosis is higher than 3 regardless of the parameters settings. All runs replicate the fat tails Volatility Clustering Financial returns exhibit clustered volatility, i.e. slow decay of autocorrelations of squared returns (LeBaron, 2006; Chen et al., 2012). We examine whether the model can reproduce price fluctuation where this volatility clustering is observed. We evaluate the volatility clustering by autocorrelation coefficients for square returns with one lag. The result reveals the autocorrelation coefficients are smaller as g gets smaller. Figure 3 shows the result when we change the parameter g from 0 to 1.5 at an interval of 0.25, under the condition d, P upd, and are set at 0.2, 0.2, and 5, respectively. The simulation results reveal that when, the clustered volatility is relatively small. Under this condition, even trend followers predict the price eventually converges to fundamental value; trend followers can be considered as quasi-fundamentalists. Thus, our results indicate there should be strong trend followers who allow current divergence between the market asset price and the fundamental value to replicate the volatility clustering. This result is consistent with previous studies findings, which show trend followers cause clustered volatility (Engle and Bollerslev, 1986). [Figure 3] Adequate Parameters In sum, the model replicates the U-shaped intraday pattern of trading volume and return volatility, gradual incorporation of fundamental information during regular-hours sessions, and the two statistically existing stylized facts, under the conditions: (1) There are strong trend followers (at least, ) 10

11 (2) There are biased traders (d>0) (3) Investors update their strategy, but not excessively (at least 0< P upd < 0.5) (4) Investors assume overnight risk at the end of regular trading sessions (b>a). From a different viewpoint, it can be said the simulation results support the view there are strong trend followers and biased traders in the market, and investors gradually update their investment strategies based on recent price movement. 3.3 The Effect of Extending Trading Hours The parameters that satisfy the conditions mentioned in Section can be regarded as adequate model parameters verified by the stylized facts, intraday volume and volatility patterns, and the gradual incorporation of fundamental information. Specifically, we show the result when we set d, g,,, and P upd at 0.2, 1.25, 5, 10, and 0.2, respectively. However, the implication of the simulation result is invariant regardless of the parameter settings as long as the settings satisfied the above-mentioned conditions. We change P e (the ratio of market participants who trade during the extended-hours session) from 0 to 1 at an interval of 0.1. Then, we examine the effect of an extension of trading hours on 1) price efficiency, which is evaluated by the deviation between stock prices and fundamental values, 2) return volatility during the regular-hours session and the U-shaped intraday pattern of return volatility, and 3) daily trading volume and trading concentration at the open and close Price Efficiency We examine whether the deviation between prices and fundamental values is lowered by extending trading hours, even if there are limited market participants during the extended-hours session. We change P e from 0 to 1 and calculate a difference between market price and fundamental value, which is defined by, during the regular-hours sessions and that during the extended-hours sessions. Table 3 and Figure 4 show the simulation results. The result, shown in Figure 4(a), reveals when there are enough market participants during the extended-hours session (when P e 0.4), the stock price is closer to the fundamental value than when there is periodic market closure. However, when there is not enough market participation during the session (especially when P e 0.2), the stock price diverges from the fundamental value more than when there is periodic market closure. These results suggest the extension of trading hours could result in lower price efficiency when there are not enough market participants during the extended-hours session. Interestingly, not only during the extended-hours session but also during regular hours, price diverges more from fundamental value if P e 0.2 than when there is a periodic market closure. The result, shown in Table 3, also reveals stock price diverges more from fundamental value during the last time step of the extended session (T24), indicating illiquid trading 11

12 has been disturbing an incorporation of fundamental information into the asset price during the extended session. 2 The divergence is narrowed after the opening bell, indicating an increase in market participants improves price efficiency. However, as shown in Figure 4(b), the widened divergence is not completely corrected at the open (T1); the negative effect of the illiquid extended-hours session on price efficiency remains even after the opening bell. [Table 3] [Figure 4] Return Volatility We examine whether return volatility during the regular-hours session (especially at the open and close) is decreased by the extension of trading hours, even if investors rarely trade during the extended-hours sessions. We calculate a ratio of return volatility at the open to that during the other regular-hours session, and a ratio of return volatility at the close to that during the other regular-hours session. We examine whether these ratios and the return volatility during the regular-hours session decreases as P e increases. If so, we can say the extension of trading hours decreases return volatility and weakens the U-shaped intraday pattern of return volatility even if there are few market participants during the extended-hours sessions. The result, shown in Table 4 & Figure 5, reveals when there are enough market participants during the extended-hours session (when P e 0.3), return volatility during the regular-hours session is lower, and return volatility at the open and close is closer to that during the other regular-hours sessions versus when there are periodical market closures. However, when there are not enough market participants during the extended-hours session (when P e < 0.3), return volatility during the regular-hours session and the ratio of return volatility at the open to that during the other regular-hours session are higher than when there are periodic market closures. As discussed in Section 3.3.1, if market participants are limited during the extended-hours session, price diverges more from the fundamental value during the session. Since this divergence from fundamental values is corrected in the opening and subsequent sessions, return volatility increases in the opening and subsequent sessions. [Table 4] [Figure 5] Trading Volume We examine whether illiquid extended-hours sessions increase daily trading volume and 2 The simulation result reveals stock price is highly volatile during the extended-hours session, consistent with the empirical findings of Barcley and Hendershott (2003). 12

13 weaken trading concentration at the open and close (the U-shaped intraday pattern of trading volume). We calculate daily trading volume, a ratio of average volume at the open to that during the other regular-hours session, and a ratio of average volume at the close to that during the other regular-hours session. The result, shown in Table 5 and Figure 6(a), reveals the volume per day increases as P e increases, indicating the extension of trading hours increases daily trading volume even if there are limited market participants during the extended-hours session. However, as shown in Figure 6(b), the ratio of average volume at the open to that during the other regular-hours session is higher than when there are periodical market closures if there are not enough market participants during the extended-hours session (P e <0.5). This result indicates trading concentration at the open is increased by the extension of trading hours if there are few market participants during the session. The increased concentration at the open could be attributed to wide divergence between market prices and their fundamental values just before the opening bell. The large gap between market prices and fundamental values just before the opening bell results in increased trading activity at the open, which results in a much smaller gap. Therefore, the trading concentration at the open is not lowered by the extension of trading hours when there are not enough market participants during the extended-hours session. [Table 5] [Figure 6] 3.4 Additional Test and Discussion An Effect of Evolutionary Updating of Strategies In our study, we run the simulation under the condition investors update the strategy according to an evolutionary fitness measure ( = 10). On the other hand, we find that, even if investors select the strategy randomly ( = 0), illiquid extended-hours trading could disturb price formation and trading activity. This result is consistent with our prediction that unstable price behavior induced by limited investor participant is a key factor for the negative impact on price efficiency and trading activity. However, we also find that if is higher, i.e., updating of strategies is more determined by evolutional fitness measure, illiquid extended-hours trading has more negative impact on price efficiency. Table 6 shows how the negative impact is affected by β. We set β at 0, 10, and 100 3, and examine how much the deviation from fundamental values and return volatility during regular session and the ratios of volatility and volume in the opening session to those during the other regular-hours session are increased by introducing illiquid extended-hours markets (P e =0.1). The results, shown in Table 6, reveals that if investors strongly follow past successful strategy ( =50), illiquid extended-hours trading (P e =0.1) increases the deviation between stock prices and fundamental values by 19.6 %, while it increases the deviation by 8.4 % if investors randomly select 3 P upd, d, g, and λ are set at 0.2, 0.2, 1.25, and 5, respectively 13

14 strategies ( =0). In addition, if is higher, illiquid extended-hours trading more increases return volatility especially during other regular-hours 4. These results suggest that evolutionary updating of strategies enhances the negative impact of illiquid extended-hours trading on price efficiency Strategy Update during Extended-hours Session To keep the model as simple as possible, we assume that investors update their forecasting rules whichever investors trade during the extended-hours session or not. However, investors who do not trade during the extended-hours session might not update their forecasting rules because they are not interested in extended-hours trading. Therefore, we also perform a simulation analysis under the condition these investors do not update their strategies during the extended-hours session. We find that the implication of the simulation analysis, i.e. the indication that illiquid extended-hours trading could have negative impact on price efficiency and trading activity, is invariant regardless of whether investors update their strategies during the extended-hours session, or not Simplified Extended-hours Setting In the U.S. stock markets, extended-hours trading occurs in two sessions: the pre-market session is from 8:00AM to 9:30AM, and the after-hours session is from 4:00PM to 6:30PM. It seems we should utilize the model where daily trading sessions consists of the regular hour trading session, the pre-market session, and the after-hours session, and there is periodic market closure after the after-hours session. However, our purpose is not to analyze on the extended-hour session of specific markets (e.g. the U.S. stock market) but to examine whether trading outside of the regular trading hours generally create more trading opportunity and price efficiency. To analyze the effect of trading outside of the regular trading hours, it is enough to utilize the simplified model where the trading sessions consist of two sessions: the regular hour trading session and the extended-hours trading session. Since it is important to keep the framework as simple as possible to enhance the understanding of financial market dynamics, we perform analysis based on the simplified model. 4. Conclusion We built a simple agent-based market model based on that of Brock and Hommes (1998), which includes the extended-hours session with limited investor participants. By utilizing the model, we examine whether the extension of trading hours is effective for creating more trading opportunity and price efficiency, even if there are few market participants during the extended-hours session. 4 On the other hand, the negative impact regarding high volume and volatility at open is not affected by. 5 The details of the results are available upon request 14

15 We find extending trading hours increases daily trading volume. However, it could increase a divergence between market prices and fundamental values not only during the extended-hours session, but also during the regular-hours session. In addition, we find the extension could result in higher return volatility during the regular-hours session and does not mitigate the U-shaped intraday pattern of return volatility when there are not enough market participants during the extended-hours session. Finally, the extension could increase trading concentration at the open. The simulation reveals the negative impact of the illiquid extended-hours session is observed if there are limited market participants during the extended-hours session (if there is less than 30% as much market participation during the extended-hours session as during the regular-hours session), and there are biased investors and strong trend followers. It seems the extension rarely causes the aforementioned negative impact on trading activity and price formation in actual stock markets. However, in terms of illiquidity during the extended-hours session, market participation during the extended-hours session is actually quite limited: there is less than 5% as much trading per time unit in after-hours sessions versus regular trading sessions in the U.S. stock market (Barclay and Hendershott, 2004). In terms of trend followers and biased traders, our analysis (and previous study) shows the stylized facts and the U-shaped intraday pattern of volume and volatility can be observed as long as there are strong trend followers and biased traders. Thus, it is highly possible the extension of trading hours has a negative impact on trading activity and price formation, in actual stock markets. In sum, our findings give important indication about increasing discussion regarding the extension of trading hours. As argued in previous studies, periodic market closures could distort trading activity and price formation, e.g. high return volatility and trade concentration especially at the open and close, and delay incorporation of fundamental information into prices. It seems this problem can easily be solved by extending trading hours. However, our results suggest the problem is not so simple; the extension of trading hours could disturb price formation and trading activity if market participation during the extended-hours session is limited. Since market participation during the extended session is limited in actual stock markets, this finding raises the possibility that the extension of trading hours has a negative impact on actual stock markets. Our finding emphasizes the importance of debating whether many investors will trade during the extended-hours session, and how to encourage investors to trade during the session before extending trading hours. References Abhyankar, A., Ghosh, D., Levin, E., and Limmack, R.J., Bid-Ask Spreads, Trading Volume and Volatility: Intra-Day Evidence from the London Stock Exchange. Journal of Business Finance and Accounting, 24,

16 Barclay, M., and Hendershott, T., Price Discovery and Trading After Hours. Review of Financial Studies, 16, Barclay, M., and Hendershott, T., Liquidity Externalities and Adverse Selection: Evidence from Trading After Hours. Journal of Finance, 59, Brock, W.A., and Hommes, C. H., A Rational Route to Randomness. Econometrica, 65, Brock, W. A., and Hommes, C. H., Heterogeneous Beliefs and Routes to Chaos in a Simple Asset Pricing Model. Journal of Economic Dynamics and Control, 22, Brock, W. A., and Kleidon, A., Periodic Market Closure and Trading Volume. Journal of Economic Dynamics and Control, 16, Chen, S. H., Chang, C. L., and Du, Y. R., Agent-based Economic Models and Econometrics. Knowledge Engineering Review, 27, Cont, R., M. Potters, and Bouchaud, J.P., Scaling in Stock Market Data: Stable Laws and Beyond. In B. Dubrulle, F. Groner & D. Sornette (eds.), Scale Invariance and Beyond. Berlin: Springer. Easlay, D., and O Hara, M., Time and the Process of Security Price Adjustment. Journal of Finance, 47, Engle R. F., and Bollerslev, T., Modelling the Persistence of Conditional Variances. Econometric Reviews, 5, Fan, Y. J., and Lai, H. N., The Intraday Effect and the Extension of Trading Hours for Taiwanese Securities. International Review of Financial Analysis, 15, Foster F. D., and Viswanathan, S., A Theory of Intraday Variation in Volume, Variance, and Trading Costs in Securities Markets. Review of Financial Studies, 3, Glosten, L., and Milgrom, P., Bid, Ask and Transaction Prices in a Specialist Market with Heterogeneously Informed Traders. Journal of Financial Economics, 14, Hamao, Y., and Hasbrouck, J., Securities Trading in the Absence of Dealers: Trades and Quotes on the Tokyo Stock Exchange. Review of Financial Studies, 8, Harris, L., A Transaction Data Study of Weekly and Intradaily Patterns in Stock Returns. Journal of Financial Economics, 16, Hong, H., and Wang, J., Trading and Returns under Periodic Market Closures. Journal of Finance, 55, Houstion, J. F., and Ryngaert, M. D., The Links between Trading Time and Market Volatility. Journal of Financial Research, 15, Jain, P.C., and Joh, G.H., The Dependence between Hourly Prices and Trading Volume. Journal of Financial and Quantitative Analysis, 23,

17 Le Baron, B., Agent-Based Computational Finance. Handbook of Computational Economics, 2, Kyle, A., Continuous Auctions and Insider Trading. Econometrica, 53, Mandelbrot, B., The Variation of Certain Speculative Prices. Journal of Business, 36, Mandelbrot, B., Fractals and Scaling in Finance. Berlin: Springer. Osaki, S., TSE Looking at Extending Cash Trading Hour Again. NRI Financial Research Paper lakyara, 78. Pagan, A., The Econometrics of Financial Markets. Journal of Empirical Finance, 3, Wood, R.A., McInish, T.H., and Ord, J.K., An Investigation of Transactions Data for NYSE Stocks. Journal of Finance, 40,

18 Table 1 Trading Concentration Table 1(a) shows average volume for each trading hour when we change the parameter P upd from 0 to 1 at an interval of 0.1, under the condition d, g,, and are set at 0.2, 1.25, 10 and 5, respectively. Table 1(a) shows average volume for each trading hour when we change the parameter d from 0 to 0.4 at an interval of 0.1 under the condition P upd, g, and are set at 0.2, 1.25, and 5, respectively. The rows ALL, T1/Other Regular Hours, T2/Other Regular Hours, and T6/Other Regular Hours represent aggregated trading volume for one trading day, and the ratios of average volume in the opening session (T1) to that during the other regular-hours session, average volume in the subsequent session (T2) to that during the other regular-hours session, and average volume in the closing session (T6) to that during the other regular-hours session, respectively. The rows T1, T2 T6 represent average volume in periods T1, T2 T6, respectively. (a) Influence of the Strategy Update Speed (P upd ) Pupd=0 Pupd=0.1 Pupd=0.2 Pupd=0.3 Pupd=0.4 Pupd=0.5 Pupd=0.6 Pupd=0.7 Pupd=0.8 Pupd=0.9 Pupd=1.0 ALL T1/Other Regular-Hours T2/Other Regular-Hours T6/Other Regular-Hours T1(Opening) T T T T T6 (Closing) (b) Influence of the Biased Traders (d) d=0 d=0.1 d=0.2 d=0.3 d=0.4 ALL T1/Other Regular-Hours T2/Other Regular-Hours T6/Other Regular-Hours T1(Opening) T T T T T6 (Closing)

19 Table 2 The U-shaped Pattern of Return Volatility Table 2 shows the average return volatility (standard deviation of returns) for each trading hour when we change the parameter P upd from 0 to 1 at an interval of 0.1, under the condition d, g,, and are set at 0.2, 1.25, 10, and 5, respectively. The rows T1/Other Regular Hours, T2/Other Regular Hours, T6/Other Regular Hours, and Regular Hours represent the ratios of average volatility in the opening session (T1) to that during the other regular-hours session, average volatility in the subsequent session (T2) to that during the other regular-hours session, average volatility in the closing session (T6) to that during the other regular-hours session, and average volatility during the regular-hours session, respectively. Rows T1, T2 T6 represent average volatility in periods T1, T2 T6, respectively. Pupd=0 Pupd=0.1 Pupd=0.2 Pupd=0.3 Pupd=0.4 Pupd=0.5 Pupd=0.6 Pupd=0.7 Pupd=0.8 Pupd=0.9 Pupd=1.0 T1/Other Regular-Hours T2/Other Regular-Hours T6/Other Regular-Hours Regular-Hours 1.68% 1.81% 1.93% 2.04% 2.13% 2.20% 2.27% 2.35% 2.40% 2.43% 2.47% T1(Opening) 3.07% 3.22% 3.29% 3.35% 3.41% 3.45% 3.46% 3.51% 3.56% 3.53% 3.58% T2 1.34% 1.55% 1.70% 1.86% 1.98% 2.10% 2.22% 2.34% 2.40% 2.49% 2.55% T3 1.04% 1.23% 1.40% 1.54% 1.64% 1.74% 1.82% 1.88% 1.92% 1.97% 1.98% T4 1.00% 1.20% 1.35% 1.50% 1.60% 1.69% 1.79% 1.85% 1.89% 1.97% 2.01% T5 1.00% 1.20% 1.36% 1.49% 1.61% 1.69% 1.79% 1.85% 1.92% 1.96% 2.00% T6 (Closing) 0.99% 1.58% 1.71% 1.83% 1.91% 1.98% 2.06% 2.14% 2.19% 2.24% 2.27% 19

20 Extended-Hours Session Regular-Hours Session Table 3 Deviation from Fundamental Value Table 3 shows the difference between the market price and fundamental value when we change the parameter P e from 0 to 1 at an interval of 0.1, under the condition P upd, d, g,, and are set at 0.2, 0.2, 1.25, 10, and 5, respectively. Rows Regular Hours and Extended Hours represent the time-series average of the deviation during the regular-hours and extended-hours sessions, respectively. Rows T1, T2 T24 represent an average value of the deviation in periods T1, T2 T24, respectively. Pe=0 Pe=0.1 Pe=0.2 Pe=0.3 Pe=0.4 Pe=0.5 Pe=0.6 Pe=0.7 Pe=0.8 Pe=0.9 Pe=1 Regular-Hours 1.65% 1.81% 1.69% 1.66% 1.63% 1.62% 1.60% 1.60% 1.58% 1.58% 1.57% Extended-Hours 2.96% 5.76% 3.78% 3.01% 2.54% 2.25% 2.03% 1.88% 1.75% 1.67% 1.58% T1(Opening) 1.90% 2.62% 2.01% 1.83% 1.72% 1.67% 1.64% 1.62% 1.60% 1.59% 1.57% T2 1.57% 1.67% 1.61% 1.60% 1.58% 1.57% 1.58% 1.58% 1.57% 1.57% 1.56% T3 1.55% 1.57% 1.58% 1.58% 1.58% 1.57% 1.57% 1.59% 1.56% 1.57% 1.56% T4 1.56% 1.57% 1.58% 1.57% 1.59% 1.58% 1.56% 1.58% 1.56% 1.56% 1.57% T5 1.57% 1.58% 1.57% 1.57% 1.58% 1.59% 1.57% 1.58% 1.57% 1.57% 1.58% T6 (Closing) 1.78% 1.82% 1.81% 1.78% 1.75% 1.73% 1.67% 1.65% 1.61% 1.61% 1.57% T7 1.94% 2.72% 2.23% 2.07% 1.96% 1.86% 1.78% 1.71% 1.65% 1.63% 1.57% T8 2.09% 3.61% 2.70% 2.35% 2.13% 1.97% 1.86% 1.76% 1.69% 1.63% 1.57% T9 2.24% 4.37% 3.09% 2.57% 2.27% 2.07% 1.92% 1.80% 1.71% 1.65% 1.57% T % 4.94% 3.36% 2.73% 2.38% 2.15% 1.97% 1.84% 1.72% 1.66% 1.57% T % 5.38% 3.59% 2.88% 2.46% 2.20% 2.00% 1.86% 1.74% 1.67% 1.58% T % 5.70% 3.74% 2.99% 2.51% 2.24% 2.03% 1.87% 1.75% 1.68% 1.57% T % 5.94% 3.87% 3.08% 2.56% 2.27% 2.05% 1.88% 1.75% 1.67% 1.57% T % 6.16% 3.94% 3.12% 2.60% 2.30% 2.05% 1.90% 1.76% 1.67% 1.58% T % 6.32% 4.01% 3.16% 2.63% 2.32% 2.06% 1.90% 1.77% 1.66% 1.58% T % 6.42% 4.06% 3.17% 2.65% 2.32% 2.07% 1.91% 1.78% 1.67% 1.58% T % 6.47% 4.11% 3.22% 2.67% 2.31% 2.08% 1.92% 1.78% 1.67% 1.58% T % 6.47% 4.15% 3.23% 2.68% 2.32% 2.10% 1.93% 1.77% 1.68% 1.59% T % 6.52% 4.17% 3.24% 2.70% 2.34% 2.11% 1.92% 1.78% 1.68% 1.58% T % 6.53% 4.20% 3.24% 2.71% 2.35% 2.11% 1.93% 1.78% 1.69% 1.58% T % 6.53% 4.19% 3.26% 2.71% 2.36% 2.11% 1.94% 1.78% 1.69% 1.59% T % 6.55% 4.18% 3.26% 2.71% 2.37% 2.12% 1.94% 1.78% 1.69% 1.58% T % 6.56% 4.18% 3.26% 2.72% 2.37% 2.11% 1.93% 1.79% 1.68% 1.58% T % 6.61% 4.20% 3.25% 2.73% 2.37% 2.12% 1.94% 1.78% 1.68% 1.58% 20

21 Extended-Hours Session Regular-Hours Session Table 4 Return Volatility Table 4 shows return volatility when we change the parameter P e from 0 to 1 at an interval of 0.1, under the condition P upd, d, g,, and are set at 0.2, 0.2, 1.25, 10, and 5, respectively. Rows T1/Other Regular Hours, T2/Other Regular Hours, T6/Other Regular Hours, Regular Hours, and Extended Hours represent the ratios of average volatility in the opening session (T1) to that during the other regular-hours session, average volatility in the subsequent session (T2) to that during the other regular-hours session, average volatility in the closing session (T6) to that during the other regular-hours session, and an average of return volatility during the regular-hours and extended-hours sessions, respectively. Rows T1, T2, T24 represent an average value of return volatility in periods T1, T2 T24, respectively. Pe=0 Pe=0.1 Pe=0.2 Pe=0.3 Pe=0.4 Pe=0.5 Pe=0.6 Pe=0.7 Pe=0.8 Pe=0.9 Pe=1 T1/Other Regular-H T2/Other Regular-H T6/Other Regular-H Regular-Hours 1.95% 2.83% 2.06% 1.78% 1.64% 1.56% 1.48% 1.44% 1.41% 1.39% 1.37% Extended-Hours 0.00% 3.17% 2.30% 1.97% 1.77% 1.64% 1.55% 1.49% 1.44% 1.40% 1.37% T1(Opening) 3.32% 5.72% 3.63% 2.79% 2.34% 2.04% 1.80% 1.65% 1.54% 1.45% 1.38% T2 1.73% 2.43% 1.81% 1.61% 1.52% 1.46% 1.42% 1.41% 1.39% 1.37% 1.36% T3 1.40% 1.52% 1.42% 1.39% 1.38% 1.37% 1.37% 1.35% 1.35% 1.36% 1.37% T4 1.36% 1.37% 1.36% 1.36% 1.35% 1.36% 1.37% 1.35% 1.35% 1.35% 1.36% T5 1.36% 1.34% 1.35% 1.37% 1.36% 1.36% 1.35% 1.36% 1.37% 1.36% 1.35% T6 (Closing) 1.71% 1.77% 1.77% 1.72% 1.65% 1.61% 1.52% 1.48% 1.44% 1.40% 1.36% T7 0.00% 2.99% 2.21% 1.91% 1.73% 1.61% 1.53% 1.47% 1.42% 1.39% 1.36% T8 0.00% 3.09% 2.28% 1.95% 1.74% 1.65% 1.55% 1.48% 1.43% 1.39% 1.37% T9 0.00% 3.09% 2.30% 1.96% 1.75% 1.63% 1.53% 1.48% 1.43% 1.40% 1.38% T % 3.10% 2.27% 1.95% 1.75% 1.63% 1.55% 1.48% 1.43% 1.39% 1.34% T % 3.13% 2.27% 1.95% 1.76% 1.63% 1.53% 1.47% 1.42% 1.40% 1.37% T % 3.15% 2.29% 1.97% 1.76% 1.62% 1.54% 1.49% 1.43% 1.39% 1.37% T % 3.15% 2.28% 1.96% 1.76% 1.64% 1.55% 1.48% 1.42% 1.39% 1.36% T % 3.15% 2.30% 1.96% 1.77% 1.65% 1.55% 1.49% 1.43% 1.40% 1.36% T % 3.20% 2.31% 1.98% 1.77% 1.64% 1.55% 1.48% 1.44% 1.40% 1.37% T % 3.19% 2.33% 1.97% 1.78% 1.63% 1.56% 1.48% 1.43% 1.40% 1.37% T % 3.17% 2.30% 1.96% 1.77% 1.66% 1.54% 1.49% 1.44% 1.39% 1.37% T % 3.21% 2.32% 1.95% 1.77% 1.64% 1.56% 1.48% 1.42% 1.41% 1.36% T % 3.21% 2.31% 1.97% 1.77% 1.63% 1.55% 1.48% 1.43% 1.39% 1.37% T % 3.17% 2.28% 1.96% 1.77% 1.65% 1.55% 1.48% 1.43% 1.40% 1.37% T % 3.21% 2.29% 1.96% 1.78% 1.65% 1.56% 1.49% 1.42% 1.40% 1.36% T % 3.17% 2.31% 1.96% 1.76% 1.64% 1.56% 1.48% 1.44% 1.41% 1.36% T % 3.24% 2.31% 1.96% 1.77% 1.64% 1.56% 1.49% 1.44% 1.38% 1.36% T % 3.19% 2.32% 1.97% 1.76% 1.64% 1.55% 1.48% 1.44% 1.39% 1.36% 21