Physica A 391 (2012) 1354–1360

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Physica A

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Intraday volatility and network topological properties in the Koreanstock marketJunghoon Lee a,1, Janghyuk Youn a,1, Woojin Chang b,∗

a Technology Management, Economics and Policy Program, Seoul National University, 599 Gwanak-ro, Gwanak-gu, Seoul 151-742, Republic of Koreab Department of Industrial Engineering, Seoul National University, 599 Gwanak-ro, Gwanak-gu, Seoul 151-742, Republic of Korea

a r t i c l e i n f o

Article history:Received 24 March 2011Received in revised form 21 July 2011Available online 5 October 2011

Keywords:High-frequency dataMinimum spanning treeVolatilityTopological property

a b s t r a c t

We examine whether the relationship between market volatility and network propertiesin the low-frequency level can be applied to the high-frequency level. For the analysis, weuse the minimum spanning tree (MST) method constructed from intraday Korean stockmarket data. The results show that the higher the market volatility is, the denser the MSTof stocks becomes. The normalized tree length shows a strong negative relationship withmarket volatility, indicating that the distances between nodes are shorterwhen themarketvolatility is high. The mean occupation layer shows the tendency of having a smaller valuein a higher volatility market. The maximum number of links becomes larger when themarket volatility increases. All these network properties support the network being denseand shrinking in high market volatility conditions; that is, the degree of co-movement infinancial market is reinforced in the intraday high-frequency level.

© 2011 Elsevier B.V. All rights reserved.

1. Introduction

The topological perspective on financial markets has received much attention in the field of econophysics [1–9].A financial market itself is generally considered as a complex system, tangled intricately. Mantegna [4] proposed thetransformation of correlation coefficients to the Euclidean distance. The correlation coefficient is calculated from log returnof two assets. A lot of research has attempted to apply the minimum spanning tree (MST) method to various financialareas: stock markets (global [10,11], developed market [1,2,9,12–17], emerging market [3,18,19]), interest rates [20,21],currencies [22–24], bonds [25], and commodities [26]. Most of this research is static analysis, such as overviews on thenetwork topology or taxonomic studies in terms of sector, region, or other characteristics. Recently, Coelho et al. [11]attempted a dynamic analysis. Onnela et al. [5] provided the basic idea that themarket condition is reflected in the topologyof networks in a financial market by comparing the topological difference between Black Monday and a day of normalconditions. Gilmore et al. [25] studied the dynamic co-movement of government bonds in 1998–2006 using hierarchicaltree and network properties. Jang et al. [22] showed that the topology of a currency network is changed in a currency crisis.However, these studies did not cover intraday level analysis. On the other hand, the importance of high-frequency intradayanalysis of the financial market has been gradually highlighted as the investment time horizon gets shorter thanks to theadvances of computing and communication technologies. Accordingly, the interest in volatility is moving from daily towardintraday level [27–29].

Our study focuses on the intraday high-frequency level analysis in the topology of a financial network. Assets are activelytraded in a short time horizon and the number of assets traded during the short time period is sufficient to construct a

∗ Corresponding author. Tel.: +82 2 880 8335; fax: +82 2 889 8560.E-mail addresses: black8@snu.ac.kr (J. Lee), fodis27@snu.ac.kr (J. Youn), changw@snu.ac.kr (W. Chang).

1 Tel.: +82 2 880 1380.

0378-4371/$ – see front matter© 2011 Elsevier B.V. All rights reserved.doi:10.1016/j.physa.2011.09.016

J. Lee et al. / Physica A 391 (2012) 1354–1360 1355

network structure. Considering these conditions, we analyze a stock market in which trading is executed in an automatedelectronic system. By this analysis, we try to extend the previous research to intraday level analysis and to find a moregeneralized relationship between the volatility of the financial market and the corresponding network properties.

This paper is organized as follows. The following section gives a brief description of the data and provides somepreliminary analysis including intraday stylized facts. Section 3 reports our findings on the network properties for variousmarket volatility levels. Finally, a summary and conclusions are presented.

2. Methodology

In this section, we describe the methodology used for the analysis of the data. In order to construct the MST followingthe method suggested by Mantegna [4], the correlation coefficient should be calculated in the first step. For the preliminarywork, the stock price return time series should be calculated from the time series of price. The logarithmic return of stock iover period τ at time t is given by

Ri (t, τ ) = ln Pi (t) − ln Pi (t − τ) , (1)

where Pi(t) is the price of stock i at time t . The correlation coefficient between stock i and stock j at time t over the n timeintervals is defined as

ρtij =

TRti · Rt

j

−

Rti · 1T

Rtj · 1T

T

Rti · Rt

i

−

Rti · 1T

2 T

Rtj · Rt

j

−

Rtj · 1T

2 , (2)

where

Rti = (Ri (t − nτ + τ) , . . . , Ri (t)) , i = 1, . . . ,N,

1T = (1, . . . , 1) , T number of 1s,

where T indicates the length of time window: T = nτ . X · Y denotes the inner product of two vectors X and Y . Through thecalculation of the correlation coefficients between all pairs of N stocks, the correlation matrix C t at time t is constructed. C t

is an N × N symmetric matrix with diagonal elements equal to unity. In the second step, we use the method proposed byMantegna [4] to construct the MST. All the elements of the correlation matrix C t are transformed to distances. The distancebetween stock i and stock j at time t is defined as

dtij =

2(1 − ρt

ij). (3)

The MST for the N stocks can be obtained through Kruskal’s algorithm from the N ×N distance matrix [30]. For the purposeof analyzing theMST, we employ some network properties which display the characteristics of the network. The normalizedtree length (NTL), which measures the closeness among the components of network, at time t , is given by

NTLt =1

N − 1

−dtij∈Θ

dtij, (4)

where Θ is the set of edges and dtij is the distance between stock i and stock j at time t in Eq. (3).There are various methods to determine the central node of a network: the highest vertex degree (number of edges), the

highest correlation coefficient-weighted vertex degree, and the betweenness centrality [7,22,25].We adopt the betweennesscentrality. The betweenness centrality is defined as

B (υ) =

−i=υ

−j=υ,j=i

πij(υ)

πij, (5)

where πij indicates the number of shortest geodesic paths from node i to node j, and πij(υ) denotes the number of shortestgeodesic paths from node i to node j through a node υ .

The level of node i, φ(υi), counts the number of edges connected to the central node through the shortest path from nodei. By definition, the level of the central node is zero: φ (υc) = 0.

Based on the centrality and the level of the node, the mean occupation layer (MOL) at time t is defined by

MOLt =1N

N−1

φ(υi). (6)

Themean occupation layer is an index for the topological structure of a network. If the value of theMOL is small, the networkappears to be dense and its nodes tend to crowd the area around the central node. The maximum number of links, kmax, isobtained by counting the number of edges of the most connected node in the network. We define the market volatility ofthe time window T at time t , σt , as the average standard deviation of N stocks. In this paper, we choose the values of τ = 1and T = 30 min.

1356 J. Lee et al. / Physica A 391 (2012) 1354–1360

0.0120 0.5 1 1.5

σt× 10–3

2 2.5 3

0 0.5 1 1.5σt

× 10–32 2.5 3 5 10 15 20 25 30 35 40 45 50

0 0.5 1 1.5σt

× 10–32 2.5 3

0.014

0.016

0.018NLT

0 0

500

1000

1500

2000

2500

3000

3500

4000

5

10

15

k max

Vc

0.02

0.022

0.024

1

2

3

4

MO

L

5

6

7

8

9a b

c d

Fig. 1. Network properties of 30 min windows (the total number of time windows is 79,360): (a) normalized tree length, (b) mean occupation layer,(c) maximum number of links (kmax), (d) frequency of a stock’s becoming the central node (Vc ) of the MST, in descending order.

3. Data

We examine the minute-by-minute price data of 50 representative stocks in the Korean stock market, included in theKorea Composite Stock Price Index (KOSPI) 50 index. The records started on 2 January 2009 and ended on 30 December2009, covering 252 trading days. The Korean Stock Exchange is open for 6 h from 9:00 am to 3:00 pm on all days of theweek except Saturdays, Sundays, and holidays declared by the Exchange in advance. The data during the last 10min of dailytrading period (2:50 pm through 3:00 pm) is excluded, since the possible trading in this period is simultaneous bids andoffers only. As a result, we get 79,359 windows of 30 min periods for the year 2009.

4. Results and discussion

The overall tendency of network properties is presented in Fig. 1. We obtained 79,359 MSTs, which were constructedby trading data from the 30 min time windows. Each MST produces the corresponding network properties: NTL, MOL,and kmax.

Fig. 1a displays the distribution of the NTL versus market volatility. It is observed that a clear negative correlation existsbetween the NTL and the market volatility. The negative linear trend (the regression slope is −3.78) in Fig. 1a implies thatthe average Euclidean distance decreases as the market volatility increases. In other words, the more stocks tend to move inthe same direction as the market volatility becomes higher. This result also indicates that network shrinkage has occurredin a crisis such as Black Monday or the Asian currency crisis, and that the shrinkage extent is proportional to the marketvolatility.

Fig. 1b shows the plots of the MOL versus market volatility. The distribution range of the MOL for given market volatilityis narrowed as the market volatility increases. When the market volatility, σt , is 0.5×10−3, the MOL has a wide range, fromaround 3 to 7. When σt is 1.0× 10−3, the MOL has a narrow range, from 2.5 to 5.5. When σt is 1.5× 10−3, the range of MOLvalue is much more narrow: from 3 to 4. This result shows that high market volatility causes network shrinkage.

The plot of the maximum number of links, kmax, is displayed in Fig. 1c. The small kmax values (kmax = 3) are distributedmostly in the range where σt is less than 0.5. On the other hand, large kmax values (kmax > 10) are distributed over the rangewhere σt is greater than 0.5.

Fig. 1d shows the frequency of a stock’s becoming the central node of theMST, in descending order. The top five stocks areHyundai Merchant Marine (3959, 5.0% of 79,359), Samsung Electronics (3534, 4.5%), POSCO (3347, 4.2%), Shinhan Financial

J. Lee et al. / Physica A 391 (2012) 1354–1360 1357

Table 1The network properties for volatility groups.

Group Num Volatility (×10−3) NTL (×10−2) MOL kmax

Mean Minimum Maximum Mean Minimum Maximum Mean Minimum Maximum Mean Minimum Maximum

1 303 0.200 0.160 0.211 2.185 2.072 2.249 4.642 3.200 6.660 4.805 4 72 3,533 0.254 0.211 0.278 2.163 2.045 2.265 4.565 2.820 8.440 4.908 3 83 15,589 0.328 0.278 0.367 2.136 2.012 2.271 4.429 2.740 8.000 4.979 3 94 22,396 0.423 0.367 0.484 2.102 1.964 2.266 4.238 2.500 7.960 5.163 3 115 19,332 0.554 0.484 0.639 2.046 1.860 2.230 4.014 2.240 7.700 5.389 3 126 11,798 0.724 0.639 0.843 1.971 1.787 2.165 3.780 2.300 7.600 5.779 3 127 4,603 0.949 0.843 1.113 1.896 1.718 2.101 3.598 2.200 6.200 6.071 4 128 1,488 1.242 1.113 1.468 1.810 1.639 1.994 3.515 2.080 6.180 6.304 4 139 303 1.596 1.469 1.920 1.735 1.588 1.863 3.473 2.380 5.720 6.261 4 10

10 14 2.160 1.960 2.425 1.573 1.480 1.652 3.146 2.620 4.160 6.429 5 8

Fig. 2. Network properties of the ten groups (categorized based on the market volatility): (a) normalized tree length, (b) mean occupation layer, (c)maximum number of links.

Group (2900, 3.7%), and LG Corporation (2548, 3.2%). The top five stocks of these 50 stocks comprise 20.5% of central nodes.There are only seven stocks which become the central node of the MST more than 2000 times. Most of the stocks (76%) aredistributed in the 1000–2000 range.

For further analysis, all the 79,359 MSTs are categorized into ten groups in accordance with the logarithm of marketvolatility during the corresponding time window; group 1 denotes the smallest volatility group and group 10 denotes thelargest volatility group. Table 1 presents the detailed description and the corresponding network characteristics of eachgroup. The MSTs belonging to medium volatility level groups outnumber those belonging to high or low volatility levelgroups.

Fig. 2 displays the characteristics of these ten groups. The NTL, in Fig. 2a, shows a clear negative relationship with thelevel of volatility; we observe amonotone decreasing pattern of NTL from group 1, the smallest volatility group, (the averageNTL is 2.185 × 10−2) through group 10, the largest volatility group, (the average NTL is 1.520 × 10−2). This shows that thestructure of the MST becomes denser as the market volatility increases. Also, this indicates that more stocks move in thesame direction when the market is more volatile. The MOL presents a similar tendency (in Fig. 2b) to the NTL in Fig. 2a. Themean MOL value of group 1 is 4.642 while that of group 10 is 3.146. This implies that the nodes in group 1 reach the centralnode by passing through three or four other nodes on average, whereas the nodes in group 10 are two or three nodes awayfrom the central node on average. kmax (in Fig. 2c) confirms the network properties observed in the NTL andMOL. The meanvalue of kmax for group 1 and group 10 is 4.805 and 6.429, respectively. This means that around five nodes are linked to thecentral node in group 1 while the number of nodes linked to the central node becomes greater than six in group 10. All theresults (Euclidean distance, layer, and connectivity of central node) in Fig. 1 support the network of the high volatility groupbeing more concentrated around the central node than that of the low volatility group. This implies that the co-movementof stocks is reinforced as the market becomes volatile.

Fig. 3 shows an example of the network structures of group 1 (left, 3a) and group 10 (right, 3b). Fig. 3a is obtained fromthe time window 9:02–9:32 of 27 February 2009, which belongs to group 1 (the lowest market volatility group). Fig. 3b isconstructed based on the time window 12:54–13:24 of 8 September 2009, which belongs to group 10 (the highest volatilitygroup). At a glance, the nodes of theMST in Fig. 3a are densely connected, unlike the nodes of theMST in Fig. 3b. For theMSTin Fig. 3a, the market volatility is 3.1 × 10−3, the NTL is 2.15 × 10−2, the MOL is 4.38, and kmax is 4. For the MST in Fig. 3b,the market volatility is 2.4 × 10−3, the NTL is 1.51 × 10−2, the MOL is 2.82, and kmax is 7.

Fig. 4 shows the distribution of the number of links per node. Comparing the distributions of group 1 and group 10, morelinks are connected to a node in group 10 than in group 1. This supports the network of group 10 being more concentratedthan that of group 1.

1358 J. Lee et al. / Physica A 391 (2012) 1354–1360

Fig. 3. Network structures for high and low volatility: (a) group 1 (low volatility), (b) group 10 (high volatility).

Fig. 4. Distribution of the number of links per node: group 1 (black line), group 10 (gray line). (For interpretation of the references to colour in this figurelegend, the reader is referred to the web version of this article.)

As the market volatility changes, topological shrinkage and expansion of the corresponding MST are observed. Fig. 5shows an example of the intraday time series of the market volatility (σt ), NTL, and MOL for 27 February 2009 (whichbelongs to volatility group 10). There are over 300 points which are generated by rolling windows with a time period of30 min. In Fig. 5a, the market volatility starts at a high level

σt = 2.56 × 10−3

, drops continuously to 0.48 × 10−3 at

around the 125th window, expands again to 1.01× 10−3, and drops again to 0.48× 10−3. The volatility gradually increasesduring the last hour, and ends at 0.95 × 10−3. The trace of the NTL in Fig. 5b is a reverse image of the market volatility inFig. 5a. Contrary to the market volatility, the NTL starts with the lowest level

NTLt = 1.51 × 10−2

of the day and records

the highest value of 2.07 × 10−2 at around the 125th window. The value of the MOL in Fig. 5c fluctuates throughout theobservation period. The movement of kmax in Fig. 5d is noisy, but a moving average value of around 5 is maintained.

Overall, the NTL best responds to the market volatility, and the MOL also responds well to the volatility. The NTL is ameasure to show how far away the nodes in the network are from each other. The MOL explains the topological structureof the network. For instance, if there are two networks having the same topological structure except for distances betweennodes, the NTL values of the two networks are different while the MOL values are same. On the other hand, as kmax is themaximum number of links to just one node, the kmax value is insufficient to explain the network structure, and it can playthe role of a complementary measure.

The results obtained from time series of σt , NTL, MOL, and kmax indicate that topological shrinking of the network occursnot only in extreme cases such as Black Monday or a currency crisis but also in times of high market volatility.

J. Lee et al. / Physica A 391 (2012) 1354–1360 1359

Fig. 5. Intraday dynamics of market volatility and network properties: (a) market volatility, (b) normalized tree length, (c) mean occupation layer, (d)maximum number of links.

5. Conclusion

We investigated the relationship between the market volatility and the network properties of the MST in the intradayhigh-frequency level dynamics using intraday Korean stock market data.

The results suggest that the network structure becomes denser as the market volatility increases. The NTL respondsnegatively to the market volatility, so the trend of the NTL series is a reverse image of that of the market volatility series.The MOL tends to decreases as the market volatility increases, and kmax shows the tendency of increasing for higher marketvolatility.

All these network properties consistently support the characteristics observed from the analysis of low-frequencydynamics (daily or longer) also being found in the intraday high-frequency dynamics. In other words, as a market becomesvolatile, the co-movement tendency of financial assets is reinforced at the intraday high-frequency level in a similar way toor longer low-frequency level.

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