Sharp counter-trend movement: a new stylized. fact in stock markets

Transcription

1 Sharp couner-rend movemen: a new sylized fac in sock markes Weihong Huang* Yu Zhang Division of Economics, 14 Nanyang Drive, Nanyang Technological Universiy, Singapore, Absrac This paper presens a new sylized fac in sock markes called he sharp counerrend movemen. I describes he phenomenon ha he sock price change agains a rend is generally faser han he price change along a rend, despie he direcion of a price rend. To verify i, wo measuremens are derived from he mehod of inverse saisics, represening he primary rend and he asymmery of price rise/fall speed of a sock price series respecively. Daa analysis of varieies of sock indices shows a clear negaive correlaion beween hese wo measures, verifying he exisence of sharp couner-rend movemen. Key words: sock indices, sylized facs, rend index, asmmery index Preprin submied o Elsevier Science 24 July 2013

2 1 Inroducion Sylized facs [3] are saisical properies of empirical daa. I is believed by some economiss ha heoreical models which can reproduce sylized facs are capable o explain he realiy beer [6]. In he las few decades, he sudy of sylized facs has araced a lo aenion. Some widely acceped sylized facs of asse reurns are volailiy clusering, heavy ails, volume/volailiy correlaion, leverage e ec, absence of auocorrelaion, auocorrelaion in absolue and squared reurns, and so on. Here, we propose a new sylized fac in sock markes depicing he phenomenon ha he price change agains a rend is normally faser han he price change along a rend. We believe his phenomenon is general in sock markes and refer o i as he sharp couner-rend movemen. The sharp couner-rend movemen may look similar o anoher sylized fac called he gain/loss asymmery in sock reurns [7], which is described by Con [3] as large drawdowns in sock prices and sock index values bu no equally large upward movemens. Bu we argue ha he gain/loss asymmery is one subse of he sharp couner-rend movemen when here is an upward rend in price series. The gain/loss asymmery ells a par of a sory, while he sharp couner-rend movemen ells a more comprehensive version. When he price rend is downward, if here is a rebound, normally he rebound is rapid, hough i may no able o mainain for a long period. Our proposiion of sharp counerrend movemen focuses more on wheher a price movemen is agains or along a rend insead of on he speci c direcions of price movemens. And he couner movemens are normally sharper, in he sense ha he price changes are faser, han movemens along he main rend. Acually, his phenomenon 2

3 has been menioned in Dow Theory [2], one of he mos imporan origins of echnical analyses in nancial markes. In his famous book, he sock marke baromeer, Hamilon [5] wroe and we quoe: Coinciden wih i, or in he course of i, is Dow s secondary movemen, represened by sharp rallies in a primary bear marke and sharp reacions in a primary bull marke. Though he Dow Theory is no longer in fashion now, his saemen abou secondary movemens sheds ligh on our work. We believe he phenomenon of sharp couner-rend movemen is general, and we need o verify i wih help from special saisical mehod. To verify he exisence of sharp couner-rend movemen, he mehod of inverse saisics is used o derive wo indices, represening he primary rend and he price rise/fall asymmery [9] respecively. By combining hese wo indices ino an index map, a clear negaive correlaion beween hem veri es he exisence of sharp couner-rend movemen naurally. Sock indices from worldwide markes and sequenial ime periods of S&P500 are sudied for parallel and hisorical comparisons. 2 Mehodology 2.1 Inverse saisics In our sudy, we use he mehod of inverse saisics o analyze sock index prices. Inverse saisics is a saisical approach iniially used in urbulence [1] o describe he spaial disribuion of uid velociy. Ghashghaie e al. [4] creaively inroduced his mehod ino he sudy of foreign exchange markes. Since hen, inverse saisics has been used o sudy nancial daa, especially 3

4 probabiliy reurn= reurn= waiing ime[days] Fig. 1. Asymmeric waiing ime disribuions of inverse saisics of S&P500 prices of sock indices [8]. Di eren from he normal saisics, which xes a ime inerval, like a day, and hen ges he disribuion of reurns; he inverse saisics works in he opposie direcion. I xes a level of reurn in advance, and hen calculaes he disribuion of he ime invesors have o wai o realize he predeermined reurn for he rs ime. Normally, he waiing ime disribuions of opposie predeermined reurns are compared and a signi can asymmery beween hem is explained as a proof of he gain/loss asymmery or he price rise/fall asymmery. Figure 1 shows an example of he asymmery of waiing ime disribuions when we use inverse saisics o analyze he price series of S&P500. 4

5 2.2 Trend index In inverse saisics, o ge he waiing ime disribuion of achieving a predeermined reurn, we need he sar poin of waiing ravels hrough he whole ime span. I happens ha, saring from some ime insans, he predeermined reurn fails o be realized before he end of daa, especially for hose ime insans which are imporan urning poins or hose ime insans close o he end of daa. Afer excluding hem, he collecion of all realized predeermined reurn forms a waiing ime disribuion. Under any predeermined reurn, here is a probabiliy of achieving i p achieve; and a probabiliy of failing o achieve i p fail;. We de ne he rend index (hereforh, TI) as following T I = p achieve;+ p achieve;, where is he absolue value of predeermined reurns. Since for any predeermined reurn, p achieve; + p fail; = 1, we can ge an alernaive expression for TI. T I = p fail; p fail;+ The value of TI falls ino he inerval ( 1; 1). When he price series displays an upward rend, naurally p achieve;+ > p achieve;, and TI is posiive. If here is a downward rend in he price series, normally p achieve;+ < p achieve;, and TI is negaive. TI equals 0 when here is no rend in price. In a word, he value of TI shows he general rend of a price series. 5

6 2.3 Asymmery index Sock price ucuaion asymmery describes he phenomenon ha price rising speed is di eren from price falling speed in sock indices. The resul of inverse saisics in Figure 1 is used as an imporan qualiaive suppor of his phenomenon. We consruc an asymmery index (hereforh, AI) o evaluae he sock price ucuaion asymmery quaniaively, which is de ned as y + y AI = + y y, where y is he amoun of price change and denoes he waiing ime for realizing a predeermined price change for he rs ime. The value of AI falls ino he inerval ( 1; 1). Similar evaluaion of asymmery can be found in Neuroscience. From he de niion of AI, we can ge he following properies. Proposiion 1 According o he de niion of AI, we know ha 1) when y + + = y, AI = 0; 2) when y + + > y, 0 < AI < 1, and AI! 1 when y + + y! 1; and 3) when y + + < y, 1 < AI < 0, and AI! 1 y + when +! 0. y Previously, we focused on reurns. So he price changes in AI formula are replaced by he reurns. Since posiive and negaive reurns share he same absolue value, i.e. jy + j = jy j, hey can be canceled ou. Furhermore, if we use o denoe he waiing ime, and subsiue he expecaions of waiing ime disribuions E ( + ) and E ( ) for + and respecively, he funcion of AI is simpli ed as AI = E ( ) E ( + ) E ( ) + E ( + ). 6

7 Accordingly, Proposiion 1 can be rephrased as following. Proposiion 2 According o he simpli ed funcion of AI, we know ha 1) when E ( ) = E ( + ), AI = 0; 2) when E ( ) > E ( + ), 0 < AI < 1, and AI! 1 when E( ) E( + )! 1; and 3) when E ( ) < E ( + ), 1 < AI < 0, and AI! 1 when E( ) E( + )! 0. A posiive value of AI indicaes ha, during all ucuaions of hisorical prices, he price rise is generally faser han he price fall. While a negaive value means he price fall is faser han he price rise. This economic inuiion works in a large sense, as he expecaions of waiing ime disribuions are quie general. 2.4 Index map The combinaion of TI and AI gives us an index map wih TI as ordinae agains AI as abscissa. For any price series, he process of previous daa analysis produces a poin in he index map, locaing a a resriced area. According o he economic meanings of values of TI and AI, poins in di eren quadrans reveal di eren general feaures of corresponding price series. Proposiion 3 1) Poins in quadran 1 and 2 indicae an upward rend in price series. Poins in quadran 3 and 4 indicae a downward rend. 2) In quadran 2 and 3, price rise is generally slower han price fall. In quadran 1 and 4, price rise is generally faser han price fall. 3) In quadran 2 and 4, price change agains he main rend is faser han price change along he main rend. In quadran 1 and 3, price change agains he main rend is slower han price change along he main rend. 7

8 TI 1 reurn= JKSE KLSE AORD BSESN KS11 HSI SSEC STI BVSP MERV IXIC ATX TWII FTSE FCHI GDAXI BFX SSMI GSPC N AEX.AS 0.4 GD.AT AI Fig. 2. Index map of 23 sock indices from 1998 o 2012 In he nex secion, hisorical prices of sock indices from worldwide markes are analyzed. 3 Resuls and discussion We analyzed sock indices from 23 sock markes. Their sock codes are MERV, BVSP, S&P 500, IXIC, AORD, SSEC, HSI, BSESN, JKSE, KLSE, N225, STI, KS11, TWII, ATX, BFX, FCHI, GDAXI, AEX.AS, SSMI, FTSE, GD.AT, and TA100. We used heir daily close prices from 1998 o 2012, and go he resul shown in Figure 2. Figure 2 shows us an overview of diverse indices. The clear negaive correlaion beween TI and AI reveals he implici relaion beween main rend 8

9 and price ucuaion asymmery of sock indices. Mos indices fall ino he rs quadran, and he fourh quadran ges second mos indices inside. Tha is, in mos sock markes, generally speaking, he sock indices show upward rends, and he price falls are faser han price rises. Even in markes wih downward rends, more frequenly, he price rises is faser han price falls. To sum up, he price change agains a rend is normally faser han he price change along a rend. Tha is, when here is a rend in a price series, he price movemen couner o he rend is faser han he price movemen along he rend. This is he reason why we named his phenomenon he sharp counerrend movemen. Besides parallel comparison among di eren indices, similar comparison beween di eren ime spans of one sock index can be made in he index map. We divided he hisorical prices of S&P500 from 1950 o 2011 ino separae daa ses of every wo years and go Figure 3. Once again, we see a clear negaive correlaion of AI and TI. Mos poins are locaed in quadran 2 and 4. From any price series, shor or long, one can always ge a rend. As long as here is a rend, price change agains he rend is normally faser han he price change along he rend. The sharp couner-rend movemen works a di eren imescales. In Figure 3, special aenion needs o be direced o poins locaed in quadran 3 and quadran 1. The region of quadran 3 demonsraes he wors siuaion. Tha is, he main rend of price is downward, and among all price movemens, he price rises are slower han price falls. Only wo poins fall ino his area, marked as 2002 and 2008, represening daa from 2002 o 2003 and from 2008 o The inerne bubble bursing during 2002 and he 2008 nancial crisis are primary causes for hem. However, quadran 1 shows 9

10 TI 1 imewindow=2 years, reurn= AI Fig. 3. Index map of every wo years daa of S&P500 from 1950 o 2011 an opposie siuaion ha he price rend is upwards and he price rises are faser han price falls. Such a good siuaion usually happens when he marke is recovering from a major price drop, like poins marked as 1958 and Or such siuaions can be so good ha bubbles are forming and hey are followed by imporan sock marke crashes, like poins marked as 1986 and In he process of daa analysis, he predeermined reurn of inverse saisics is he only facor ha can a ec he locaions of poins in he index map. The predeermined reurn should no be large, especially when he ime span of price series is shor. Because a large reurn can cause he reurn fails o be realized from any saring ime poin, and hen no waiing ime disribuion can be gained. In principle, he shorer he daa, he smaller he predeermined reurn should be. Bu heir relaion is no linear. Figure 2 and 3 show wo proper 10

11 choices of predeermined reurn under di eren ime spans. If we furher lower he predeermined reurn, he locaions of poins may change, especially hose close o coordinae axes. Bu, mos poins sill locae a quadran 2 and 4, and he negaive correlaion beween TI and AI mainains. Therefore, he sharp couner-rend movemen is robus. 4 Conclusion In his paper, a new sylized fac named as he sharp couner-rend movemen is proposed. I describes he phenomenon ha he sock price change agains a rend is generally faser han he price change along a rend. I is veri ed by he mehod of inverse saisics and wo measures represening primary rend and asymmery of ucuaions of sock prices respecively. Daa analyses of varieies of sock indices suppor he exisence of his new sylized fac in sock markes. The appearance of he sharp couner-rend movemen in oher nancial markes, such as exchange markes, will need more daa analyses o con rm. Besides, he origin of his sylized fac is unclear. I may relae o invesors behavior under marke senimen. These problems show us direcions for fuure work. References [1] Biferale, L., Cencini, M., Lanoe, A., Vergni, D., Vulpiani, A., Inverse saisics of smooh signals: he case of wo dimensional urbulence. Physical Review Leers 87 (12), [2] Brown, S. J., Goezmann, W. N., Kumar, A., The Dow Theory: William 11

12 Peer Hamilon s rack record re-considered. The Journal of Finance 53, [3] Con, R., Empirical properies of asse reurns: sylized facs and saisical issues. Quaniaive Finance 1, [4] Ghashghaie, S., Breymann, W.,Peinke, J.,Talkner, P., Dodge, Y., Turbulen cascades in foreign exchange markes. Naure 381, [5] Hamilon, W. P., The sock marke baromeer: a sudy of is forecas value based on Charles H. Dow s heory of he price movemen. Barron s, New York. [6] Hommes, C. H., Modeling he sylized facs in nance hrough simple nonlinear adapive sysems. Proceedings of he Naional Academy of Sciences 99, [7] Huang,W., Zheng, H., Chia, W.-M., Asymmeric reurns, gradual bubbles and sudden crashes. The European Journal of Finance 19 (5), [8] Jensen, M. H., Johansen, A., Simonsen, I., Inverse saisics in economics: he gain-loss asymmery. Physica A 324, [9] Zhang, Y., Li, H., Invesors risk aiudes and sock price ucuaion asymmery. Physica A 390,