Fractal Analysis on the Stock Price of C to C Electronic Commerce Enterprise Ming Chen

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1 6th Interntionl Conference on Electronic, Mechnicl, Informtion nd Mngement (EMIM 2016) Frctl Anlysis on the Stock Price of C to C Electronic Commerce Enterprise Ming Chen Soochow University, Suzhou, Chin cm192@163.com Keywords: Frctl nlysis; C to C; Electronic commerce; Stock price Abstrct. To void the sttus tht the stock vlue of C to C electronic commerce enterprises chnge rdiclly, we hve induced the frctl method to mke n empiricl nlysis on the typicl cse of listed C to C e-commerce enterprises, nd studied the structurl fctors which behind them nd influence the mrket pricing of them. Bsed on the frctl nlysis of EBAY, our study hs found tht the stock price voltility of C to C e-commerce enterprise ppers s wek trend with cyclicl fluctutions. And furthermore, we hve provided new method for this seprtion. Introduction In the most influentil listed C to C electronic commerce enterprises worldwide, EBAY is typicl one. We choose EBAY s the typicl cse. Using its historicl stock price dt, crry through demonstrtion nlysis by the frctl method, to inquire bout the structurl fctors which influence its stock price movements, nd get deepen understnding of the pricing of C to C e-commerce enterprise. EBAY Inc. ws founded in It is one of the world fmous Internet intermediry mrket nd the world's leding online uction site. Its trding vrieties re involving C to C ntiques nd trding, books, udio nd video, coins, stmps, collectibles, toys, jewelry, computer, electronic products, cermics, toys nd other items. EBAY is listed on ASDAQ tionl Mrket; its stock code is EBAY. EBAY ws listed on ASDAQ in September 24, Its dily stock closing price voltility from September 24, 1998 is shown in Fig. 1. Dt Source: Figure 1. Voltility of EBAY Dily Stock Closing Price The Bised Rndom Wlk nd the Frctl Anlysis Mny empiricl studies find tht the distribution of stock return devites significntly from the Guss distribution, present kind of "ft til" chrcteristic [1]. The probbility in the region of high yield nd high loss is greter thn the probbility of Guss distribution [2]. It reflects the herd behvior in finncil mrkets, nd the herd behvior leds to the "ft til" chrcteristic of stock return distribution. The unexpected events of finncil mrket not only hs multi fctor, nonliner, but lso hs uncertinty [3]. Some studies show, in the finncil mrkets, people re not equl to digest nd confirm informtion. It often leds to the bised rndom wlk of stock price, or clled The uthors - Published by Atlntis Press 475

2 Frctionl Brown Motion (FBM) [4]. So, we hve no reson to believe tht mrkets must be blnced nd effective [5]. Finncil mrket my perfectly possible come to the frctl structure. The word "Frctl" is proposed for the first time in 1975 by Benoti B. Mndelbrot [6]. Its originl mening is "no rules, score, reduced to frgments". The mthemticl definition of frctl is: If set in Eucliden spce whose Husdorff dimension (DU) [7] [8] is lwys greter thn its topologicl dimension (DT), i.e. DU>DT, we cll the set s "frctl set", nd "frctl" for short. Common ground sys, thing s components similr to the whole body in certin wy clled frctl. Contrry to the efficient mrket hypothesis, the finncil mrket is not liner, blnced or orderly, but non equilibrium, nonliner, nd disordered. It is chotic, nd obey frctl Brown motion [9]. Rely on the frctl geometry lnguge; we cn get more in-depth understnding of finncil mrket. The frctl theory of stock mrket suggests, the stock mrket is not, t lest not lwys generl equilibrium efficient mrket. The chnges of stock price re with the chrcteristics of long-term memory of pst history. It is not completely rndom wlk, but "bised rndom wlk"[10].this process is highly dependent on the initil conditions. Its direction of chnge is uncertin. The finl price is not unique price which sty in generl equilibrium stte, but stochstic equilibrium price which comes from the polymorphic equilibrium mrket determined by multi-ttrctors. Frctl Sttistics Anlysis of EBAY Stock Price We now nlyze the stock price voltility of EBAY by the R/S method (i.e. The Rescled Rnge Anlysis Method) founded by H.E. Hurst. Assume time series {xi}, the number of observtions is M. The bsic ide of R/S nlysis method is: divide the time series {xi} whose length is M into An djcent subintervls whose length re (2 L). L is the length of mximum subintervl, A=M. Ech subintervl is nmed I, (=1, 2,, A ). Ech xi of Ech I is nmed xi,, (k=1, 2,,). Assume the mens of {xi} of I is e, We hve: x k, R I k ( xi, e ), i1 x minx mx k, 1k k 1,2,..., k, 1k 2 ( xk, e ) k 1 SI (1) Where is the length of subintervl, xk, is the ccumultion devition of subintervl, SI is the stndrd devition of the subintervl I, RI is the rnge of the subintervl I, RI become big with incresing. In order to compre different time series, Hurst used the stndrd devition of the observed vlue divided by its rnge, tht is: R divided by S. He hs the following eqution: A 1 H ( R / S) ( RI / SI ) ( ) A 1 (2) In eqution (2), A is the constnt; H is the Hurst Index, nd 0 H 1, 2 L. There re 3 different types of Hurst Index: (1)H=0.5, (2) 0 H 0.5, (3) 0.5 H 1. According to the sttistic knowledge, when H=0.5, the originl series is stndrd rndom wlk. The present will not ffect the future, nmely, the originl series is lck of sttisticl correltion in long-term. When 0 H 0.5, the originl series is with nti-persistence. It is often clled the men-reverting, tht is, the pst increment is negtive correltion with the increment of the future. If series is go up in the erlier period, then, it will probbly go down in the next period. On the other hnd, if it is go down, then in the next period, it will probbly go up. When 0.5 H 1, the originl series is with persistence, tht is, its pst increment is positively relted to the incrementl of the future. If series is go up 476

3 (down) in the erlier period, then, it will continue to go on (down) in the next period. The lrger vlue of H, the less of noise in this series, the series hs stronger persistence nd more obvious trend. The persistence level cn be mesured through H minus 0.5. There re mny of persistent series in nture; the stock mrket is one of them. Tke the logrithm on both ends of the eqution (3): ln( R / S) H ln( ) ln 2,3,...,L (3) The clcultion of Hurst index generlly hs two methods, both bsed on eqution (3). The first method is tking the length of ech subintervl in turn bsed on the length of the smple, for exmple, =50,100,,M. Performing the regression by the OLS method ccording to eqution (3), we cn get the vlues of plurlity of H. These estimtes re ssocited with L: the length of mximum subintervl, denoted s HL. Using {HL} series, the mximum vlue corresponding to L is the verge cycle length. If HLM=MxHL, then the verge cycle length is LM, the Hurst index is HLM. The second method is tking L = M, then compute HM by the first method. The vlue of HM is the Hurst index. But using this method we re unble to obtin the verge cycle length. For series, in order to describe the present effects on the future, B.B.Mendelbrot[11] hs introduced correltion mesurement index: (2H 1) C M 2 (4) In eqution (4), CM expresses the correltion during intervl M. Thus, independent series (H =0.5) hs no correltion; Series with persistence (H>0.5) hs positive correltion, nd series with nti-persistence (H <0.5) hs negtive correltion. In this study, we use the first method to clculte the Hurst index. In the estimtion nd regression process, we use the logrithmic rte of stock return of EBAY. Tht is, mke: St ln( pt / pt 1) (5) Where pt is the stock price t time T; St is the logrithmic stock return rte t time T, For R/S nlysis, Logrithmic rte of return is more suitble thn the percentge chnge in price. In R/S method, the rnge is the cumultive devition to the verge vlue. The sum of logrithmic return rte is equl to the cumultive return rte, while the percentge chnge is not. In order to remove the liner correltion of St Series, before the R/S nlysis, we first conduct first-order regression to St, to obtin the residul series: xt St ( bt) (6) Where nd b re liner regression coefficients of time series St, {xt} is the residul series, s show in Fig. 2. Y X Vrible 1 Line Fit Plot Figure 2. First-order Regression Residul Series of St Figure 3. ln ---ln (R/S) Line Grph 477

4 Crries on R/S nlysis to {xt} series, respectively tking t=50,100,150,...,600 trding dys, Clculting ccording to the bove formuls, We got the following, ln, R/S, ln (R/S) sttistic tble: Tble 1, ln, R/S, ln (R/S) Sttistic Tble ln R/S ln(r/s) The ln ---ln (R/S) line grph, s shown in Fig. 3. We cn see from Figure. 3, the line shows n lmost liner growth when ln 6.2, while it shows n obvious turning point when ln>6.2, ln= 6.2 corresponds to =500. Therefore, we cn drw conclusion tht the effective "long-term memory" cycle of EBAY s stock price voltility is bout 500 trding dys. Then, we perform OLS regression use the dt to estimte the vlue of H, the result is: ln( R / S) ln ( ) ( ) [ ] [ ] The regression prmeters nd vrince nly-sis re shown in Tble 2: Tble 2 The Regression Prmeters nd Vrince Anlysis of R/S Anlysis (7) Its liner fitting chrt is shown in Fig. 4: 478

5 Figure 4. ln---ln(r/s) Liner Fitting Chrt Regress results shows, H= The higher R2 (0.97), higher F vlue (237.2), closing to 2 of DW vlue(1.95) nd lower stndrd devition (0.08) show good fitting degree of this regression. Through the vlue of H, we cn clculte the vlue of correltion metric index CM is 1.096, it indictes tht the voltility of EBAY stock price hs frctl structure nd persistence to certin degree. It is not rndom wlk process, but bised rndom wlk process. However, the H vlue is not devite gretly from 0.5, lso show tht there exist lrge noise components. Different from the ordinry rndom wlk process, the stock return series is bised rndom wlk process, becuse it hs "long-term memory" in ply. The long-term memory is not infinite, but is limited. According to the clcultion bove, the verge long-term memory cycle of stock price voltility of EBAY is bout 500 trding dys. R/S nlysis shows tht, in the long-term memory cycle, the voltility of stock return rte of EBAY is persistent series with frctl probbility distribution. It follows bised rndom wlk process. The mrket shows some tendency strengthens behvior, rther thn the men reverting behvior, but there re still lrge noise components in it. When exceeding the long-term memory cycle, the voltility of stock return rte is no longer reflected the tendency strengthens behvior. Acknowledgements Thnks for the Philosophy nd Socil Sciences Foundtion of Jingsu, Chin (EYB022). References [1] McIntire E J B, RompréG, Severns P M. Bised correlted rndom wlk nd fory loop: which movement hypothesis drives butterfly metpopultion? [J]. OECOLOGIA, 2013, 172(1): [2] Swetdri Smdder M, Koushik Ghosh D, Tpsendr Bsu D. FRACTAL AALYSIS OF PRIME IDIA STOCK MARKET IDICES[J]. FRACTALS-COMPLEX GEOMETRY PATTERS AD SCALIG I ATURE AD SOCIETY, 2013, 21(1): [3] Peters E. Frctl Mrket Anlysis: Applying Chos theory to Investment nd Economics [J]. JOURAL OF ECOOMIC BEHAVIOR & ORGAIZATIO, 1998, 34(2): [4] Frezz M. Goodness of fit ssessment for frctl model of stock mrkets [J]. Chos, 2014, 66(1): [5] Feng Z, Run-tong Z, Ling-yun H, et l. The Anlysis of Frctl Fetures nd Long-run Memory Mechnism in the Shnghi nd Shenzhen Stock Mrket Composite Index Systems [J]. Journl of Beijing Jiotong University (Socil Sciences Edition),

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