jei jei A Bootstrap Analysis of the Nikkei 225 Abstract

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1 A Boosrap Analysis of he Nikkei 225 Journal of Economic Inegraion A Boosrap Analysis of he Nikkei 225 James J. Kung Ming Chuan Universiy Andrew P. Carverhill Universiy of Hong Kong Absrac This sudy inends o find ou wheher or no he Nikkei 225 evolves over ime in accordance wih he following four widely used processes for deermining sock prices: random walk wih a drif, AR(1), GARCH(1,1), and GARCH(1,1)-M. Given he fac ha, in acualiy, we have bu one sample of ime series daa, he moivaion of his sudy is o make use of he boosrap echnology o deal wih his one-sample problem. Specifically, we use he boosrap echnique o generae 2,000 arificial Nikkei series from each process and compue he reurn from he rading rule for each of he 2,000 arificial Nikkei series. Then, we consruc a 95% boosrap percenile inerval wih hese 2,000 reurns and deermine if i conains he reurn compued from he acual Nikkei series. If i does, we claim ha reurns from he arificial Nikkei series are in agreemen wih hose from he acual Nikkei series. Our resuls show ha, of he four processes, GARCH(1,1)-M generaes reurns ha are mos agreeable wih hose compued from he acual Nikkei series. An imporan implicaion of his sudy is ha a proper model for pricing Nikkei-relaed derivaives is one ha uses he GARCH(1,1)-M process o depic he dynamics of he Nikkei reurn series. JEL Classificaions: C15, C22, G15 Key Words: Nikkei 225, Boosrap mehod, Simple Moving Average, Reurn-generaing processes, Boosrap percenile inerval * Corresponding Auhor: James J. Kung; School of Managemen, Ming Chuan Universiy, 250 Chung Shan N. Road, Secion 5, Taipei 111, Taiwan, Tel: , Fax: , fnjames@mail.mcu.edu.w; Co-Auhor: Andrew P. Carverhill; School of Business, Universiy of Hong Kong, Pokfulam road, Hong Kong, Tel: (852) , Fax: (852) , carverhill@business.hku.hk. Acknowledgemens: We are graeful o Alan Wong of he Hong Kong Bapis Universiy and Kalok Chan, Susheng Wang, and John Wei of he Hong Kong Universiy of Science & Technology for useful commens and suggesions. c 2012-Cener for Economic Inegraion, Sejong Insiuion, Sejong Universiy, All Righs Reserved. 487

2 James J. Kung, Andrew P. Carverhill I. Inroducion Over he years, he Japanese sock marke has been consisenly classified as a developed marke by FTSE Group s lis, MSCI lis, Dow Jones lis, Russell Global lis, he Inernaional Finance Corporaion, and he Global Sock Markes Facbook. As a maer of fac, he Japanese sock marke has undergone a series of reforms since he 1990s. In paricular, in November 1996, Prime Miniser Ryuaro Hashimoo iniiaed a comprehensive reform package, ofen referred o as he Japanese Big Bang 1, wih he objecive of creaing a free, fair, and global financial marke. According o he prime miniser, free means creaing a marke in which marke principles prevail; fair means enhancing he fairness and ransparency of he marke hrough clearly defined accouning and supervisory rules; and global means reforming he marke in line wih inernaional sandards. Thanks o he reforms implemened during he 1990s, many sudies (e.g., Cajueiro and Tabak, 2004; Worhingon and Higgs, 2006; Lim, 2007; Chong and Chan, 2008) have provided empirical evidence ha he Japanese sock marke has become more efficien. The above said, i is imporan o sudy how he Japanese sock marke one of he larges markes in he world in erms of rading volume and marke capializaion - evolved before and afer hose financial reforms were implemened in he 1990s. Specifically, his sudy aims o find ou wheher or no he Nikkei 225 (henceforh referred o as he Nikkei), he benchmark indicaor of he Japanese sock marke, evolves over ime in accordance wih some widely used random processes for sock prices. Accordingly, by means of a simple moving average rading rule, we employ he boosrap mehod of Efron (1979, 1982) o invesigae how closely reurns from he Nikkei are in agreemen wih reurns generaed from he following four widely used random processes for sock prices: a random walk wih a drif RW(α )), an auoregressive process of order one (AR(1)), and wo generalized auoregressive condiional heeroscedasic models - (GARCH(1,1) and GARCH(1,1)-M). Given he imporance of he Nikkei, he implicaion of his sudy is ha, for paricipans (e.g., invesors, speculaors, and hedgers) in he Japanese sock marke, a beer knowledge of he dynamics of he index is crucial for appropriaely managing sock marke risk and correcly pricing Nikkei-relaed derivaives 2. However, for ime series daa, we are consrained by he fac ha we have only one hisorical sample. Hence, i is hardly surprising ha previous empirical findings 3 based on differen 1 One major focus of he Japanese Big Bang reform was on insiuional changes. These changes included he following wo broad areas: (1) Improving he efficiency and compeiiveness of domesic financial insiuions (e.g., foreign financial companies were allowed o operae more freely in Japan); and (2) Abolishing monopoly powers previously enjoyed by domesic banks, securiies companies, and insurance firms (e.g., domesic and foreign financial insiuions would compee on an equal fooing for Japan s muli-rillion pension fund business). For deails, see Craig (1998) and Hall (1998). 2 Some Nikkei-relaed derivaives include he Nikkei 225 index opions raded on he Osaka Securiies Exchange and he Singapore Exchange, and he Nikkei 225 warrans raded on he American and Torono Sock Exchanges. 3 For example, Alexander (1961, 1964), Jensen and Beningon (1970), Bollerslev (1986), Engle, e al. (1987), French, e al. (1987), Chou (1988), Conrad and Kaul (1988, 1989), Bollerslev, e al. (1992), and Tsay (2005). 488

3 A Boosrap Analysis of he Nikkei 225 ime series daa are, o a cerain exen, divided on asse price dynamics. Accordingly, he moivaion of his sudy is o make use of boosrap echnology o deal wih his one sample problem. Simply pu, in he conex of his sudy, he boosrap enables us o generae a random a large number of samples (2,000 samples in his sudy) hrough replacemen wih each of he above four random processes such ha each of hese so-called boosrap samples (i.e., arificial Nikkei series) will possess he same saisical properies as he acual Nikkei series. Using hese 2,000 arificial Nikkei series for consrucing a 95% boosrap percenile inerval, we can invesigae how closely reurns from he acual Nikkei series mach reurns generaed from each of he four processes. The logic of our boosrap implemenaion can beer be undersood from anoher perspecive. Suppose he Nikkei lierally evolves hrough ime according o, say, a random walk wih a drif (i.e., RW(α )). Then, he acual Nikkei series is simply a sample drawn from his RW(α ) process. Hence, i is highly likely ha he reurn compued from his acual Nikkei series will fall wihin he 95% boosrap percenile inerval consruced using he 2,000 reurns generaed from his RW(α ) process. The res of he paper proceeds as follows: Secion II describes he Nikkei price series and gives a descripion of he simple moving average rading rule. Secion III describes our boosrap mehod, presens reasons for using he four reurn-generaing processes, and uses an illusraion o show how o implemen our boosrap by which arificial Nikkei price series are generaed for each of he four processes. In Secion IV, we presen and discuss our empirical resuls. In Secion V, we conclude his sudy and ouline he implicaions of his research. II. Daa and Simple Moving Average Trading Rule A. The Nikkei Daa Series The daa used are daily closing prices of he Nikkei (see Figure 1) from January 1, 1971, o December 31, 2010 a oal of 10,436 observaions. They were rerieved from he DaaSream daabase. Since no major financial reform was implemened in Japan before he 1990s, we pariion he enire sample period ino wo equal sub-periods 4 : and We compue he daily reurn as he naural log difference of he Nikkei prices. Tha is, R (1) = log( P ) log( P 1 ) 4 The sub-period is he ime when he Japanese sock marke moved forward a full speed; he sub-period is he ime which is ofen referred o as he wo los decades of Japan. 489

4 James J. Kung, Andrew P. Carverhill where P and 1 for he ime from day -1 o day. P are he closing prices of he Nikkei on day -1 and day, and R is he reurn In an efficien marke, securiy reurns are independen of one anoher over ime because new informaion comes o he marke in a random and unpredicable manner, and securiy prices respond insanly and accuraely o his new informaion. Hence, he magniude of he auocorrelaion in securiy reurns can offer some clues as o he efficiency of he marke. Simply pu, auocorrelaion should be insignifican if he marke is efficien. From Table 1, we noe ha he daily auocorrelaion for is saisically significan a he 1% level a lags 1, 2, 3, and 5, whereas ha for is saisically significan a he 1% level only a lag 1. In oher words, for he earlier sub-period, he reurn on day is likely o depend on reurns on days -1, -2, -3, and -5; whereas for he laer sub-period, he reurn on day is likely o depend only on he reurn on day -1. Hence, he auocorrelaions provide a rough indicaion ha he Japanese sock marke displayed relaively greaer efficiency for he sub-period. B. The Trading Rule The rading rule used in his sudy is he Simple Moving Average 5 (SMA). The n-day Moving Average (MA) on day is 1 1 M +, n = Pk = n+ 1 n n k = n+ 1 n [ P + P + + P P ] where P k is he closing price of he Nikkei on day k. According o SMA rules, a buy signal is generaed when he closing price rises above he n- day MA and a sell signal is generaed when he closing price falls below he n-day MA. Tha is, an invesor would ake a long posiion in he Nikkei when a buy signal is generaed and, conversely, a shor posiion in he Nikkei when a sell signal is generaed. When a signal is generaed, SMA rules require ha he posiion be mainained unil he closing price peneraes he n- day MA again. A commonly used SMA rule is 1-100, where he MA is 100 days. In his sudy, we use he following SMA rules: (1, 20), (1, 50), (1, 100), and (1, 200). Each rule is evaluaed wih bands of 0% and 1%, making a oal of eigh SMA rules. A band is used o reduce he number of imes an invesor would have o move ino and ou of he marke. For example, Brock e al. (1992), Bessembinder and Chan (1998), and Siegel (2002) all use a 1% band for heir SMA rules. (2) 5 See Edwards, e al. (2007) for deails on he simple moving average and oher rading rules. 490

5 A Boosrap Analysis of he Nikkei 225 III. Boosrap Implemenaion In his secion, we firs give a simple descripion of he boosrap mehod 6 cusomized for his sudy; hen, we poin ou why i is jusifiable o use he four random processes for generaing arificial Nikkei series; and finally, we make use of an illusraion o show how o implemen our boosrap by which arificial Nikkei price series are generaed for each of he four processes. A. The Boosrap Mehod For our boosrap implemenaion, we apply each of he four random processes o he acual Nikkei reurn series o obain heir respecive esimaed parameers (e.g., α in RW(α )) and residuals. The residuals are hen redrawn wih replacemen o form a scrambled residual series which is hen used wih he esimaed parameers o generae arificial Nikkei reurn series such ha each of hese so-called boosrap samples (i.e., arificial Nikkei series) will possess he same saisical properies as he acual Nikkei series. In his sudy, he relevan saisic is he reurn from he 2000SMA rading rule compued from he Nikkei seriesseries. Specifically, we use he boosrap o generae 2,000 arificial Nikkei series 7 from each of he four processes and compue he reurn from he SMA rading rule for each of hese 2,000 arificial Nikkei series. Then, following on from Efron and Tibshirani (1993), we consruc a 95% boosrap percenile inerval wih hese 2,000 reurns. If he reurn from he SMA rule compued from he acual Nikkei series falls wihin his 95% percenile inerval, hen we claim ha his reurn agrees wih hose generaed from he arificial Nikkei series and, furhermore, infer ha he acual Nikkei series is in agreemen wih hose arificial Nikkei series generaed for a given random process. In oher words, he acual Nikkei series is like a sample drawn from his process. B. The Four Reurn-Generaing Processes The four random processes 8 for generaing arificial Nikkei reurn series are as follows: RW(α ) AR(1) GARCH(1,1) R = α + ε (3) R = α + βr + ε (4) 1 R α + ε = (5) u 6 See Efron and Tibshirani (1993) for deails. 7 According o Efron and Tibshirani (1993), 2,000 arificial Nikkei series are more han enough for esimaion accuracy purposes. 8 Alernaively, we can wrie he four processes in a compac form as R = u + ε, where u = α in RW(α ) and GARCH(1,1); α + βr in AR(1); and u = α + βσ 2 in GARCH(1,1)-M. = 1 491

6 James J. Kung, Andrew P. Carverhill GARCH(1,1)-M 9 R = α + βσ + ε 2 (6) For equaions (3) and (4), ε is independenly and idenically disribued. For equaions (5) and (6), σ = + + a bε 1 cσ 1, ε = σ z, and z ~ N( 0,1). Tha is, R is he reurn on day, ε is normally disribued and serially uncorrelaed, z is normally 10 disribued wih zero mean and 2 uni variance, and σ is a linear funcion of he square of he las period s error (i.e., ε 1) and 2 of he las period s condiional variance (i.e., σ 1). Noe ha if β in equaion (6) is posiive and saisically significan, hen increased risk (as measured by an increase in he condiional variance σ ) resuls in an increase in reurn 2 R. Hence, β can be regarded as he amoun of risk. We choose he four random processes because hey have been found by numerous sudies o bes characerize he dynamics of asse/sock reurns. These four encompass a wide range of random processes commonly used for asse/sock reurns. A his juncure, some relaed references are in order. For he RW(α ) process, Fama (1995) claims ha he empirical evidence o dae provides srong suppor for he random walk model. For he AR(1) process 11, Conrad and Kaul (1989) show ha a firs-order auocorrelaion of 0.20 is found for a value-weighed porfolio of he larges companies over he period, and ha higher order auocorrelaion beyond a lag of one day is basically zero. For he GARCH (1,1) process, Brooks (2008) saes ha in general a GARCH(1,1) model will be sufficien o capure he volailiy clusering in he daa, and rarely is any higher order model esimaed or even enerained in he academic finance lieraure. For he GARCH(1,1)-M process, Chou (1988) fis such a process o he weekly reurns of he NYSE value-weighed index over he period and finds he exisence of changing equiy premiums. For deails, see Alexander (1961, 1964), Fama (1965, 1970, 1995), and Jensen and Beningon (1970) for random walks; Conrad and Kaul (1988, 1989) and Tsay (2005) for he AR(1) process; and Bollerslev (1986), Engle e al. (1987), French e al. (1987), Chou (1988), Bollerslev e al. (1992), and Brook (2008) for he wo GARCH processes. C. An Illusraion As an illusraion, we use an AR(1) process o demonsrae how o implemen our boosrap by aking he following seps. (i) Based on he acual Nikkei reurn series, we esimae he wo parameers in equaion (4) using he ordinary leas squares mehod and obain he 9 The GARCH(1,1)-M process can also be expressed in such a form ha he condiional mean is linear in he condiional sandard 2 deviaion σ raher han in he condiional variance σ. 10 An alernaive o a sandard normal disribuion is o assume ha z follows a sandard Suden s disribuion, in which case he densiy has more probabiliy mass in he ails. 11 In his sudy, a major reason for using an AR(1) process is ha he auocorrelaions for daily reurns from he Nikkei over he wo sub-periods are boh saisically significan a he 1% level a lag 1. See Table

7 A Boosrap Analysis of he Nikkei 225 wo esimaes α and β. (ii) We compue he residual as e = R R, where = 1, 2,, N and R = α + β R 1. Hence, we obain a series of residuals; ha is, { e 1, e 2,, e N }. (iii) For each j, we randomly draw a residual wih replacemen from he residual series and form R j = α + β R j 1 + e j(where j = 1, 2,, N) o generae an arificial AR(1) Nikkei reurn series. (iv) We conver each arificial AR(1) Nikkei reurn series ino an arificial AR(1) Nikkei price series using equaion (1). (v) Similar o compuing he daily reurn from he SMA rule using acual Nikkei price series, we compue he daily reurn from he SMA rule for buy and for sell using arificial AR(1) Nikkei price series. (vi) Repeaing seps (i) (v), we obain 2,000 daily b b b s s s reurns (denoed by R1, R2,..., R2000) for buy and 2,000 daily reurns (denoed by R1, R2,..., R2000) for sell compued respecively from 2,000 arificial AR(1) Nikkei price series. Following Efron and Tibshirani (1993), we consruc a 95% boosrap percenile inerval such ha he 2.5h percenile and 97.5h percenile of he 2,000 daily reurns (for buy and for sell) compued from arificial Nikkei price series are, respecively, he lower and upper limis for he inerval. Specifically, arranging he 2,000 daily reurns in ascending order such ha b b b b s s s s R ( 1) R(2)... R(1999) R(2000) for buy and R ( 1) R(2)... R(1999) R(2000) for sell, we find ha he 95% boosrap inerval 12 is [R b b (51), R (1950) ] for buy and [R s s (51), R (1950) ] for sell. Tha said, we deermine if he daily reurn (for buy and for sell) from each SMA rule compued from he acual Nikkei price series lies wihin his 95% boosrap inerval. IV. Empirical Resuls We use he ordinary leas squares mehod o esimae he parameers of he RW(α) and AR(1) processes, and he maximum likelihood mehod o esimae he parameers of he GARCH(1,1) and GARCH(1,1)-M processes. Table 2 presens he parameer esimaes for he four processes over he wo sub-periods. The parameers are esimaed using he RATS economeric package. Table 3 shows he daily reurns from he eigh SMA rules based on acual Nikkei price series. For , each of he eigh rules resuls in a posiive daily reurn for buy and a negaive daily reurn for sell, suggesing ha he marke ended o move upward over his sub-period. For , each of he eigh rules resuls in a daily reurn for buy smaller han ha for sell, implying ha he marke ended o move downward over his sub-period. In he following secions, we will deermine if he daily reurns from he eigh SMA rules in Table 3 compued from he acual Nikkei price series lie wihin heir respecive 95% boosrap percenile inervals under each of he four processes. In Tables 4-7, for each of he eigh SMA rules, Mean is he average value of he 2, b b Tha is, R ( 51) and R are he 2.5h and 97.5h perceniles of he 2,000 daily reurns for buy; ( 1950) R s s (51) and R are he 2.5h and ( 1950) 97.5h perceniles of he 2,000 daily reurns for sell. 493

8 James J. Kung, Andrew P. Carverhill daily reurns compued from arificial Nikkei price series, R (51) ( and R denoe he 2.5h (1950) and 97.5h perceniles of he 2,000 daily reurns for buy and for sell. For example, considering he (1, 20, 0%) rule for buy in he firs hree rows of Table 4, is he average value of he 2,000 daily reurns, and and are he 2.5h and 97.5h perceniles of he 2,000 daily reurns. Tha is, [ , ] is a 95% boosrap confidence inerval. For visual clariy, hose 95% inervals are shaded if he daily reurns in Table 3 compued from he acual Nikkei price series lie wihin heir respecive inervals. A. Resuls based on Arificial RW(α ) Nikkei Series Table 4 shows he daily reurns from he eigh SMA rules based on he arificial Nikkei price series generaed from he random walk process for he wo sub-periods. For , none of he eigh SMA rules for buy resuls in he daily reurns compued from he acual Nikkei series lying wihin heir respecive 95% inervals; bu hree of he eigh SMA rules for sell resul in he daily reurns compued from he acual Nikkei series lying wihin heir respecive 95% inervals. Tha is, he (1, 20, 0%), (1, 50, 0%), and (1, 100, 0%) rules for sell resul in he daily reurns of , , and (see Table 3) from he acual Nikkei series lying wihin [ , ], [ , ], and [ , ], respecively. For , four SMA rules for buy and five SMA rules for sell resul in he daily reurns compued from he acual Nikkei series lying wihin heir respecive 95% inervals İn comparison, daily reurns compued from he acual Nikkei series appear more likely o have been generaed from he RW(α ) process for he sub-period han for he sub-period. Given he fac ha he Japanese sock marke has become more efficien (see Secion I) afer a series of financial reforms were implemened during he 1990s, i is no surprise ha he Nikkei evolved over he second sub-period as if i were a sample likely drawn from he RW(α ) process. B. Resuls based on Arificial AR(1) Nikkei Series The AR(1) process is used o deec wheher he resuls from he SMA rules are caused by daily auocorrelaions in he series. If he reurns are posiively auo- correlaed, a higher (lower) reurn oday will end o be followed by a higher (lower) reurn on he following day; if he reurns are negaively auocorrelaed, a higher (lower) reurn oday will end o be followed by a lower (higher) reurn on he following day. The parameer esimaes for he AR(1) process in Table 2 indicae some degree of posiive auocorrelaion for (where β = ) and some degree of negaive auocorrelaion for (where β = ). 494

9 A Boosrap Analysis of he Nikkei 225 Table 5 shows he daily reurns from he eigh SMA rules based on he arificial Nikkei price series generaed from he AR(1) process for he wo sub-periods. For , four SMA rules for buy and seven SMA rules for sell resul in he daily reurns compued from he acual Nikkei series lying wihin heir respecive 95% inervals. For , none of he eigh SMA rules for boh buy and sell resuls in he daily reurns compued from he acual Nikkei series lying wihin heir respecive 95% inervals. In comparison, daily reurns compued from he acual Nikkei series appear more likely o have been generaed from he AR(1) process for he sub-period han for he sub-period. Given he fac ha he sock marke has become more efficien as a resul of he reforms implemened since he 1990s, i is no surprising ha he dynamics of he Nikkei exhibied no obvious sign of auocorrelaion over he second sub-period. C. Resuls based on Arificial GARCH(1,1) Nikkei Series The GARCH(1,1) process allows he condiional variance o be dependen on one previous variance and one lagged squared error. Table 6 shows he daily reurns from he eigh SMA rules based on he arificial Nikkei price series generaed from he GARCH(1,1) process. For , wo SMA rules for buy and six SMA rules for sell resul in he daily reurns compued from he acual Nikkei series lying wihin heir respecive 95% inervals. For , only one SMA rule for buy bu hree SMA rules for sell resul in he daily reurns compued from he acual Nikkei series lying wihin heir respecive 95% inervals. In comparison, daily reurns compued from he acual Nikkei series appear more likely o have been generaed from he GARCH(1,1) process for he firs sub-period han for he second sub-period. D. Resuls based on Arificial GARCH(1,1)-M Nikkei Series Finance heory claims ha invesors should be rewarded a higher reurn for bearing addiional risk. The GARCH(1,1)-M process is designed o model such a phenomenon, where he condiional variance of asse reurns is included in he reurn equaion (see equaion (6)). Table 7 shows he daily reurns from he eigh SMA rules based on he arificial Nikkei price series generaed from he GARCH(1,1)-M process. For , five SMA rules for buy and all eigh SMA rules for sell resul in he daily reurns compued from he acual Nikkei series lying wihin heir respecive 95% inervals. For , four SMA rules for buy and six SMA rules for sell resul in he daily reurns compued from he acual Nikkei series lying wihin heir respecive 95% inervals. Of he four random processes, he GARCH(1,1)-M process appears o have generaed daily reurns ha are mos likely in agreemen wih hose from he acual Nikkei series. 495

10 James J. Kung, Andrew P. Carverhill V. Conclusion and Implicaion This sudy aims o find ou wheher or no reurns from he Nikkei are agreeable wih hose generaed from four widely used random processes for sock prices. Given he fac ha we have only one sample of any ime series daa, he moivaion of his sudy is o use he boosrap o deal wih his one-sample problem To proceed, we use he boosrap o generae 2,000 arificial Nikkei series for each process and compue he reurn from he SMA rading rule for each of hese 2,000 arificial Nikkei series. Then, we se up a 95% boosrap percenile inerval wih hese 2,000 reurns and deermine if he inerval conains he reurn compued from he acual Nikkei series. If i does, we claim ha his reurn agrees wih hose generaed from he arificial Nikkei series and, moreover, we infer ha he acual Nikkei series is in agreemen wih hose arificial Nikkei series generaed for a given process. Our empirical resuls indicae ha, of he four random processes, GARCH(1,1)-M generaes reurns ha are mos agreeable wih hose compued from he acual Nikkei series. Given he imporance of he Japanese sock marke in he world, a beer grasp of he dynamics of he Nikkei is indispensable for appropriaely handling Japanese sock marke risk and correcly pricing Nikkei-relaed derivaives. A relevan case in poin is he Nikkei 225 index opions, which are acively raded on he Osaka Securiies Exchange and he Singapore Exchange. Given our resuls ha boh he random walk wih a drif 13 and he GARCH(1,1) process 14 are inadequae for depicing he Nikkei reurn series, an imporan implicaion of his sudy is ha a more appropriae model for pricing Nikkei 225 index opions is one ha uses he GARCH(1,1)-M process o characerize he dynamics of he Nikkei reurn series. Received 06 April 2011, Revised 28 April 2012, Acceped 30 May 2012 References Alexander, S.S. (1961), Price Movemens in Speculaive Markes: Trends or Random Walks, Indusrial Managemen Review, Vol. 2, pp Alexander, S.S. (1964), Price Movemens in Speculaive Markes: Trends or Random Walks, Number 2, Indusrial Managemen Review, Vol. 5, pp Bessembinder, H., Chan, K. (1998), Marke Efficiency and he Reurns o Technical Analysis, Financial Managemen, Vol. 27, No. 2, pp The coninuous-ime analog of he random walk wih drif in equaion (3) is he arihmeic Brownian moion, which means ha he Nikkei price process in equaion (1) is a geomeric Brownian moion (GBM). Our resuls imply ha he well-known Black-Scholes opion pricing model (1973), which assumes a GBM for he price process, is no appropriae for pricing Nikkei 225 opions. 14 Our resuls also imply ha he GARCH opion pricing model of Duan (1995) is no appropriae for pricing Nikkei 225 opions. 496

11 A Boosrap Analysis of he Nikkei 225 Black, F., Scholes, M. (1973), The Pricing of Opions and Corporae Liabiliies, Journal of Poliical Economy, Vol. 81, pp Bollerslev, T. (1986), Generalized Auoregressive Condiional Heeroskedasiciy, Journal of Economerics, Vol. 31, pp Bollerslev, T., Chou, R.Y., Kroner, K.F. (1992), ARCH Modeling in Finance: A Review of he Theory and Empirical Evidence, Journal of Economerics, Vol. 52, pp Brock, W., Lakonishok, J., LeBaron, B. (1992), Simple Technical Trading Rules and he Sochasic Properies of Sock Reurns, Journal of Finance, Vol. 47, No. 5, pp Brooks, C. (2008), Inroducory Economerics for Finance, 2nd ediion, Cambridge Universiy Press, Cambridge, UK. Cajueiro, D.O., Tabak, B.M. (2004), Ranking Efficiency for Emerging Markes, Chaos, Solions and Fracals, Vol. 22, pp Chong, T. T., Chan, S. T. (2008), Srucural Change in he Efficiency of he Japanese Sock Marke afer he Millennium, Economics Bullein, Vol. 7, No. 7, pp Chou, R.Y. (1988), Volailiy Persisence and Sock Valuaions: Some Empirical Evidence using GARCH, Journal of Applied Economerics, Vol. 3, pp Conrad, J., Kaul, G. (1988), Time-Variaion in Expeced Reurns, Journal of Business, Vol. 61, No. 4, pp Conrad, J., Kaul, G. (1989), Mean Reversion in Shor-Horizon Expeced Reurns, Review of Financial Sudies, Vol. 2, No. 2, pp Craig, V.V. (1998), Financial Deregulaion in Japan, FDIC Banking Review, Vol. 11, No. 3, pp Duan, J.C. (1995), The GARCH Opion Pricing Model, Mahemaical Finance, Vol. 5, No. 1, pp Edwards, R.D., Magee, J., Bassei, W.H.C. (2007), Technical Analysis of Sock Trends, 9h ediion, AMA- COM, New York. Efron, B. (1979), Boosrap Mehods: Anoher Look a he Jackknife, Annals of Saisics, Vol. 7, No. 1, pp Efron, B. (1982), The Jackknife, he Boosrap, and Oher Resampling Plans, Sociey for Indusrial and Applied Mahemaics, Philadelphia. Efron, B., Tibshirani, R. (1993), An Inroducion o he Boosrap, Chapman and Hall, New York. Engle, R.F., Lilien, D.M., Robins, R.P. (1987), Esimaing Time-Varying Risk Premia in he Term Srucure: The ARCH-M Model, Economerica, Vol. 55, No. 2, pp Fama, E.F. (1965), The Behavior of Sock Marke Prices, Journal of Business, Vol. 38, pp Fama, E.F. (1970), Efficien Capial Markes: A Review of Theory and Empirical Work, Journal of Finance, Vol. 25, pp Fama, E.F. (1995), Random Walks in Sock Marke Prices, Financial Analyss Journal, January-February, pp French, K.R., Schwer, G.W., Sambaugh, R.F. (1987), Expeced Sock Reurns and Volailiy, Journal of Financial Economics, Vol. 19, No. 1, pp

12 James J. Kung, Andrew P. Carverhill Hall, M.J.B. (1998), Financial Reform in Japan: Causes and Consequences, Edward Elgar, Norhampon. Jensen, M., Beningon, G. (1970), Random Walks and Technical Theories: Some Addiional Evidence, Journal of Finance, Vol. 25, pp LeRoy, S.F. (1982), Expecaions Models of Asse Prices: A Survey of Theory, Journal of Finance, Vol. 37, pp Lim, K.P. (2007), Ranking Marke Efficiency for Sock Markes: A Nonlinear Perspecive, Physica A, Vol. 376, pp Siegel, J.J. (2002), Socks for he Long Run, 3rd ediion, McGraw-Hill, New York. Tsay, R.S. (2005), Analysis of Financial Time Series, 2nd ediion, John Wiley & Sons, Hoboken, New Jersey. Worhingon, A.C., Higgs, H. (2006), Weak-Form Marke Efficiency in Asian Emerging and Developed Equiy Markes: Comparaive Tess of Random Walk Behavior, working paper, Universiy of Wollongong, Ausralia. Figure 1. The Nikkei 225 (1971~2010) Table 1. Summary Saisics for Daily Reurns on Acual Nikkei Series 1971~ ~2010 Number of Observaions Average Daily Reurn Daily Sandard Deviaion Esimaed auocorrelaions: Lag ** ** Lag ** * Lag ** * Lag * * Lag ** Noe: Numbers wih * (**) are significan a 5% (1%) level for a wo-ailed es. 498

13 A Boosrap Analysis of he Nikkei 225 Table 2. Parameer Esimaes for he Four Reurn-Generaing Processes Process Parameer 1971~ ~2010 RW(α ) α ( ) ( ) AR(1) α ( ) ( ) β ( ) ( ) GARCH(1,1) α ( ) ( ) α ( ) ( ) b ( ) ( ) c ( ) ( ) GARCH(1,1)-M α ( ) ( ) β ( ) ( ) α ( ) ( ) b ( ) ( ) c ( ) ( ) Noe: Parameers are esimaed using RATS. Numbers in parenheses are sandard -raios. 499

14 James J. Kung, Andrew P. Carverhill Table 3. Daily Reurns from Simple Moving Average Rules based on Acual Nikkei Series 1971~ ~2010 Rule Buy Sell Buy Sell (1, 20, 0%) (1, 50, 0%) (1,100,0%) (1,200,0%) (1, 20, 1%) (1, 50, 1%) (1,100,1%) (1,200,1%) Average Noe: Simple Moving Average rules are idenified as (shor, long, band), where shor and long are he lenghs of shor and long moving averages respecively, and band is he percenage difference required o generae a buy or sell signal. 500

15 A Boosrap Analysis of he Nikkei 225 Table 4. Daily Reurns from Simple Moving Average Rules based on Arificial RW(α) Nikkei Series 1971~ ~2010 Rule Buy Sell Buy Sell (1, 20, 0%) Mean R (51) R (1950) (1, 50, 0%) Mean R (51) R (1950) (1,100,0%) Mean R (51) R (1950) (1,200,0%) Mean R (51) R (1950) (1, 20, 1%) Mean R (51) R (1950) (1, 50, 1%) Mean R (51) R (1950) (1,100,1%) Mean R (51) R (1950) (1,200,1%) Mean R (51) R (1950) Average Noe: Simple Moving Average rules are idenified as (shor, long, band), where shor and long are he lenghs of shor and long moving averages respecively, and band is he percenage difference required o generae a buy or sell signal. Mean is he average value of he 2,000 daily reurns. R (51) and R (1950) are he 2.5h and 97.5h perceniles of he 2,000 reurns for buy and for sell. Shaded R (51) and R (1950) are he 95% boosrap inervals ha conain he daily reurn from he acual Nikkei series. 501

16 James J. Kung, Andrew P. Carverhill 502 Table 5. Daily Reurns from Simple Moving Average Rules based on Arificial AR(1) Nikkei Series 1971~ ~2010 Rule Buy Sell Buy Sell (1, 20, 0%) Mean R (51) R (1950) (1, 50, 0%) Mean R (51) R (1950) (1,100,0%) Mean R (51) R (1950) (1,200,0%) Mean R (51) R (1950) (1, 20, 1%) Mean R (51) R (1950) (1, 50, 1%) Mean R (51) R (1950) (1,100,1%) Mean R (51) R (1950) (1,200,1%) Mean R (51) R (1950) Average Noe: Simple Moving Average rules are idenified as (shor, long, band), where shor and long are he lenghs of shor and long moving averages respecively, and band is he percenage difference required o generae a buy or sell signal. Mean is he average value of he 2,000 daily reurns. R (51) and R (1950) are he 2.5h and 97.5h perceniles of he 2,000 reurns for buy and for sell. Shaded R (51) and R (1950) are he 95% boosrap inervals ha conain he daily reurn from he acual Nikkei series.

17 A Boosrap Analysis of he Nikkei 225 Table 6. Daily Reurns from Simple Moving Average Rules based on Arificial GARCH(1,1) Nikkei Series 1971~ ~2010 Rule Buy Sell Buy Sell (1, 20, 0%) Mean R (51) R (1950) (1, 50, 0%) Mean R (51) R (1950) (1,100,0%) Mean R (51) R (1950) (1,200,0%) Mean R (51) R (1950) (1, 20, 1%) Mean R (51) R (1950) (1, 50, 1%) Mean R (51) R (1950) (1,100,1%) Mean R (51) R (1950) (1,200,1%) Mean R (51) R (1950) Average Noe: Simple Moving Average rules are idenified as (shor, long, band), where shor and long are he lenghs of shor and long moving averages respecively, and band is he percenage difference required o generae a buy or sell signal. Mean is he average value of he 2,000 daily reurns. R (51) and R (1950) are he 2.5h and 97.5h perceniles of he 2,000 reurns for buy and for sell. Shaded R (51) and R (1950) are he 95% boosrap inervals ha conain he daily reurn from he acual Nikkei series. 503

18 James J. Kung, Andrew P. Carverhill 504 Table 7. Daily Reurns from Simple Moving Average Rules based on Arificial GARCH(1,1)-M Nikkei Series 1971~ ~2010 Rule Buy Sell Buy Sell (1, 20, 0%) Mean R (51) R (1950) (1, 50, 0%) Mean R (51) R (1950) (1,100,0%) Mean R (51) R (1950) (1,200,0%) Mean R (51) R (1950) (1, 20, 1%) Mean R (51) R (1950) (1, 50, 1%) Mean R (51) R (1950) (1,100,1%) Mean R (51) R (1950) (1,200,1%) Mean R (51) R (1950) Average Noe: Simple Moving Average rules are idenified as (shor, long, band), where shor and long are he lenghs of shor and long moving averages respecively, and band is he percenage difference required o generae a buy or sell signal. Mean is he average value of he 2,000 daily reurns. R (51) and R (1950) are he 2.5h and 97.5h perceniles of he 2,000 reurns for buy and for sell. Shaded R (51) and R (1950) are he 95% boosrap inervals ha conain he daily reurn from he acual Nikkei series.

19 A Boosrap Analysis of he Nikkei