INTERNATIONAL JOURNAL OF SCIENTIFIC & TECHNOLOGY RESEARCH VOLUME 3, ISSUE 12, December 2014 ISSN

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1 INTERNATIONAL JOURNAL OF SCIENTIFIC & TECHNOLOGY RESEARCH VOLUME 3, ISSUE 2, Deember 204 ISSN An Appliation Of Generalised Johansen-Ledoit- Sornette (GJLS) Model To Detet Size Of Rational Speulative Bubble In DJIA Stok Market During Year 997 Devendran Indiran, Nurfadhlina Abdul Halim, Wan Muhamad Amir W. Ahmad Abstrat: Rational speulative bubble an be defined as transient upward movements of stok pries above fundamental value due to speulative investors. The Generalised Johansen-Ledoit-Sornette (GJLS) model have been developed as a flexible tool to identify the size of rational speulative bubble. This model is ombines the eonomi theory of rational expetation bubbles with finite-time singular rash hazard rates, behavioural finane on imitation and herding of investors and traders as well as mathematial statistial physis of bifurations and phase transitions. It has been employed suessfully to a large variety of stok bubbles in many different markets. The purpose of this study is to predition bubble size of DJIA during eonomi risis 997. Keywords: Bubbles, Intrinsi Value, GJLS, Eonomi Crises, DJIA, Predition I. INTRODUCTION Rational speulative bubbles an be defined as positive aeleration of pries above fundamental value []-[3], [6]. A sudden rise in the prie of a ontinuous proess also an be named as rational speulative bubble [4]. Rational speulative bubbles are one of the serious issue that give negative impliations to the development of ountry s eonomy. This is beause of eonomi bubble formation and dramati bursts in finanial markets [5]. Many reent theories portraits that eonomi bubbles an be reated beause of positive feedbak trading by noise traders, heterogeneous beliefs of investors together with a limitation on arbitrage and synhronization failures among rational traders. Researhes done by [7]-[3] proved that the ombined effets of heterogeneous beliefs and short-sales onstrained may lead large movements in asset. In this kind of models whih assume heterogeneous beliefs and short-sales, the asset pries are determined at equilibrium to the extent that they reflet the heterogeneous beliefs about payoffs, but short sales boundaries fore the pessimisti investors disappear from the market, leaving only optimisti investors and thus magnified asset prie levels. However, when short sales limitations no longer tie investors, then pries fall bak downwards. In another lass of models, the role of noise traders in fostering positive feedbak trading has been emphasized. The term noise trader was proposed first by [4] and [5] to portray irrational investors. These noise positive feedbak traders purhase seurities when pries rises and sell when pries drop. Due to this positive feedbak mehanism, the divergene between the market prie and the intrinsi value has been bloated [6]-[9]. The empirial evidenes on this theory are mainly from the studies on momentum trading strategies. Stoks whih performed poorly in the past will perform better in a long-term perspetive (over the next three to five years) than stoks whih performed well in the past [20]. In ontrast, at intermediate horizon (three to twelve months), the stoks whih performed well previously will still perform better [2]. Apart from that, the fators that ause inflation have relationship to eonomi bubbles. The following figure portraits the relation between inflation and eonomi bubbles. The inrease in demand for goods and servies and at the same time lak in the supply of goods and servies Inrease in the prie of goods and servies drastially Eonomi Bubbles Devendran Indiran, Nurfadhlina Abdul Halim, Wan Muhamad Amir W. Ahmad Universiti Malaysia Terengganu, Malaysia, sirdev82@yahoo.om Universiti Malaysia Terengganu, Malaysia, lina@umt.edu.my Universiti Malaysia Terengganu, Malaysia, wmamir@umt.edu.my The value of urreny slumps and the prie of onsumer goods and servies are inreasing. Inflation Figure : The relation between inflation and eonomi bubble 309

2 INTERNATIONAL JOURNAL OF SCIENTIFIC & TECHNOLOGY RESEARCH VOLUME 3, ISSUE 2, Deember 204 ISSN However, identifying the existene of eonomi bubbles remains an unsolved problem in standard eonometri and finanial eonomi methods [24], [25]. This is due to the fat that the intrinsi value is in general poorly onstrained and it is impossible to differentiate between exponentially growing bubble pries. Diagnosing the bubble ex-ante ould help to take several ations to stop from bubble bursting. But none of the theories mentioned above an diagnose bubble ex-ante. This may be due to the fat that all these theories annot distinguish between intrinsi and bubble prie and annot give a prie dynamis whih leads to a rash. The Standard Johansen-Ledoit-Sornette (SJLS) model or Johansen-Ledoit-Sornette Model was developed by Sornette and his olleagues has potential to desribe the prie dynamis during a bubble regime by analysing the umulative human behaviour in new perspetives. It also has the ability to predit the most probable rash time after a bubble ex-ante. Generalized Johansen-Ledoit-Sornette (GJLS) Models have been developed as flexible tools to detet bubble size by prediting fundamental value by [26]. This study foused on estimating bubble size that formed in DJIA stok market during year 997. II. Generalised Johansen LeoditSornette Model The JLS model of finanial bubbles and rashes is an extension of the rational expetation bubble model proposed by [7]. A finanial bubble is modelled as a regime of aelerating or super-exponential power law growth puntuated by short-lived orretions organized aording the symmetry of disrete sale invariane [8]. The super-exponential power law is argued to result from positive feedbak resulting from noise trader deisions that tend to enhane deviations from fundamental valuation in an aelerating spiral. We firstly onsider the purely speulative asset that pays no dividends, so that we do not take into aount the interest rate, information asymmetry, risk aversion, and the market learing ondition. The rational expetations are simply orresponding to the familiar martingale hypothesis in (). Et pt pt t t () where p t denotes the prie of the asset at time t and E t. indiates the expetation onditional on information revealed up to time t. Then lets the umulative distribution funtion (df) of the time of rash is alled Q t, the probability density funtion (pdf) is hazard rate is ht q t Qt dq qt and the dt. The hazard rate is the probability per unit of time that the rash will happen in the next instant if it has not happened yet. In the JLS model, the stok market dynamis is desribed as (2). dp ( t) dt dj (2) p where p is the stok market prie and the term dj indiates a disontinuous jump suh that dj 0 before the rash and dj after the rash happens. The parameter determined the loss amplitude assoiated with the ourrene of a rash. The time-dependent drift t is hosen so that the prie proess satisfies the martingale ondition given as (3) and (4), respetively. E t dp t pt dt pt ht dt 0 (3) t h t (4) And (5) is orresponding to the prie. p log p t t 0 t h t dt (5) t 0 This gives the logarithm of the prie as the relevant observable. The higher the probability of a rash, the faster the prie grow (onditional on having no rash) in order to obey the martingale ondition. Intuitively, investors must be remunerated by a higher return in order to be indued to hold an asset that might rash. The sensitivity of the market reation to news or external influenes aelerate on the approah to this transition in a speifi way haraterized by a power law divergene at the ritial time t of the form F t z ( t t), where z is alled a ritial exponent. This form amounts to the following property of (6). d ln f d ln t t z (6) (6) is a onstant, namely that the behaviours of the observable F beome self-similar lose to t. The symmetry of self-similarity in the present ontext refers to df the fat that the relative variations d ln F of the F observable with respet to relative variations d t t d ln t t of the time-to-rash are independent t t of time t, as expressed by the onstany of the exponent z. The rash hazard rate follow the same dependene as (7). m t B t t h (7) where B is a positive onstant and t is the ritial point or theoretial date of the bubble end. The term m must in the range of 0 m for an important eonomi reason s otherwise; the prie would go infinity when approahing t (if the bubble has not rashed yet). The first order expansion for (7) (the hazard rate) is given by (8). 30

3 INTERNATIONAL JOURNAL OF SCIENTIFIC & TECHNOLOGY RESEARCH VOLUME 3, ISSUE 2, Deember 204 ISSN h t B t m m t os ln t t t t The rash hazard rate now displays log-periodi osillations. This an easily see by taking the exponent z to be omplex with a non-zero imaginary part, sine the real (8) zi m part of t t is t t os lnt t. The evolution of the prie before the rash and ritial date is then given by (9). ln E[ pt ] A Bt t m t os lnt t C t m The generalised Johansen LeoditSornette Model is formed by inferring fundamental value of stok in eq.(9).extension of (9) is said to be GJLS Model that proposed by [26]. The prie dynamis of an asset as (9) p p F LPPL t (5) do not onsider the ase whih would give an eonomially non-sensible behaviour, namely the prie diverges in finite time before the rash hazard rate itself diverges. In summary, [26] onsidered a model as shown below. p exp( F LPPL ( t)), (6) The final model (6) was applied to the DJIA stok index to identify the size of bubble that appeared during the year 997. This study is to obtain fundamental value of stok prie and followed by identifiation of bubble size as well. Time interval that seleted to analyse are as shown in the figure 2 below. t pdt t pdw p p dj dp (0) where the t pdt t pdw desribes the statistial geometrial Brownian motion and the third term is the jump. When the rash ours at some time t * (indiate t* dj ), the prie drops abruptly by amplitude t* p t * p.where, the prie drops from ( t * ) p t * p to p. The prie hanges from its value just before rash to a fixed well-defined valuation p. Inferring no-arbitrage ondition E dp 0 t p kp p ht (2) t to (0) leads to Conditional on the absene of a rash, the dynamis of the expeted prie obeys the equation dp t pdt kp p ht dt (3) and the fundamental prie must obey the ondition p min pt. For, the solution is pt p F LPPL t ln (4) where solution is LPPLt F is given by the (9); For,0, the 3

4 INTERNATIONAL JOURNAL OF SCIENTIFIC & TECHNOLOGY RESEARCH VOLUME 3, ISSUE 2, Deember 204 ISSN Figure 2: Time interval of DJIA Stok Market III. RESULT Figure 3: Fitted urve seleted DJIA index data,

5 INTERNATIONAL JOURNAL OF SCIENTIFIC & TECHNOLOGY RESEARCH VOLUME 3, ISSUE 2, Deember 204 ISSN TABLE : Bubble Size and Fundamental Value of DJIA Stok During 997 Stok Market Time Interval Market Value Fundamental value Bubble Size DJIA 30/08/995-28/03/ ,29% The above table shows the values obtained for DJIA stok market during the interval 30/08/995-28/03/997. The DJIA market stopping time from the historial data is This meant that the rational speulative bubble start to flat from the value The fundamental value obtained is ; this value explains that the market value deviated from its intrinsi value about 2337 or 29%. This phenomena is alled as bubble phase whih aused risis eonomy during year 997. The deviation between fundamental value and market value is a size of the appeared bubble. IV. CONCLUSION This paper examines the possible size of existene rational speulative bubble in the DJIA stok market on the 997. The GJLS model was suessfully employed to the data to ahieve our goal of study. It is essential needs for researher to study on eonomi bubble. It is beause the eonomi bubbles are one of the serious issue that give negative impliations to the development of eonomy whih is the fator leads to an eonomy risis. V. ACKNOWLEDGMENT We would like to aknowledge Universiti Malaysia Terengganu for providing failities. VI. REFERENCES [] J. Galbraith, The great rash, 929, Mariner Books, 997 [2] D. Sornette, Critial market rashes, Physis Reports 378 (2003) 98 [3] C. Kindleberger, Manias, Panis and Crashes: A Historyof Finanial Crises, 4th Edition, Wiley, [4] Kindleberger, Charles P.: Manias, Panis and Crashes: A History of Finanial Crises. New York: Basi Books, 978. [5] Statman, M. (998). Investor psyhology and market ineffiienies,eqiuty market and valuation methods. The Institute of Chartered Finanial Analysts, California [6] R.J.Shiller, Irrational Exuberane, 2nd Edition, Prineton University Press, 2005 [7] Lintner, John, 969, The aggregation of investors diverse judgments and preferenes in purely ompetitive seurity markets, Journal of Finanial and Quantitative Analysis 4, [8] Miller, Edward, 977, Risk, Unertainty and Divergene of Opinion, Journal of Finane 32, [9] M. Harrison, D. Kreps, Speulative investor behaviour in astok market with heterogeneous expetations, Quarterly Journal of Eonomis 92 (978) [0] M. Harrison, D. Kreps, Speulative investor behaviour in astok market with heterogeneous expetations, Quarterly Journal of Eonomis 92 (978) [] J. Chen, H. Hong, J. C. Stein, Breadth of ownership and stok returns, Journal of Finanial Eonomis 66 (2002) [2] J. Sheinkman, W. Xiong, Overonfidene and speulative bubbles, Journal of Politial Eonomy (2003) [3] D. Duffie, N. Garleanu, L. H. Pedersen, Seurities lending, shorting, and priing, Journal of Finanial Eonomis 66 (2002) [4] Kyle, Albert S., 985, Continuous Autions and Insider Trading, Eonometria 53, [5] Blak, Fisher; Noise, The Journal of Finane,Vol. 4, No. 3, Papers and Proeedings of the Forty- Fourth Annual Meeting of the Ameria Finane Assoiation, New York, New York, Deember 28-30, 985. (Jul., 986), pp [6] Shleifer, A. Summers, DeLong, J. B., L. H. and Waldmann, R. J. (990). Noise trader risk in finanial markets. Journal of Politial Eonomy, 98, [7] N. Barberis, A. Shleifer, R. Vishny, A model of investor sentiment, Journal of Finanial Eonomis 49(3) (998)

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