International Journal of Finanial Eonomis Vol. 4, No., 205, -7 Estimating Fundamental Value and the Size of Rational Speulative Bubbles of Hong Kong Stok Market during the Year 2008 Devendran Indiran, Nurfadhlina Abdul Halim 2, Wan Muhamad Amir W. Ahmad 3 Abstrat Rational speulative bubble an be well-defined as transient upward movements of stok pries above fundamental value due to speulative investors. The Generalised Johansen-Ledoit-Sornette (GJLS) model has been developed as a flexible devie to diagnose the size of rational speulative bubble. This model is ombines the eonomi theory of rational expetation bubbles with finite-time singular rash hazard rates, behavioral 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 predit intrinsi value and size of rational speulative bubble of Hong Kong stok market during global eonomi risis 2008. The intrinsi value predited by using the seleted time interval is 20997.03 whih shows that the market value is deviated about 33.63% from its fundamental value. This deviation is alled as size of the speulative rational bubble that formed during global eonomi risis 2008. By using the predited intrinsi value, we found that the rational speulative bubble start to form and grow in Hong Kong stok market from 5/06/2006 to. There are two bubble phases found in the period of seleted time interval. It is essential needs for researher to study on finanial bubbles. 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. Keywords: Bubble size, Intrinsi value, GJLS. Introdution A positive aeleration of pries above intrinsi value is said to be a rational speulative bubbles[]- [3], [6]. An unexpeted rise in the prie of a ontinuous proess also an be named as rational speulative bubble [4]. Rational speulative bubbles are one of the severe issue that give negative onsequenes to the growth of ountry s eonomy. This is beause of eonomi bubble development and dramati bursts in finanial markets [5]. Many reent onepts desribes that eonomi bubbles an be produed beause of heterogeneous beliefs of investors together with a limitation on arbitrage and synhronization failures among rational traders positive feedbak trading by noise 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. BS. Finanial Mathematis, Universiti Malaysia Terengganu (UMT), Kuala Terengganu, Terengganu (203), MS. Mathematial Sienes, Universiti Malaysia, Terengganu (UMT), Kuala Terengganu, Terengganu (at present) 2 leturer in Shool of Informatis and Applied Mathematis, Universiti Malaysia Terengganu (UMT). 3 Senior leturer in Shool of Informatis and Applied Mathematis, Universiti Malaysia Terengganu (UMT). 205 Researh Aademy of Soial Sienes http://www.rassweb.om
D. Indiran et al. In another lass of models, the role of noise traders in fostering positive feedbak trading has been highlighted. The term noise trader was proposed first by [4] and [5] to show irrational investors. These noise positive feedbak traders purhase seurities when pries inreases and sell when pries drop. Due to this positive feedbak mehanism, the deviation 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]. However, prediting the burst of eonomi bubbles remains an unsolved problem in standard eonometri and finanial eonomi methods [22], [23]. This is due to the fat that the fundamental value is in general poorly onstrained and it is impossible to differentiate between exponentially growing bubble pries. Deteting the bubble ex-ante ould help to take some 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 differentiate between intrinsi and bubble prie and annot give a prie dynamis whih leads to a rash. Generalized Johansen-Ledoit-Sornette (GJLS) Models have been developed as flexible tools to predit bursting of rational speulative bubble [23]. This study speially onduted to estimate fundamental value and size of rational speulative bubble of Hong Kong stok marketof the year 2008. 2. Generalised Johansen Ledoit Sornette Model The GJLS model of eonomi 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 () t where p 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 dq q t density funtion (pdf) is qt and the hazard rate is ht. The hazard rate is the probability dt Qt 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 2
International Journal of Finanial Eonomis 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 t h (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 z of the form Ft ( t t), where z is alled a ritial exponent. This form amounts to the following property of (6). d ln f z (6) d lnt t (6) is a onstant, namely that the behaviors of the observable F beome self-similar lose to t. The df symmetry of self-similarity in the present ontext refers to the fat that the relative variations d ln F F d of the observable with respet to relative variations t t d ln t t of the time-to-rash are t t independent of time t, as expressed by the onstany of the exponent z. The rash hazard rate follows 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). ' m ' m t B t t os ln t h t t t ' (8) The rash hazard rate now displays log-periodi osillations. This an easily see by taking the zi exponent z to be omplex with a non-zero imaginary part, sine the real part of t t is m t t os lnt t. The evolution of the prie before the rash and ritial date is then given by (9). 3
D. Indiran et al. m m t ] A Bt t Ct t os lnt t ln E[ p (9) The generalised Johansen Ledoit Sornette Model is formed by inferring fundamental value of stok in eq.(9). Extension of (9) is said to be GJLS Model that proposed by [23]. The prie dynamis of an asset as 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 * t* (indiate dj ), the prie drops abruptly by amplitude t* where, the prie drops from ( t * ) before rash to a fixed well-defined valuation p. Inferring no-arbitrage ondition Et dp 0 p t * p. () * p too pt p to (0) leads to t p kp p ht (2). The prie hanges from its value just 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 p. For, the solution is ln pt p F LPPL t (4) where F LPPLt is given by the (9); For,0 p p F LPPL t t, the solution is (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 Hang Seng Index to estimate the fundamental value and size of rational speulative bubble that appeared during the year 2008. 3. Results and Disussion As a first step, we test few time intervals to predit the index value at market stopping time in order to hoose a most appropriate time window to predit fundamental value of HSI. Table shows the results obtained for index value at market stopping time. 4
International Journal of Finanial Eonomis Table : Predited index value at market stopping time Time Intervals Index value at market Predited stopping time index value Differenes 4/04/2003-3638.22 26888.84 5% 0/05/2004-3638.22 2769.28 4.0% 24/0/2005-3638.22 28029.43.4% 5/06/2006-3638.22 28548.5 9.7% MSE.047636E- 05.589367E- 08 5.97643E- 06 3.47953E- 07 There are four different time intervals seleted to predit index value at market stopping time. The most nearest index value obtained is for the time interval 5/06/2006-. This time interval was used to foreast the fundamental value of HSI during the year 2008. The fitted Hang Seng index with the GJLS model is shown in figure. Figure : Fitted urve of HSI, 2008 Market HSI Table 2: Predited intrinsi value of HSI, 2008 Market Value Intrinsi Time Interval Before Crash Value 5/06/2006-3638.22 20997.03 Bubble Size 064.9, 33.63% The intrinsi value predited by using the seleted time interval is shown in Table 2. The obtained intrinsi value is 20997.03 whih shows that the market value is deviated about 33.63% from its fundamental value. This deviation is alled as size of the speulative rational bubble that formed during global eonomi risis 2008. By using the predited intrinsi value, we found that the rational speulative 5
D. Indiran et al. bubble start to form and grow in Hong Kong stok market from 5/06/2006 to. There are two bubble phases found in the period of seleted time interval. The summary of the phases are shown in the Table 3 and illustrated in figure 2. Figure 2: Bubble Phases of Hong Kong Stok Market, 2008 Table 3: Bubble Phase and Size of the Bubble Formed in Hong Kong Stok Market, 2008 Bubble Phases Starts Ends Bubble Size Date Market Value Date Market Value Duration Min Max % % 5/06/2007 207.05 5/08/2007 2375.72 62 0.0.79 20/08/2007 2595.63 2850.0 3 2.85 33.63 The maximum size of rational speulative bubble formed in Hong Kong stok market is 33.63% and appeared about 3 days before rash. Aording to the Table 3, we an summarize that the longer the duration the bigger the size of the bubble formed. 4. Conlusion In a onlusion, this paper examines the intrinsi value and size of rational speulative bubble of HSI stok market during the year 2008. The GJLS model was suessfully employed to the data to ahieve our goal of study. It is essential needs for researher to study on finanial bubbles. 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. Aknowledgment We would like to aknowledge Universiti Malaysia Terengganu for providing fund and failities. 6
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