Current log-periodic view on future world market development
Stanislaw Drozdz, Jaroslaw Kwapien, Pawel Oswiecimka, Josef Speth
aa r X i v : . [ q -f i n . S T ] J un Current log-periodic view on future world marketdevelopment
Stanis law Dro˙zd˙z , , Jaros law Kwapie´n , Pawe l O´swie¸cimka ,Josef Speth , Institute of Nuclear Physics, Polish Academy of Sciences, PL–31-342 Krak´ow,Poland Institute of Physics, University of Rzesz´ow, PL–35-310 Rzesz´ow, Poland Institut f¨ur Kernphysik, Forschungszentrum J¨ulich, D–52425 J¨ulich, GermanyApplicability of the concept of financial log-periodicity is discussed andencouragingly verified for various phases of the world stock markets devel-opment in the period 2000-2010. In particular, a speculative forecastingscenario designed in the end of 2004, that properly predicted the worldstock market increases in 2007, is updated by setting some more preciseconstraints on the time of duration of the present long-term equity mar-ket bullish phase. A termination of this phase is evaluated to occur inaround November 2009. In particular, on the way towards this dead-line, aSpring-Summer 2008 increase is expected. On the precious metals marketa forthcoming critical time signal is detected at the turn of March/April2008 which marks a tendency for at least a serious correction to begin.In the present extended version some predictions for the future oil priceare incorporated. In particular a serious correction on this market is ex-pected in the coming days.PACS numbers: 05.45.Pq, 52.35.Mw, 47.20.Ky
Keywords: Complex systems, Financial markets, Fundamental laws ofnature
E-mail address: [email protected]
1. Introduction
The financial dynamics is a multiscale phenomenon and therefore thequestion which of its properties are scale invariant and which are scale char-acteristic refers to the essence of this phenomenon. There exists strong (1)
Drozdz printed on September 5, 2018 related evidence that at least a large portion of the financial dynamics isgoverned by phenomena analogous to criticality in the statistical physicssense [1]. In its conventional form criticality implies a scale invariance [2]of a properly defined function Φ( x ) characterizing the system:Φ( λx ) = γ Φ( x ) . (1)A constant γ then describes how the properties of the system change underrescaling by the factor λ . The general solution to this equation reads [3]:Φ( x ) = x α Π(ln( x ) / ln( λ )) , (2)where the first term represents a standard power-law that is characteristicof continuous scale-invariance with the critical exponent α = ln( γ ) / ln( λ )and Π denotes a periodic function of period one. This general solution canbe interpreted in terms of discrete scale invariance. Due to the second termthe conventional dominating scaling acquires a correction that is periodic inln( x ). Imprints of such oscillations can often be identified in the financialdynamics [4, 5, 6, 7]. To make a proper mapping one defines x = | T − T c | ,where T denotes the clock time labelling the original price time series andrepresents a distance to the critical point T c . The emerging sequence ofspacings between the corresponding consecutive repeatable structures at x n - as seen in the linear scale - forms a geometric progression according to therelation ( x n +1 − x n ) / ( x n +2 − x n +1 ) = λ . The time points T c thus correspondto the accumulation of such oscillations and, in the context of the financialdynamics such points mark a reversal of the trend. An important relatedelement, for a proper interpretation and handling of the financial patternsas well as for consistency of the theory, is that such log-periodic oscillationsmanifest their action self-similarly through various time scales [7]. Thisapplies both to the log-periodically accelerating bubble market phase aswell as to the log-periodically decelerating anti-bubble phase.Fig. 1 schematically illustrates the relevant structures on one particulartime scale. The thick solid line corresponds to the first term ( x α ) in Eq. 2and it represents the global market trends on both sides of T c , increasingand decreasing respectively. On both these sides the log-periodic oscillationsare superimposed, accelerating and decelerating correspondingly. These os-cillations are generated by the second term in Eq. 2. A specific form ofthe periodic function Π in this Equation is as yet not provided by any firstprinciples. Since in the corresponding methodology the oscillation struc-ture carries the most relevant information about the market dynamics, fortransparency one uses the simplest representations for such a function. Onereasonable possibility is the first term of its Fourier expansion,Π(ln( x ) / ln( λ )) = A + B cos( ω π ln( x ) + φ ) . (3) rozdz printed on September 5, 2018 x n x n+1 x n+2 x’ n+2 x’ n+1 x’ n x n+1 - x n x n+2 - x n+1 x’ n - x’ n+1 x’ n+1 - x’ n+2 = λ = λ T c For the financial markets λ = λ =2 Fig. 1. Schematic illustration of the possible structures, relevant for the financialmarkets and consistent with the Eq. 2. The thick - cusp-shaped at T c - solidline corresponds to the conventional phase transition and in the financial contextit reflects an overall market trend. Superimposed on top of this trend are three- cosine, cosine modulus and saw-like - possible practical representations for theoscillation pattern periodic in the logarithm of the distance x from the critical time T c ( x = | T − T c | ). Log-periodicity means that the ratios λ and λ of the distancesbetween the consecutive repeatable structures x n are constant. Furthermore, forthe financial markets λ = λ = 2. This implies that ω = 2 π/ ln( λ ). Already such a simple parametrizationallows to properly reflect the contraction of oscillations, especially on thelarger time scales. On the smaller time scales just replacing the cosine by its modulus equally well describes oscillations and in addition it ofteneven better follows departures of the market amplitude from its averagetrend. Another simple - from the market perspective an even more realistic- representation of Π is to use an asymmetric saw-like function. Such threepossibilities are indicated in Fig. 1. What they all have however to obey isthe same contraction ratio (preferred scaling factor) expressed by λ and λ .For the real markets more and more evidence is collected that the preferredscaling factor λ ≈ λ = λ = λ = 2. Univer- Drozdz printed on September 5, 2018 sality of the λ , establishes very valuable and in fact crucial constraint onpossible forms of the analytic representations of the market trends and ofthe oscillation patterns, including the future ones. This greatly amplifiesa predictive power of the corresponding methodology. Also very helpful inthis respect is the requirement of self-similarity which helps to clarify thesignificance of a given pattern and allows to determine on what time scaleit operates. The present contribution provides further arguments in thisfavour.
2. World stock market since 2003
As far as the trends are concerned the contemporary stock market indicesworld-wide are remarkably synchronized. This is one of the characteristicsreflecting the world globalization [10]. For the recession period since 2000this effect is illustrated in ref. [9]. This period ends approximately in thefirst months of 2003 and the stock markets synchronously enter the bullishperiod. As is shown in Fig. 2 all the indices assume the up trend. There ofcourse are some differences in the magnitude of oscillation amplitudes butthe common similarity of this oscillation pattern is clearly visible. Due tothis similarity in the following we shall concentrate mostly on the AmericanS&P500 index because it represents the world largest market and is thusexpected to provide the most reliable indicator of the global world markettrends.An early (made in February 2007, shown also at the time of FENS07Conference) attempt to log-periodically grasp the large scale S&P500 pat-terns and to provisionally estimate duration of its present increasing phase,before it enters recession of comparable magnitude as the one between mid2000 and early 2003,is shown in Fig. 3. The corresponding critical time T c points to the turn of October/November 2009. This scenario thus in-dicates that until this time the market, on average, should preserve its uptrend. An extra argument in favour of this scenario is that it was predictingan intermediate sizeable correction in around the period November 2007-February 2008 and it took place indeed. It however also demands that inaround the end of February 2008 (time of writing this contribution) thiscorrection terminates and the index starts rising till at least late Summerbefore it starts correcting again. The reason for ignoring in this scenariothe mid 2004 correction seen in the S&P500 and comparable in magnitudeto the one (relevant) in May-June 2006 is that this former correction is tobe interpreted a remnant of the bear, since September 2000 log-periodicallydecelerating market component. The related part of this component is alsodrawn in the panel (b) of Fig. 3. This panel includes the S&P500 data up topresent and, accordingly, the global trend is updated, but the log-periodic rozdz printed on September 5, 2018 Time -3-2-10123 N o r m a li ze d L og a r it h m i c I nd i ce s S&P500 NIKKEI FTSEHang-Seng NASDAQ DAXWIG
Fig. 2. Time series of the logarithm of the most important indices world-widein the period 2002-2007. The time series have been appropriately normalized, P ( t ) → P ( t ) −
σ ( P ( t )) , where < ... > denotes the average over the period underconsideration and σ is the standard deviation. structure remains unchanged.The above updated scenario, including the real data up to present, forthe three pairs of indices, S&P500 and Nasdaq, DAX and FTSE, HangSengand WIG (Poland), is shown in the three panels of Fig. 4. The log-periodiccomponent remains here the same as the one in Fig. 3. The main reasonfor such a selection of indices presented in Fig. 4 is to show some fromamong the world most important ones whose oscillation patterns remainin a satisfactory agreement with the same common scenario, as well asthose (HangSeng and WIG) whose blind relating to such a scenario mayseem pointless. Let us recall here however a phenomenon of the ”super-bubble” [8]. This is an effect that from time to time takes place in thefinancial dynamics and whose identification appears relevant for a properinterpretation of the financial patterns with the same universal value ofthe preferred scaling factor λ = 2. This phenomenon of a ”super-bubble”is a local boost, itself evolving log-periodically, superimposed on top of along-term bubble and seen as an extra acceleration of the price increase.Such a ”super-bubble” then crashes and the system returns to a normalbubble state that eventually crashes at the time determined by the long-term Drozdz printed on September 5, 2018 year i nd ex v a l u e T c = 1 Nov 2009 S&P500 (a) year i nd ex v a l u e T c = 1 Nov 2009 T c = 1 Sep 2000 (b) Fig. 3. (a) Time series of the S&P500 from 1.1.2001 until 23.2.2007 versus the early(23.2.2007), presented at the FENS2007 Conference, optimal log-periodic represen-tation (solid line) with λ = 2 and T c =1.11.2009 for the market bull phase. (b) Thesame log-periodic representation including the S&P500 data up to 23.2.2008. Onlythe global trend is corrected as compared to (a). The dashed line indicates thelog-periodically decelerating structure that started on 1.9.2000. patterns. Two spectacular examples of such a phenomenon are providedby the Nasdaq in the first quarter of 2000 and by the gold price in thebeginning of 1981 [8]. It seems very likely to us that the fast increasesand then even faster decreases that we see in the last two indices of Fig. 4during the period June 2006 - February 2008 constitute further examples ofthe ”super-bubbles” and the corresponding markets have just returned totheir normal long-term (since March 2003) up trend and an ultimate criticaltime T c may very well coincide with the same common scenario.In ref. [11] a speculative scenario for the stock market (S&P500) devel-opment in the time period 2000 - 2010 was invented by representing themarket index as a superposition of the two components: one declining andlog-periodically decelerating since September 2000 and the second one risingand thus log-periodically accelerating. Based on the longest possible timescale (since 1800) log-periodic representation [8] for this index, the criti-cal time for this second component was provisionally estimated in aroundSeptember 2010. Time verified that so far this scenario makes a lot ofsense. In particular, a spectacular increase in 2007 as well as the reverse ofthe increasing trend by the end of the year was predicted. Recall that thisscenario was presented in November 2004. These facts encourage further itselaboration. Since, as discussed above, we now have a more precise estimate rozdz printed on September 5, 2018 S & P S&P500 N A S D A Q NASDAQ D AX DAX
Time W I G WIG F T S E FTSE H ang - S e ng Hang-Seng T C = (a)(b)(c) Fig. 4. (a) Time series from 1.1.2002 until 15.2.2008 for the three pairs of indices:(a) S&P500 and NASDAQ, (b) DAX and FTSE and (c) HangSeng and WIG, versusthe optimal log-periodic representation for the world bull market phase with λ = 2and T c =1.11.2009. for the end of the present long-term bull market phase the scenario underconsideration can be improved. An accordingly updated variant of this sce-nario is shown in Fig. 5. Interestingly, during the year 2007 this theoreticalmarket representation opened room for an even stronger increase than whatthe S&P500 has performed. Some other markets, like the ones shown in thepanel (c) of Fig. 4, made however use of this freedom and executed a moreproportional detour.
3. Precious metals market
Some preliminary evidence already exists that the log-periodic oscilla-tions may also accompany the commodities market dynamics during somemore speculative periods. Such patterns have been identified on the goldmarket [8] during the time period 1978-1982, in this case including a veryspectacular ”super-bubble” before the ultimate reverse of the long-termtrend, and on the oil futures market in the years 1998-2004 [11]. In this
Drozdz printed on September 5, 2018 year i nd ex v a l u e T c = 1 Nov 2009 S&P500 T c = 1 Sep 2000 Fig. 5. A log-periodic scenario, represented by the solid line, for the S&P500 de-velopment until end 2009. The critical time is fixed as T c = 1 . . λ = 2 components (dashed lines): log-periodically deceleratingsince 1.9.2000 and log-periodically accelerating towards 1.11.2009, correspondingly. connection it needs to be also emphasized that the same preferred scalingfactor λ ≈ λ = 2) structure which points to theturn of March/April 2008 as the critical time T c . Whether this indicatesan ultimate reversal of the precious metals present price trends in aroundthis time or only a sizeable (20-30%) correction cannot be settled at this rozdz printed on September 5, 2018 Time S il ve r XAU01.2003 02.2004 01.2005 02.2006 03.2007 02.2008 G o l d XAG P l a ti nu m XPT T c =26 March Fig. 6. Time series of the gold (red), silver (grey) and platinum (cyan) prices (inUSD) since 1.1.2003 until 22.2.2008, versus their optimal log-periodic representa-tion with λ = 2. The corresponding critical time T c =26.3.2008. stage of the development. If the later possibility is to occur, i.e., after sucha correction the market starts returning to the previous highs, then - in thespirit of self-similar log-periodicity [7] - it is expected to assume increasetowards significantly higher levels and this market phase should last foranother 3-4 years in order to complete the full log-periodic cycle on the ap-propriately longer time horizon, such that the whole log-periodic structureseen in Fig. 6 constitutes its self-similarly nested substructure. It seemsalso likely that the last few months accelerated increases, especially on theplatinum market, can be interpreted in terms of a ”super-bubble”. In anycase however at the T c indicated above one may expect the beginning of asignificant correction on the precious metals market.
4. Oil market (Note added on June 23, 2008) Time satisfactorily verified the aboveprediction for the precious metals market. This encourages applying thesame methodology to another commodity market - the oil market - wherethe prices expressed in terms of the USD went up almost a factor of threeover the past one year. Our log-periodic interpretation of the underlyingprice dynamics over the time period 2000-2010 is illustrated in Fig. 7. Thepast year sharp increase turns out to be transparently log-periodic with λ = 2 and the related critical time corresponds to the first decade of July Drozdz printed on September 5, 2018 U S D Time E U R t c =11Jul2008 T c =19Sep2010 Brent Crude Oil (Barrel)
Fig. 7. Time series of the oil price in USD (grey line represents the oil price inEuro) since mid 1999 versus its optimal log-periodic representation with λ = 2.The critical time T c =September 2010 corresponds to an ultimate reverse of thelong term (since 1999) oil price up trend while t c =11.7.2008 sets an upper limit forthe end of the present ”super-bubble”. λ = 2 opens room for the continuationof increases to similar or even somewhat higher levels and the estimatedcritical time setting the dead-line for this long term increase corresponds tothe late Summer in 2010, thus several months after the stock market entersa serious recession.JS thanks the Foundation for Polish Science for financial support throughthe Alexander von Humboldt Honorary Research Fellowship.REFERENCES [1] D. Sornette, Why Stock Markets Crash: Critical Events in Complex FinancialSystems , (Princeton University Press, Princeton, 2003). rozdz printed on September 5, 2018