Log-periodic power law and genelized hurst exponent analysis in estimating an asset bubble bursting time

Wątorek, Marcin; Stawiarski, Bartosz Article Log-periodic power law and genelized hurst exponent analysis in estimating an asset bubble bursting time e-Finanse: Financial Internet Quarterly Provided in Cooperation with: University of Information Technology and Management, Rzeszów Suggested Citation: Wątorek, Marcin; Stawiarski, Bartosz (2016) : Log-periodic power law and genelized hurst exponent analysis in estimating an asset bubble bursting time, e-Finanse: Financial Internet Quarterly, ISSN 1734-039X, University of Information Technology and Management, Rzeszów, Vol. 12, Iss. 3, pp. 49-58, https://doi.org/10.1515/fiqf-2016-0001 This Version is available at: https://hdl.handle.net/10419/197439 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc-nd/3.0 www.e-fi nanse.com University of Information Technology and Management in Rzeszów 49 Marcin Wątorek1, Bartosz Stawiarski2 Abstract We closely examine and compare two promising techniques helpful in es� ma� ng the moment an asset bubble bursts. Namely, the Log-Periodic Power Law model and Generalized Hurst Exponent approaches are considered. Sequen� al LPPL fi � ng to empirical fi nancial � me series exhibi� ng evident bubble behavior is presented. Es� ma� ng the cri� cal crash-� me works sa� sfactorily well also in the case of GHE, when substan� al „decorrela� on” prior to the event is visible. An extensive simula� on study carried out on empirical data: stock indices and commodi� es, confi rms very good performance of the two approaches. 1 Ins� tute of Nuclear Physics, Polish Academy of Sciences, email: mwatorek@i� .edu.pl. 2 Cracow University of Technology Faculty of Physics, Mathema� cs and Computer Science, email: bstawiar[email protected]. LOG-PERIODIC POWER LAW AND GENERA LIZED HURST EXPONENT ANALYSIS IN ESTIMATING AN ASSET BUBBLE BURSTING TIME Financial Internet Quarterly „e-Finanse” 2016, vol.12/ nr 3, s. 49-58 DOI: 10.1515/fi qf-2016-0001 JEL classifi ca� on: C22, C53, C61 Keywords: asset bubble, crash, Log-Periodic Power Law, Generalized Hurst Exponent, mul� fractality, forecas� ng, burs� ng � me es� ma� on Received: 31.10.2015 Accepted: 14.09.2016 www.e-fi nanse.com University of Information Technology and Management in Rzeszów 50 „e-Finanse” 2016, vol. 12 / nr 3 Marcin Wątorek, Bartosz Stawiarski Log-periodic power law and generalized hurst exponent analysis in estimating an asset bubble bursting time Introduction Specula� ve bubbles have been occurring all throughout the history of fi nancial markets, irrespec� ve of the asset classes involved. One of the most pronounced pioneer bubbles was the Dutch tulip mania in 1635-1637, followed by a huge crash that wiped out large fortunes. Whereas the early stages of bubble forma� on usually pass unno� ced, the ripe phases of these anomalies can be detected by a number of techniques, eg. augmented Dickey-Fuller tests for unit root. In addi� on, taking present market fundamentals simultaneously into account usually makes the work more successful. Bubbles inevitably burst, leading to severe price downturns or outright crashes of magnitude corresponding to the scale of the preceding overvalua� on. The exact moment of this burs� ng, called also a crash-� me or rupture point, draws a clear line between two dis� nct regimes for price dynamics. As far as investment effi ciency is concerned, predic� ng the crash-� me tc poses a fi nancially vital and mathema� cally challenging research problem. Its importance is associated with large fi nancial bets put at stake, especially shortly before the crucial peak. Several approaches have been proposed to model the price dynamics prior to and right a� er the bust. One of the powerful tools has been developed and expanded for nearly two decades by D. Sorne� e, who employs a LogPeriodic Power Law for modeling the asset price dynamics (Johansen, Ledoit & Sorne� e, 2000). Another precursor of this approach is S. Drożdż (Drożdż, Grummer, Ruf & Speth, 2003). Importantly, although the very � me tc can be easily determined ex-post, one should rather focus upon its reliable interval es� ma� on as the whole process of bubble burs� ng can be interpreted as a phase transi� on. The method has been proved successful on a number of occasions (Zhang et al., 2016), e.g. spectacularly precise predic� on of crude oil bubble burs� ng � me in 2008 (Drożdż, Kwapień & Oświęcimka, 2008). Another promising tool for detec� ng the end of the specula� ve bubble is analysis of long range dependence using the Hurst exponent. Specifi c decorrela� on envisaged in investor behavior can be found in numerous papers such as those by: Kristoufek (2010), Grech and Pamuła (2008), Morales, Di Ma� eo, Grama� ca, Aste (2012). Although there exist other concurrent tools devised to tackle this topic (dynamic systems evolu� on, smooth transi� on models), in this paper we focus on the two above, applica� onally vital approaches, useful both from an academic and business point of view. Log-Periodic Power Law Model In our fi rst approach to detect specula� ve bubbles we are using the LPPL model. Based on Johansen et al. (2000) we assume that in a bubble regime price follows a stochas� c diff eren� al equa� on: (1) where p = p(t) is the asset stock price, µ(t) - dri� , σ(t) - vola� lity, dW is the increment of a standard Wiener process and dj represents a discon� nuous jump such that j = 0 before the crash and j = 1 a� er the crash. Each successive crash corresponds to a unit jump of j. The parameter κ quan� fi es the amplitude of the crash when it occurs. Denote Ft - fi ltra� on generated by the price process p(t), namely Ft = σ{p(s): s ≤ t}. The jump’s dynamics are governed by a crash hazard rate h(t). Since h(t)dt is the probability that the crash occurs between t and t + dt condi� onally on the fact that it has not yet happened, we have (2) The JLS model assumes that two types of agents are present on the market: a group of traders with ra� onal expecta� ons and a group of noise traders who exhibit herding behavior that may destabilize the asset price. According to this model, the ac� ons of noise traders are quan� fi ed by the following dynamics of the hazard rate (Johansen et al., 2000): (3) where B’,C’ denote amplitude parameters; ω,φ’ - phase parameters and tc is the cri� cal � me marking the end of the bubble. The power law behavior (tc −t)m−1 embodies the mechanisms of posi� ve feedback at the origin of the bubble forma� on. The log-periodic func� on cos(ω ln(tc−t)−φ’) takes into account the existence of a possible hierarchical cascade of panic accelera� on causing the bubble to pop. In the JLS model the ra� onal agent is risk neutral and has ra� onal expecta� ons. Thus, the asset price p(t) follows a mar� ngale process: . www.e-fi nanse.com University of Information Technology and Management in Rzeszów 51 „e-Finanse” 2016, vol. 12 / nr 3 Marcin Wątorek, Bartosz Stawiarski Log-periodic power law and generalized hurst exponent analysis in estimating an asset bubble bursting time Under these assump� ons the no-arbitrage condi� on is just .Accordingly, the excess return is propor� onal to the crash hazard rate, namely Thus, solving (1) under the condi� on that no crash has occurred yet leads to the following log-periodic power law (LPPL) equa� on for the log-price expecta� on: (4) where and . It should be noted that solu� on (4) describes the dynamics of the average log-price only up to the cri� cal � me tc and cannot be used beyond it. This crash-� me tc corresponds to the termina� on of the bubble and indicates the change to another regime, which could be either a large crash with accelera� ng oscilla� ons (nega� ve bubble – Wątorek, Drożdż & Oświęcimka, 2016) or decelera� ng oscilla� ons (an� -bubble – Johansen & Sorne� e, 1999) or a change of the average growth rate. The LPPL model (4) is described by 3 linear parameters and 4 nonlinear parameters . These parameters are subject to the following constraints, described in Sorne� e, Woodard, Jiang, Zhou (2013): ; ; , , . To fi t the LPPL func� on (4) to empirical data we employed a procedure proposed by Filimonov and Sorne� e (2013), which reduces the es� ma� on to just three nonlinear parameters . The key idea of this method is to decrease the number of nonlinear parameters and simultaneously to eliminate the interdependence between the phase and the angular log-frequency . Let us rewrite (4) by expanding the cosine term as follows: (5) Now, we introduce two new parameters: (6) and rewrite the LPPL equa� on (4) as (7) As seen from (7), the LPPL func� on has now only 3 nonlinear and 4 linear parameters, and the two new parameters and contain the former phase . The resul� ng model is calibrated on the data using the Ordinary Least Squares method, providing es� mators of all the parameters: within a given � me window subject to analysis. GHE approach In our second approach, we aim at connec� ng “decorrela� on” (long memory tapering) and mul� fractality growth with the burs� ng of the specula� ve bubble. To achieve that, we employed the no� on of Generalized Hurst Exponent, henceforward GHE, based on Di Ma� eo (2007). This exponent is a tool for studying directly the scaling proper� es of the data via the q-th order moments of the distribu� on of the � me series increments with , namely: (8) where . GHE is then obtained from the scaling behavior of func� on (8) when the following rela� on holds: (9) and hence we calculate GHE via regression from the following func� on: (10) Processes exhibi� ng this scaling behavior can be divided into two classes: 1) Processes with , i.e. independent of These processes are unifractal, which means that their scaling behavior is uniquely determined by the constant H, known as Hurst exponent or self-affi ne index (Di Ma� eo, 2007). 2) Processes with non-constant are called mul� scaling (or mul� fractal) and each moment scales with a diff erent exponent. Previous works have pointed out how fi nancial � me series exhibit scaling behaviors which are not simply fractal, but rather mul� fractal, e.g. Di Ma� eo (2007). Depending on q, the exponents are associated with special features. For instance, when , describes the scaling behavior of the absolute values of the increments. This exponent value is expected to be closely related to the original Hurst exponent indeed scaling the absolute spread within the increments. The exponent value corresponding to is associated with the scaling of the autocorrela� on func� on and is related to the power spectrum index (Di Ma� eo, 2007). Quite intui� vely, the recent past is more important www.e-fi nanse.com University of Information Technology and Management in Rzeszów 52 „e-Finanse” 2016, vol. 12 / nr 3 Marcin Wątorek, Bartosz Stawiarski Log-periodic power law and generalized hurst exponent analysis in estimating an asset bubble bursting time than the remote past. To incorporate that we can assume that the informa� onal impact of observa� ons decays exponen� ally. This smoothing can be a� ained by defi ning weights as: (11) where is the weights characteris� c � me. Introducing an exponen� al decay factor , the parameter is given by Pozzi, Di Ma� eo, Aste, (2012) as (12) Hence the weighted GHE (abbrev.: GHEw) is obtained by replacing normal averages in (8) with weighted averages (13) as described in the scaling rela� on (9), so we get (14) As an indicator of degree of mul� fractality we consider the quan� ty: (15) following the paper of Morales, Di Ma� eo, Aste (2014). Mul� fractality in � me series can be interpreted as a consequence of fat-tailed behavior. Study given in Barunik, Aste, Di Ma� eo, Liu (2012) shows that temporal correla� ons have the eff ect of reducing the measured mul� fractality. In Moreales et al. (2012) GHEw was used as a tool to detect unstable periods within fi nancial � me series. In our analysis we want to link mul� fractality increase at the end of the specula� ve bubble both with autocorrela� ons decay and fat-tailed distribu� ons. According to Fractal Market Hypothesis, mul� fractality in � me series may result from the existence of mul� ple market players having diff erent � me horizons (Weron & Weron, 2000). Capturing the dynamics of investors’ interac� ons can be carried out by using GHE, which measures the autocorrela� on func� on decay rate. Empirical results LPPL approach In order to fi t the LPPL func� on described above we have to select the ini� al parameters . Next, we need to calculate linear parameters by OLS method and then minimize the cost func� on using nonlinear least squares method. In previous works random choice of the ini� al parameters was proposed, see e.g. Filimonov and Sorne� e (2013), using local peak detec� on Pele (2012) or constant – Drożdż et al. (2003). In our work we decided to test all possible values of startup parameters with step 0.05, [2,22] with step 0.5, or with step 10, depending on the data length. All calcula� ons were performed in Matlab package. We minimized the cost func� on by using Region-Trust algorithm, which in Bree and Joseph (2013) was proposed as an improvement to the tradi� onally used Levenberg-Marquardt algorithm. To get more robust results, we carried out the analysis on empirical data with moving star� ng point with step either 5 or 10 trading days in a shrinking � me window and moving end point with 5 trading days step in an expanding � me window , similar as in Jiang, Zhou, Sorne� e, Woodard (2010). For each � me window approximately 8000 combina� ons of the prescribed ini� al parameters were taken and a� er the nonlinear op� miza� on we got the same amount of parameters with the sum of squared residuals. Lowest SSR points at the best t within each � me window. During the fi � ng process ge� ng a stable value of tc is essen� al, therefore we compared SSR’s from each � me window by coun� ng mean squared error. Finally, we calculated an 80% confi dence interval based on 5% tc with the lowest MSE. We tested our approach on 10 historical asset bubbles and then applied it on current, real fi nancial � me series. The results are presented in the table below: www.e-fi nanse.com University of Information Technology and Management in Rzeszów 53 „e-Finanse” 2016, vol. 12 / nr 3 Marcin Wątorek, Bartosz Stawiarski Log-periodic power law and generalized hurst exponent analysis in estimating an asset bubble bursting time Here t1, tend, n denote the � me series beginning, end and length, respec� vely; tc interval - 80% confi dence interval based on 5% tc’s with the lowest MSE. Figures 1–3 below show three stock indices from Table 1, namely WIG20, DAX and Shanghai Composite. Best LPPL func� on is depicted in red and the 80% tc confi dence interval in green. Table 1: LPPL fi � ng results data t1tend nbest tctc interval peak SHX 1/13/2014 10/2/2015 421 347 319-371 346 DAX 8/22/2011 10/2/2015 1046 925 930-1004 923 DJI 9/15/1981 8/21/1987 1502 1477 1464-1524 1504 DJI 7/23/2002 10/5/2007 1312 1359 1260-1360 1314 DJI 12/15/1920 9/4/1929 2259 2245 2244-2336 2259 WIG20 10/3/2001 10/29/2007 1526 1446 1422-1598 1526 Nasdaq 10/22/1998 3/13/2000 345 376 366-426 379 Nikkei 10/9/1981 12/29/1989 2043 2043 2036-2129 2043 Gold 2/2/2001 8/31/2011 2666 2666 2659-2748 2669 HSX 9/30/2002 10/30/2007 1258 1240 1235-1269 1258 Silver 10/14/2008 4/28/2011 649 654 623-661 649 CL 10/6/2006 7/3/2008 441 472 421-543 441 Source: Authors’ own computati ons Figure 1: WIG20 3.10.2001-29.10.2007; best fi t: tc=1446, m=0.8001, ω=14.0023 Source: Authors’ own computati ons www.e-fi nanse.com University of Information Technology and Management in Rzeszów 54 „e-Finanse” 2016, vol. 12 / nr 3 Marcin Wątorek, Bartosz Stawiarski Log-periodic power law and generalized hurst exponent analysis in estimating an asset bubble bursting time Figure 2: DAX 22.08.2011-02.10.2015; best fi t: tc=925, m=0.2999, ω=2.4988 c Source: Authors’ own computati ons Figure 3: SHX 13.01.2014-02.10.2015; best fi t: tc=347, m=0.5498, ω=5.9967 Source: Authors’ own computati ons www.e-fi nanse.com University of Information Technology and Management in Rzeszów 55 „e-Finanse” 2016, vol. 12 / nr 3 Marcin Wątorek, Bartosz Stawiarski Log-periodic power law and generalized hurst exponent analysis in estimating an asset bubble bursting time It is clearly seen that in all historical cases fi � ng the LPPL func� on leads to good tc predic� ons. Results are comparable with those of Johansen and Sorne� e (2010). Especially worth no� ng, in April 2015 we achieved a perfect bubble peak detec� on for DAX, when the regime change occurred just 2 days prior to our es� mated tc. We also made post-hoc analysis of a recent bubble burst in the Shanghai Composite index, obtaining almost perfect predic� on. The above results convincingly show that the LPPL approach is a very reliable tool for ex-ante predic� ng the bubble peak moment tc. This technique is undergoing further dynamic development. One of the possible generaliza� ons could be higher-order expansion of LPPL func� ons, addressed also to high frequency data analysis (on an intraday basis crucial news can some� mes lead to major reversals). GHE approach In our second approach we calculated GHEw from equa� on (14), using the open source algorithm of T. Aste (Moreales et al., 2014) in the Matlab package. We set up , according to Di Ma� eo (2007). The � me windows used for successive GHEw calcula� ons contain 250 data points (approximately T = 250 working days in a year), each window is shi� ed by one trading day, t = 1. According to Pozzi et al., (2012), we used exponen� al smoothing lag equal to = 83 days (around three months). Working with log-prices, the resul� ng GHEw es� mates correspond to the log-returns. We used our second approach on the same empirical data and we obtained the following results: Table 2: GHEw results data t1 tend n min H(1) H(1) max H H(1)-H(2) peak crash SHX 1/13/2014 10/2/2015 421 389 0,4943 355 0,083 346 351 DAX 8/22/2011 10/2/2015 1046 1003 0,4766 948 0,0523 923 1017 DJ 9/6/1983 12/31/1987 1093 1012 0,3818 972 0,0735 1004 1033 DJI 7/19/2004 3/31/2009 1185 729 0,2664 894 0,081 814 1059 DJI 12/16/1924 11/25/1929 1323 1273 0,4588 1241 0,0857 1259 1305 WIG20 10/3/2003 5/30/2008 1169 1013 0,3506 1032 0,0652 1026 1080 Nasdaq 9/10/1998 3/16/2001 635 356 0,3506 354 0,0357 379 395 Nikkei 1/8/1986 4/5/1990 1106 1048 0,3829 1009 0,041 1043 1079 Gold 2/2/2001 12/30/2011 1252 976 0,343 1135 0,0916 1169 none HSX 10/6/2003 2/25/2009 1081 843 0,4091 904 0,0793 758 973 Silver 10/14/2008 11/11/2011 790 576 0,3432 652 0,0917 649 654 CL 10/6/2006 10/23/2008 519 483 0,3491 432 0,0426 441 501 Source: Authors’ own computati ons www.e-fi nanse.com University of Information Technology and Management in Rzeszów 56 „e-Finanse” 2016, vol. 12 / nr 3 Marcin Wątorek, Bartosz Stawiarski Log-periodic power law and generalized hurst exponent analysis in estimating an asset bubble bursting time Successive columns contain respec� vely: � me series star� ng date t1, its ending date tend, data length n, date of local H(1) minimum, local minimal value of H(1), � me of mul� fractality local maximum - max H, local mul� fractality maximum, H(1) − H(2), peak � me, crash � me. For the past bubbles maximal mul� fractality values were obtained together with minimal GHEw prior to the peak, which stands in accordance with the aforemen� oned decorrela� on phenomenon. The fi gure below presents the specifi c case study with: H(1) value red (local minimum marked); mul� fractality - green (local maximum marked): Figure 4: WIG20 3.10.2003-29.10.2007, GHEw analysis Source: Authors’ own computati ons GHEw low: 10.10.2007, mul� fractality peak: 7.11.2007, peak: 29.10.2007, crash: 21.01.2008. Figure 5: DAX 22.08.2011-02.10.2015, GHEw analysis Source: Authors’ own computati ons GHEw low: 04.08.2015, mul� fractality peak: 08.05.2015, peak: 10.04.2015, crash: 24.08.2015.