Discrete Dynamics in Nature and Society

Volume 2017, Article ID 8017510, 15 pages

https://doi.org/10.1155/2017/8017510

## New JLS-Factor Model versus the Standard JLS Model: A Case Study on Chinese Stock Bubbles

College of Finance and Statistics, Hunan University, Hunan, China

Correspondence should be addressed to Chao Li; moc.qq@207814727

Received 10 September 2016; Revised 30 November 2016; Accepted 14 December 2016; Published 18 January 2017

Academic Editor: Vincenzo Scalzo

Copyright © 2017 Zongyi Hu and Chao Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

In this paper, we extend the Johansen-Ledoit-Sornette (JLS) model by introducing fundamental economic factors in China (including the interest rate and deposit reserve rate) and the historical volatilities of targeted and US equity indices into the original model, which is a flexible tool to detect bubbles and predict regime changes in financial markets. We then derive a general method to incorporate these selected factors in addition to the log-periodic power law signature of herding and compare the prediction accuracy of the critical time between the original and the new JLS models (termed the JLS-factor model) by applying these two models to fit two well-known Chinese stock indices in three bubble periods. The results show that the JLS-factor model with Chinese characteristics successfully depicts the evolutions of bubbles and “antibubbles” and constructs efficient end-of-bubble signals for all bubbles in Chinese stock markets. In addition, the results of standard statistical tests demonstrate the excellent explanatory power of these additive factors and confirm that the new JLS model provides useful improvements over the standard JLS model.

#### 1. Introduction

As our understanding of the topic deepens, people are gradually realizing that the stock market is a complex system that has many participants with different characteristics that influence each other. In addition, no stock market is completely independent of another; each connects with others to form a larger system to some extent. For this reason, the stock market shows a nonlinear mechanism in its operation process. Thus, using an equilibrium model to study the stock market is not a suitable approach.

In consideration of the limitations of classical financial theory, more and more researchers are using nonlinear dynamic systems to research financial markets in the new discipline termed “econophysics.” For the bubble problem in stock markets, financial physicists combine rational expectations theory in economics with the self-organizing critical phenomenon in statistical physics to diagnose the plausible times at which the bubble burst. Among them, the most representative scholars are Sornette and Johansen and coworkers, who stated that bubbles are not characterized by an exponential increase in prices but rather by a faster-than-exponential growth in prices. They also argued that most financial crashes are the climax of the so-called log-periodic power law signatures (LPPLS) associated with speculative bubbles [1].

The predictive power of LPPLS was first discovered in acoustic emissions prior to rupture and in identifying the precursors of earthquakes [2, 3]. However, the introduction of LPPLS to predict the bursting of speculative bubbles in financial markets goes back to the pioneering works of Feigenbaum and Freund [4] and Sornette et al. [5], who independently of each other disclosed LPPLS structures prior to the crash of the S&P 500 in October 1987. Beyond that, by means of extending the renormalization group approach to the second order of perturbation, Sornette and Johansen [6] proposed the second-order LPPL “Landau” model to capture the evolution of stock prices over a longer precrash period, say seven to eight years, compared with the original LPPL model. Then, they used it to fit the Dow Jones index prior to the 1929 crash and the S&P 500 index prior to the 1987 crash.

Since then, a large body of empirical evidence supporting this proposition has been presented ([7–15] (hereafter the LPPL model is also referred to as the JLS (i.e., Johansen-Ledoit-Sornette) model)). All this research has shown that the time evolutions of the stock market index prior to the 1929, 1987, and 1998 crashes on Wall Street and the 1997 crash in Hong Kong are in very good agreement with the predictions of the LPPL model.

As for the cause of the stock market crash, Johansen and Sornette [13] argued that a large crash is mainly caused by local self-reinforcing imitation between investors and their herding behavior. Under the action of the positive price-to-price feedback mechanism, this self-reinforcing imitation process eventually leads to a bubble. Specifically, if the tendency for traders to “imitate” their “friends” increases up to the “critical” point (or critical time), many traders may place the same order (sell) at the same time, thus causing a crash [13]. However, it is also worth noting that imitation between investors and their herding behavior lead not only to speculative bubbles with accelerating overvaluations of financial markets possibly followed by crashes but also to “antibubbles” with decelerating market devaluations following all-time highs. For this, Johansen and Sornette [16] proposed a third-order Landau expansion in which demand decreases slowly with barriers that progressively reduce, leading to a power law decay of the market price accompanied by decelerating log-periodic oscillations. In addition, they documented this behavior on the Japanese Nikkei stock index from 1990 and on the gold future prices after 1980, both after their all-time highs [16]. Besides, a remarkable similarity in the behavior of the US S&P 500 index from 1996 to August 2002 and of the Japanese Nikkei index from 1985 to 1992 (11-year shift) was presented by Sornette and Zhou [17]. Several other examples have been described in the Russian stock market (Sornette et al., 1999) and in emerging and western markets [17–19]. Among them, authors have analyzed 39 world stock market indices from 2000 to the end of 2002, finding that 22 are in an antibubble regime (owing to the criterion of obtaining at least a solution in the fitting procedure, they have found no evidence of an antibubble in China during that period). However, although this third-order LPPL Landau model is suitable for describing the evolution of bubbles and antibubbles almost twelve years, some deviations may appear when using the second-order LPPL Landau model to calibrate antibubbles, implying that much higher-order Landau models are useless.

In addition to the third-order LPPL Landau model, Sornette and coworkers expanded the original LPPL model, including (a) constructing the Weierstrass-type LPPL model [20, 21], (b) extending the JLS model with second-order harmonics [17], (c) proposing the JLS-factor model in which the LPPL bubble component is augmented by fundamental economic factors [22], (d) inferring the fundamental value of the stock and crash nonlinearity from bubble calibration [23] and detecting market rebounds [24], (e) extending the JLS model to include an additional pricing factor called the “Zipf factor,” which describes the diversification risk of stock market portfolio [25], (f) reducing the JLS model to a function of only three nonlinear parameters [26], (g) presenting a volatility-confined LPPL model to describe and diagnose situations when excessive public expectations of future price increases cause prices to be temporarily elevated [27], (h) introducing quantile regression to the LPPLS detection problem and defining the so-called DS LPPLS Confidence and Trust indicators that enrich considerably the diagnosis of bubbles [28] (recently, Zhang et al. [29] added two new indicators (DS LPPLS Bubble Status and End-of-Bubble) into this system), (i) employing a rigorous likelihood approach and providing interval estimates of the parameters, including the most important critical times of market regime changes [30], and (j) presenting a plausible microfounded model for the previously postulated power law finite time singular form of the crash hazard rate in the JLS model of rational expectation bubbles [31]. In addition, the research team at ETH Zurich has further developed an LPPLS-based bubble detection system, as described in Sornette et al. [32] and Zhang et al. [28].

However, although the JLS model has been extended to many different forms, the original model remains a powerful and flexible tool with which to detect financial bubbles and crashes in various markets, especially Chinese stock bubbles. Examples include the antibubble in China’s stock market that started in August 2001 [33], two bubbles and subsequent market crashes in two important indices of Chinese stock markets (Shanghai Stock Exchange Composite (SSEC) index and Shenzhen Stock Exchange Component (SZSC) index) between May 2005 and July 2009 [34], two well-known Chinese stock bubbles, from August 2006 to October 2007 (bubble 1) and from October 2008 to August 2009 (bubble 2) [25], and the bubble regime that developed in Chinese stock markets in mid-2014 and started to burst in June 2015 [32].

Unfortunately, the JLS model is based on the assumption that crashes are the outcome of the interactions of market players resulting in herding behavior (i.e., an endogenous origin), while exogenous shocks (due to changes in market fundamentals) rarely play an important role, only serving as trigger factors, which is at odds with standard economy theory [35]. Although Zhou and Sornette [22] presented a general methodology under which to incorporate fundamental economic factors (e.g., interest rate, interest spread, historical volatility, implied volatility, and exchange rates) into the theory of herding to describe bubbles and antibubbles, the most surprising result is that the best model is the second-order LPPL model without any factors. Indeed, the evolution of a complex system is the result of an entangled combination of endogenous organization as well as a response to external news and exogenous shocks, especially for an equity market such as the Chinese stock market that is still immature and heavily influenced by exogenous shocks. Compared with western markets, the Chinese stock market has its own characteristics, including the following: (i) being dominated by individual investors, (ii) being highly volatile, (iii) nontradability of more than two-thirds of the shares, (iv) short sale constraints, (v) response to exogenous information such as government policies, a firm’s accounting information, and stock exchange announcements, (vi) stronger imprints of herding, (vii) overspeculation, (viii) overvaluation of markets, (ix) widely taken short-term positions, (x) insider trading, and (xi) a distempered regulation system [33, 36]. These specific features make the market exhibit strong idiosyncrasies and puzzles in addition to the more common behavior of mature stock markets [37]. Therefore, in addition to the herding behavior between traders, we think that taking into account the influence of government policies and other exogenous shocks on the evolution of Chinese stock bubble may be more advisable, especially the impact from changes in monetary policy and fluctuations in international stock markets.

Given the foregoing, we generalize the standard JLS model by incorporating two fundamental economic factors in China (i.e., the interest rate and deposit reserve rate) and the historical volatility of targeted and US equity indices into the original model. We then present an ex-post analysis of what Jiang et al. [34] and Sornette et al. [32] earlier identified as being the three significant bubbles developing in major Chinese stock markets: the first bubble ran from July 2005 to October 2008, the second one ran from November 2008 to August 2009, and the third one ran from March 2014 to July 2015. We also compare the prediction accuracy of the critical time between the original and the new JLS model (i.e., the JLS-factor model). The empirical results show that the JLS-factor model with Chinese characteristics successfully diagnoses all the well-documented bubbles in Chinese stock markets. Further, by comparing the prediction accuracy of the original and new JLS models, we find that the critical time estimated by the new JLS model for Chinese stock market bubbles during 2005–2015 is closer to the actual time than the original JLS model in general, which demonstrates the excellent explanatory power of our proposed JLS-factor model. In addition, the results of different standard statistical tests show that the new JLS model is superior to the original JLS model. Moreover, the results of significance testing provide indirect evidence on the key role of fundamental economic and exogenous factors in China affecting the evolution of Chinese stock bubbles (except the 2008–2009 bubbles, which were probably punctuated by a vanishingly small change in some endogenous factors).

The remainder of this paper is structured as follows. In Section 2, we summarize the mathematical formulation of the original JLS model, while the extended JLS model is introduced in Section 3. Section 4 presents the construction of the JLS-factor model with Chinese characteristics. Section 5 describes the tests of six documented Chinese stock bubbles by using the new JLS model and the original model and compares their respective predictive power. Section 6 shows the significance test results of the original and new JLS models. Section 7 summarizes our conclusions.

#### 2. Mathematical Formulation of the JLS Model

In this section, we recall the formation of the original JLS model, which provides a flexible framework within which to detect bubbles and predict regime changes in the price time series of a financial asset. It combines (i) the economic theory of rational expectation bubbles, (ii) behavioral finance on imitation and herding of noise traders, and (iii) the mathematical and statistical physics of bifurcations and phase transitions. The model considers the faster-than-exponential (power law with finite time singularity) increase in asset prices accompanied by accelerating oscillations as the main diagnostic of bubbles. It thus embodies a positive feedback loop of higher return anticipations competing with the negative feedback spirals of crash expectations [23].

Within the JLS framework, expected price conditioned on no crash occurring is obtained as follows (see Zhou and Sornette [22], for the concrete derivation of the model):where , is the interest rate, is the price volatility, is the market price of risk of the stochastic discount factor, and is the crash hazard rate, namely, the probability per unit time that the crash will happen in the next instant if it has not yet happened.

For and , we have

Johansen et al. [14] proposed that a crash may be caused by local self-reinforcing imitation processes between noise traders that can be quantified by the theory of critical phenomena developed in the physical sciences. Hence, they assumed that the aggregate effect of noise traders can be quantified by the following dynamics of the crash hazard rate:where , is the critical time (i.e., the most probable time for the bursting of the bubble), is the angular log-frequency, and is an initial phase determining the unit of the time. Generalizing the definition of into allows for the critical time to lie anywhere within the time series, which has the advantage of introducing a degree of flexibility into the search space for with little additional cost [17].

The power law behavior embodies the mechanisms of positive feedback at the origin of the formation of a bubble, while the cosine term on the RHS of (3) takes into account the existence of a possible hierarchical cascade of panic acceleration punctuating the course of the bubble, resulting from either a preexisting hierarchy in noise trader sizes and/or the interplay between market price impact inertia and nonlinear fundamental value investing [23].

Substituting (3) into (2) and integrating yields the LPPL equation for the price:where , which gives the terminal log-price at the critical time . and , respectively, control for the amplitude of the power law acceleration and the log-periodic oscillations. The exponent quantifies the degree of superexponential growth. is the angular log-frequency. is another phase different to that contains two ingredients: information on the mechanism of the interactions between investors and a rescaling of time. The power law with exponent captures the faster-than-exponential growth in the price and the term describes the accelerating oscillation decorating the accelerating price. Further, although additional constraints emerge from a compilation of a significant number of historical bubbles that can be summarized as , [26], Zhang et al. [28] found larger search ranges and from their research on sixteen historical bubbles.

A more general JLS model can be expressed as

Theoretically, the order parameter can be the price or the logarithm of price , while which one is reasonable to be the dependent variable is dependent on the following criterion. Zhou and Sornette [33] proposed that the observed price is the sum of a fundamental price and of a bubble or an antibubble . They had when and when . In fact, based on the rational bubble model of Johansen et al. [14] and Johansen et al. [38], if the magnitude of the crash is proportional to the price increase only associated with the contribution of the bubble, then the correct proxy is the price itself; on the contrary, if the magnitude of the crash is proportional to the price, then the correct proxy is the logarithm of the price [39]. From a theoretical view point, this is unsurprising: the rational expectation model of bubbles and crashes shows that, depending on whether the size of the crash is proportional to the price itself or that of the increase due to the bubble, either the logarithm of the price or the price itself is the correct quantity characterizing the bubble [12].

For the sake of simplicity, let us rewrite (5) in the following form:where

#### 3. Extended JLS Model

The common JLS model is the specific form when , as mentioned above. However, and do not equal zero in the real world. Thus, we extend the original JLS model locally. Similar to Zhou and Sornette [22], we assume that is a constant , which does not change over time. Moreover, we take the true values of and to calibrate the model. Specifically, we specify as the risk-free interest rate and employ the historical volatility of the targeted asset as a proxy for the volatility factor . All daily data come from the iFinD database.

Through the above extension, the original JLS model can be converted into the following form:where is the day logarithm yield of the targeted asset, is the average yield, represents the risk-free interest rate, and denotes the historical volatility of the targeted asset.

However, the above model is only a local extension in view of the general JLS model. For Chinese stock markets, the influence of macroeconomic factors, national policy, and the international economic situation on the stock market must be accounted for, except for the impact of the positive feedback effect caused by investors’ herding behavior. Therefore, we study the important factors that affect the volatility of Chinese stock indices in the next section and add these to the extended JLS model to construct a new JLS model (i.e., the JLS-factor model), which is suitable for China.

#### 4. JLS-Factor Model with Chinese Characteristics

Owing to their inherent characteristics and drawbacks, Chinese stock markets are more easily affected by changes in monetary policy and fluctuations in international stock markets than mature markets. Hence, in this section, we analyze the impact of these two factors on Chinese stock market volatility and then construct a new JLS-factor model to calibrate the well-known Chinese stock bubbles.

First, the effect of monetary policy, mainly implemented by adjusting the interest rates, deposit reserve rate, and money supply, and so on, on a country’s capital market has long been examined in agroscientific research globally. Theoretically, in the context of the transmission mechanism, monetary policy affects stock prices mainly through both the traditional interest rate channel [40] and the credit channel [41]. A number of empirical studies have applied different proxy variables to assess the effects of monetary policy shocks on stock market volatility, including the discount rate [42–45] (Mercer and Johnson, 1996), Federal Funds rate [46–54], interest rate [55–62] (Octavioet al., 2013), money supply [63–66], and Federal Funds futures [67–75].

Compared with foreign scholars’ research, abundant works in China have assessed the relationship between the deposit reserve rate and stock market (e.g., [76–87]). All this research argues that, after changing the supply of money, the variation in the deposit reserve rate causes stock prices to change in the following four ways: (i) the effect of the market interest rate, (ii) the effect of credit scale, (iii) the effect of market structure, and (iv) the effect of stock market announcements. On the one hand, the variation in the deposit reserve rate tends to directly affect the money supply of the whole society and thus changes the capital supply to the stock market and, ultimately, the evolution of stock prices. On the other hand, as a policy signal, the adjustment in the deposit reserve rate significant affects investors’ psychological expectations and thus their investment strategies and, ultimately, the evolution of stock prices. Unfortunately, early research into the relationship between swings in the deposit reserve rate and fluctuation in stock prices provided mixed results, finding no consistent relationship between these two variables and that the nature of such dynamics was unstable. From the above, a change in the deposit reserve rate is thus a factor that can affect asset prices. Therefore, we take into account its effect on the evolution of Chinese stock bubbles.

Second, as China has gradually opened up its stock market to foreign investments and cross-border listings, the comovement between the Chinese and the international stock market is increasingly strengthening. Indeed, in extreme cases such as the global financial crisis, such comovement is significantly enhanced given the deterioration of global economic fundamentals and the risk contagion among international financial markets [88, 89]. However, among all international stock markets, domestic research shows that stock price fluctuations in the United States have a more remarkable impact on that in China than others (e.g., [90–96], Hu, 2010). In particular, Zhang et al. [97] and Pan and Liu [98] found that the volatility of US stock indices can be used to predict the trend of Chinese stock prices.

Given the foregoing, we take into account the impact of the deposit reserve rate and US stock index volatility on the evolution of Chinese stock bubbles when constructing the new JLS model. Therefore, we havewhere is the risk-free interest rate, represents the deposit reserve rate, and and are specified as the volatility of the targeted index and NASDAQ, respectively (the NASDAQ Composite Index is a barometer of market value changes in each industrial category, as it includes more than 5000 companies, which is more than any other single securities market. As a result, the NASDAQ Composite Index is more representative than the S&P 500 index and Dow Jones Industrial Average).

Because the specific function forms of , , , and cannot be determined, we use the trapezoid scheme to integrate , , , and in practice, following Zhou and Sornette [22]. That is, we let

#### 5. Results of the Original and New JLS Models

To visually compare the prediction accuracy of the results of the original and new JLS models, in this section, we calibrate the evolutions of two well-known Chinese stock indices (SSEC and SZSC) in three time periods, as selected by two published papers (i.e., [32, 34]) that applied the original JLS model to fit the tendency of these two indices in the corresponding periods. By observing the minimum and maximum values among the targeted indices within the specified periods, we find that the average annual growth rate is 244.55%. This finding implies that the fundamental price should be much less than the bubble according to Zhou and Sornette’s [33] assumption. In the next step, we employ our JLS-factor model presented in (10) to fit these two indices within the same three periods with .

Each calibration uses the algorithm of Universal Global Optimization provided by 1stOpt (First Optimization) software, which has been independently developed by 7D-Soft High Technology Inc. to solve any constrained or unconstrained linear and nonlinear equation(s). This software allows us to estimate all parameters (, , , , , , , , , , and ) in a given time window of analysis without inputting the initial values. The main results of our calibrations for the evolutions of the 2005–2008, 2008-2009, and 2014-2015 bubbles are illustrated in Figure 1.