Log-periodic Power Law Singularity Model Research Articles

Detection of financial bubbles using a log‐periodic power law singularity (LPPLS) model

AbstractThis article provides a systematic review of the theoretical and empirical academic literature on the development and extension of the log‐periodic power law singularity (LPPLS) model, which is also known as the Johansen–Ledoit–Sornette (JLS) model or log‐periodic power law (LPPL) model. Developed at the interface of financial economics, behavioral finance and statistical physics, the LPPLS model provides a flexible and quantitative framework for detecting financial bubbles and crashes by capturing two salient empirical characteristics of price trajectories in speculative bubble regimes: the faster‐than‐exponential growth of price leading to unsustainable growth ending with a finite crash‐time and the accelerating log‐periodic oscillations. We also demonstrate the LPPLS model by detecting the recent bubble status of the S&P 500 index between April 2020 and December 2022, during which the S&P 500 index reaches its all‐time peak at the end of 2021. We find that the strong corrections of the S&P 500 index starting from January 2022 stem from the increasingly systemic instability of the stock market itself, while the well‐known external shocks, such as the decades‐high inflation, aggressive monetary policy tightening by the Federal Reserve, and the impact of the Russia/Ukraine war, may serve as sparks.This article is categorized under: Applications of Computational Statistics > Computational Finance Algorithms and Computational Methods > Computational Complexity Statistical Models > Nonlinear Models

Why topological data analysis detects financial bubbles?

We present a heuristic argument for the propensity of Topological Data Analysis (TDA) to detect early warning signals of critical transitions in financial time series. Our argument is based on the Log-Periodic Power Law Singularity (LPPLS) model, which characterizes financial bubbles as super-exponential growth (or decay) of an asset price superimposed with oscillations increasing in frequency and decreasing in amplitude when approaching a critical transition (tipping point). We show that whenever the LPPLS model is fitting with the data, TDA generates early warning signals. As an application, we illustrate this approach on a sample of positive and negative bubbles in the Bitcoin historical price.

A finite-time singularity in the dynamics of the US equity market: Will the US equity market eventually collapse?

Fitting Dow Jones 30 index data for the 1790–1999 period into a log-periodic power-law singularity (LPPLS) model, the seminal paper by Johansen and Sornette (2001) was the first to show that the US equity growth rate is accelerating such that the market is growing as a power law toward a spontaneous singularity. Their model suggests that the US equity market will reach this critical point in the year 2052 ± 10 years, signaling an abrupt transition to a new regime. This study re-examines this important issue using (i) a novel approach to calibrate the LPPLS model and (ii) a different data set including >20 years of additional data. The extended data account for the dot.com bubble burst (2000), the Global Financial Crisis period (2008–2009), the COVID−19 crisis (2020−2022), and the ongoing Russian–Ukrainian war (starting in 2022), which are all events with severe consequences for the global economy. The calibrated LPPLS model suggests that the US equity market will reach a singularity condition by June 2050.

Dissecting the 2015 Chinese stock market crash

AbstractWe perform a novel analysis of the 2015 Chinese stock market crash by calibrating the log‐periodic power law singularity (LPPLS) model to two important Chinese stock indices, SSEC and SZSC, from early 2014 to June 2015. Our analysis indicates that the LPPLS model can readily detect the bubble behaviour of the faster‐than‐exponential increase corrected by the accelerating logarithm‐periodic oscillations in the crash. The existence of the log‐periodicity is identified by applying the Lomb spectral analysis on the detrended residuals. The Ornstein–Uhlenbeck property and the stationarity of the LPPLS fitting residuals are confirmed by the Phillips–Perron test and the Dickey–Fuller test. We find that the actual critical day of bubble crash can be well predicted by the LPPLS model as far back as 2 months before the actual crash. We have shown that the covariance matrix adaptation evolution strategy (CMA‐ES) can be used as an alternative optimization algorithm for the LPPLS model fit. Furthermore, the change rate of the prediction end time gap ( ) can be used as an additional indicator along with the key indicator to improve the prediction of bubble burst.

The 2021 Bitcoin Bubbles and Crashes—Detection and Classification

In this study, the Log-Periodic Power Law Singularity (LPPLS) model is adopted for real-time identification and monitoring of Bitcoin bubbles and crashes using different time scale data, and the modified Lagrange regularization method is proposed to alleviate the impact of potential LPPLS model over-fitting to better estimate bubble start time and market regime change. The goal here is to determine the nature of the bubbles and crashes (i.e., whether they are endogenous due to their own price evolution or exogenous due to external market and/or policy influences). A systematic market event analysis is performed and correlated to the Bitcoin bubbles detected. Based on the daily LPPLS confidence indictor from 1 December 2019 to 24 June 2021, this analysis has disclosed that the Bitcoin boom from November 2020 to mid-January 2021 is an endogenous bubble, stemming from the self-reinforcement of cooperative herding and imitative behaviors of market players, while the price spike from mid-January 2021 to mid-April 2021 is likely an exogenous bubble driven by extrinsic events including a series of large-scale acquisitions and adoptions by well-known institutions such as Visa and Tesla. Finally, the utilities of multi-resolution LPPLS analysis in revealing both short-term changes and long-term states have also been demonstrated in this study.

The 2021 Bitcoin Bubbles and Crashes – Detection and Classification

In this study, we adopted the Log-Periodic Power Law Singularity (LPPLS) model for real-time identification and monitoring of Bitcoin bubbles and crashes using different time scale data and proposed the modified Lagrange regularization method to alleviate the impact of potential LPPLS model over-fitting to better estimate bubble start time and market regime change. We also aimed to determine the natures of the bubbles and crashes – be it endogenous due to its own price evolution or exogenous due to external market and/or policy influences. We performed a systematic market event analysis and correlated which to Bitcoin bubbles detected. Based on the daily LPPLS confidence indictor from December 1, 2019 to June 24, 2021, we found that the Bitcoin boom from November 2020 to mid-January 2021 is an endogenous bubble, stemming from the self-reinforcement of cooperative herding and imitative behaviors of market players, while the price spike from mid-January 2021 to mid-April 2021 is likely an exogenous bubble driven by extrinsic events including a series of large-scale acquisitions and adoptions by well-known institutions such as Visa and Tesla. We have also demonstrated the utilities of multi-resolution LPPLS analysis in revealing both short-term changes and long-term states.

Detection of Chinese stock market bubbles with LPPLS confidence indicator

We present an advance bubble detection methodology based on the Log Periodic Power Law Singularity (LPPLS) confidence indicator for the early causal identifications of positive and negative bubbles in the Chinese stock market using the daily data on the Shanghai Shenzhen CSI 300 stock market index from January 2002 through April 2018. We account for the damping condition of the LPPLS model in the search space and implement the stricter filter conditions for the qualification of the valid LPPLS fits by considering the maximum relative error, performing the Lomb log-periodic test of the detrended residual, and the unit-root tests of the logarithmic residual based on both the Phillips–Perron test and the Dickey–Fuller test to improve the performance of the LPPLS confidence indicator. Our analysis shows that with forward prediction, the LPPLS detection strategy based on the LPPLS confidence indicator can diagnose both the positive and the negative bubbles corresponding to well-known historical events, demonstrating its outstanding performance to foretell the bubbles in advance. We find that the probability density distribution of the estimated beginning time of bubbles appears to be skewed and the mass of the distribution is concentrated on the area where the price starts to have an obvious super-exponential growth. This study is the first work in literature that identifies the existence of bubbles in the Chinese stock market using the daily data of CSI 300 index with the advance bubble detection methodology of the LPPLS confidence indicator. By forecasting the prospective positive and negative bubbles and their impending crashes ahead of time, one can potentially help limit the bubble sizes and eventually minimize the damages from the bubble crashes.

Real-time prediction of Bitcoin bubble crashes

In the past decade, Bitcoin as an emerging asset class has gained widespread public attention because of their extraordinary returns in phases of extreme price growth and their unpredictable massive crashes. We apply the log-periodic power law singularity (LPPLS) confidence indicator as a diagnostic tool for identifying bubbles using the daily data on Bitcoin price in the past two years. We find that the LPPLS confidence indicator based on the daily Bitcoin price data fails to provide effective warnings for detecting the bubbles when the Bitcoin price suffers from a large fluctuation in a short time, especially for positive bubbles. In order to diagnose the existence of bubbles and accurately predict the bubble crashes in the cryptocurrency market, this study proposes an adaptive multilevel time series detection methodology based on the LPPLS model and a finer (than daily) timescale for the Bitcoin price data. We adopt two levels of time series, 1 h and 30 min, to demonstrate the adaptive multilevel time series detection methodology. The results show that the LPPLS confidence indicator based on this new method is an outstanding instrument to effectively detect the bubbles and accurately forecast the bubble crashes, even if a bubble exists in a short time. In addition, we discover that the short-term LPPLS confidence indicator being highly sensitive to the extreme fluctuations of Bitcoin price can provide some useful insights into the bubble status on a shorter time scale – on a day to week scale, while the long-term LPPLS confidence indicator has a stable performance in terms of effectively monitoring the bubble status on a longer time scale – on a week to month scale. The adaptive multilevel time series detection methodology can provide real-time detection of bubbles and advanced forecast of crashes to warn of the imminent risk in not only the cryptocurrency market but also other financial markets.

Dissection of Bitcoin's multiscale bubble history from January 2012 to February 2018.

We present a detailed bubble analysis of the Bitcoin to US Dollar price dynamics from January 2012 to February 2018. We introduce a robust automatic peak detection method that classifies price time series into periods of uninterrupted market growth (drawups) and regimes of uninterrupted market decrease (drawdowns). In combination with the Lagrange Regularization Method for detecting the beginning of a new market regime, we identify three major peaks and 10 additional smaller peaks, that have punctuated the dynamics of Bitcoin price during the analysed time period. We explain this classification of long and short bubbles by a number of quantitative metrics and graphs to understand the main socio-economic drivers behind the ascent of Bitcoin over this period. Then, a detailed analysis of the growing risks associated with the three long bubbles using the Log-Periodic Power-Law Singularity (LPPLS) model is based on the LPPLS Confidence Indicators, defined as the fraction of qualified fits of the LPPLS model over multiple time windows. Furthermore, for various fictitious ‘present’ times t2 before the crashes, we employ a clustering method to group the predicted critical times tc of the LPPLS fits over different time scales, where tc is the most probable time for the ending of the bubble. Each cluster is proposed as a plausible scenario for the subsequent Bitcoin price evolution. We present these predictions for the three long bubbles and the four short bubbles that our time scale of analysis was able to resolve. Overall, our predictive scheme provides useful information to warn of an imminent crash risk.

Are Bitcoin bubbles predictable? Combining a generalized Metcalfe's Law and the Log-Periodic Power Law Singularity model.

We develop a strong diagnostic for bubbles and crashes in Bitcoin, by analysing the coincidence (and its absence) of fundamental and technical indicators. Using a generalized Metcalfe’s Law based on network properties, a fundamental value is quantified and shown to be heavily exceeded, on at least four occasions, by bubbles that grow and burst. In these bubbles, we detect a universal super-exponential unsustainable growth. We model this universal pattern with the Log-Periodic Power Law Singularity (LPPLS) model, which parsimoniously captures diverse positive feedback phenomena, such as herding and imitation. The LPPLS model is shown to provide an ex ante warning of market instabilities, quantifying a high crash hazard and probabilistic bracket of the crash time consistent with the actual corrections; although, as always, the precise time and trigger (which straw breaks the camel’s back) is exogenous and unpredictable. Looking forward, our analysis identifies a substantial but not unprecedented overvaluation in the price of Bitcoin, suggesting many months of volatile sideways Bitcoin prices ahead (from the time of writing, March 2018).

China’s exchange rate regimes revisited and bubble detection in 2005-2015

The International Monetary Fund’s decision in 2015 to add the Renminbi to its Special Drawing Rights currency basket has in effect granted the Chinese Yuan the international reserve currency status. This study revisits China’s five exchange rate regimes in 2005-2015 and investigates the relationships among the currencies of the Chinese Yuan, the U.S. Dollar and the Euro. The Log-Periodic Power Law Singularity model is proposed as a generic parameterization to capture the super-exponential behavior in the tetrad of exchange rates of these currency pairs. The transient non-sustainable faster-than-exponential acceleration of the “CNY against USD” from August 2005 to July 2008 suggests the existence of an apparent bubble with an under-valuation. The clustered signals of LPPLS Confidence indicators for the “CNY against USD” from May 2011 to March 2012 suggest the termination of its long-term growth. The signals for the “EUR against USD” and “CNY against EUR” emerging from January 2014 to August 2015 show a tightly coupled relationship among these currency pairs. Moreover, the empirical estimates of the most possible critical time by the standard Ordinary Least Squares method and the Quantile Regression method for the time series of “CNY against USD” tends to be later than that of “CNY against EUR” in the five regimes.

The bubble and anti-bubble risk resistance analysis on the metal futures in China

This paper investigates the risk resistance ability of China’s metal futures before and after the financial crises in 2008 and 2015. The log periodic power law singularity (LPPLS) model is utilized to measure the financial bubble and anti-bubble during these periods. In addition, the risk resistance ability is quantitatively measured by the two derived parameters in the model. They control the super-exponential growth of price and describe the log-periodic acceleration of volatility, respectively. Further, we select the Shanghai Zinc (SZ) futures and the Shanghai Aluminum (SA) futures in Shanghai Futures Exchange as the objectives to empirically study their risk resistance ability during the above mentioned periods. Then, some interesting findings are concluded as follows: (1) For the China’s metal futures, the bubble risks last longer than the anti-bubble risks before and after the two financial crises. (2) The China’s metal futures have risk resistance ability in anti-bubble, but the ability is relatively weaker in bubble before and after the two financial crises. (3) The LPPLS model can efficiently predict and measure the bubble and anti-bubble of the financial products. It also gives us an access to observe the risk resistance ability of financial products by the derived parameters.

Dissection of Bitcoin's Multiscale Bubble History

We present a detailed bubble analysis of the Bitcoin to US Dollar price dynamics from January 2012 to February 2018. We introduce a robust automatic peak detection method that classifies price time series into periods of uninterrupted market growth (drawups) and regimes of uninterrupted market decrease (drawdowns). In combination with the Lagrange Regularisation Method for detecting the beginning of a new market regime, we identify 3 major peaks and 10 additional smaller peaks, that have punctuated the dynamics of Bitcoin price during the analyzed time period. We explain this classification of long and short bubbles by a number of quantitative metrics and graphs to understand the main socio-economic drivers behind the ascent of Bitcoin over this period. Then, a detailed analysis of the growing risks associated with the three long bubbles using the Log-Periodic Power Law Singularity (LPPLS) model is based on the LPPLS Confidence Indicators, defined as the fraction of qualified fits of the LPPLS model over multiple time windows. Furthermore, for various fictitious present analysis times t2, positioned in advance to bubble crashes, we employ a clustering method to group LPPLS fits over different time scales and the predicted critical times tc (the most probable time for the start of the crash ending the bubble). Each cluster is argued to provide a plausible scenario for the subsequent Bitcoin price evolution. We present these predictions for the three long bubbles and the four short bubbles that our time scale of analysis was able to resolve. Overall, our predictive scheme provides useful information to warn of an imminent crash risk.

Are Bitcoin Bubbles Predictable? Combining a Generalized Metcalfe's Law and the LPPLS Model

We develop a strong diagnostic for bubbles and crashes in bitcoin, by analyzing the coincidence (and its absence) of fundamental and technical indicators. Using a generalized Metcalfe’s law based on network properties, a fundamental value is quantified and shown to be heavily exceeded, on at least four occasions, by bubbles that grow and burst. In these bubbles, we detect a universal super-exponential unsustainable growth. We model this universal pattern with the Log-Periodic Power Law Singularity (LPPLS) model, which parsimoniously captures diverse positive feedback phenomena, such as herding and imitation. The LPPLS model is shown to provide an ex-ante warning of market instabilities, quantifying a high crash hazard and probabilistic bracket of the crash time consistent with the actual corrections; although, as always, the precise time and trigger (which straw breaks the camel’s back) being exogenous and unpredictable. Looking forward, our analysis identifies a substantial but not unprecedented overvaluation in the price of bitcoin, suggesting many months of volatile sideways bitcoin prices ahead (from the time of writing, March 2018).

Modified profile likelihood inference and interval forecast of the burst of financial bubbles

We present a detailed methodological study of the application of the modified profile likelihood method for the calibration of nonlinear financial models characterized by a large number of parameters. We apply the general approach to the Log-Periodic Power Law Singularity (LPPLS) model of financial bubbles. This model is particularly relevant because one of its parameters, the critical time signalling the burst of the bubble, is arguably the target of choice for dynamical risk management. However, previous calibrations of the LPPLS model have shown that the estimation of is in general quite unstable. Here, we provide a rigorous likelihood inference approach to determine , which takes into account the impact of the other nonlinear (so-called ‘nuisance’) parameters for the correct adjustment of the uncertainty on . This provides a rigorous interval estimation for the critical time, rather than the point estimation in previous approaches. As a bonus, the interval estimates can also be obtained for the nuisance parameters (, damping), which can be used to improve filtering of the calibration results. We show that the use of the modified profile likelihood method dramatically reduces the number of local extrema by constructing much simpler smoother log-likelihood landscapes. The remaining distinct solutions can be interpreted as genuine scenarios that unfold as the time of the analysis flows, which can be compared directly via their likelihood ratio. Finally, we develop a multi-scale profile likelihood analysis to visualize the structure of the financial data at different scales (typically from 100 to 750 days). We test the methodology successfully on synthetic price time series and on three well-known historical financial bubbles.

Birth or burst of financial bubbles: which one is easier to diagnose?

Abreu and Brunnermeier (2003) have argued that bubbles are not suppressed by arbitrageurs because they fail to synchronise on the uncertain beginning of the bubble. We propose an indirect quantitative test of this hypothesis and confront it with the alternative according to which bubbles persist due to the difficulty of agreeing on the end of bubbles. We present systematic tests of the precision and reliability with which the beginning t_1 and end t_c of a bubble can be determined. For this, we use a specific bubble model, the log-periodic power law singularity (LPPLS) model, which represents a bubble as a transient noisy super-exponential price trajectory decorated by accelerated volatility oscillations. Generalising the estimation procedure to endogenise the beginning of the fitting time interval, we quantify the uncertainty on the calibrated t_1 and t_c (as well as the other model parameters) via the eigenvalues of the Hessian matrix, which characterise the shape of the calibration cost function in the different directions in parameter space, on many synthetic data and four historical bubble cases. We find overwhelming evidence that the beginning of bubbles is much better constrained that their end. Our results are robust over all four empirical bubbles and many synthetic tests, as well as when changing the time of analysis (the present) during the development of the bubbles. As a bonus, we find that the two structural parameters of the LPPLS model, the exponent m controlling the super-exponential growth of price and the angular log-periodic frequency omega describing the log-periodic acceleration of volatility, are very rigid according the Hessian matrix analysis, which supports the LPPLS model as a reasonable candidate for describing the generating process of prices during bubbles.

Modified Profile Likelihood Inference and Interval Forecast of the Burst of Financial Bubbles

We present a detailed methodological study of the application of the modified profile likelihood method for the calibration of nonlinear financial models characterised by a large number of parameters. We apply the general approach to the Log-Periodic Power Law Singularity (LPPLS) model of financial bubbles. This model is particularly relevant because one of its parameters, the critical time tc signalling the burst of the bubble, is arguably the target of choice for dynamical risk management. However, previous calibrations of the LPPLS model have shown that the estimation of tc is in general quite unstable. Here, we provide a rigorous likelihood inference approach to determine tc, which takes into account the impact of the other nonlinear (so-called nuisance) parameters for the correct adjustment of the uncertainty on tc. This provides a rigorous interval estimation for the critical time, rather than a point estimation in previous approaches. As a bonus, the interval estimations can also be obtained for the nuisance parameters (m,w, damping), which can be used to improve filtering of the calibration results. We show that the use of the modified profile likelihood method dramatically reduces the number of local extrema by constructing much simpler smoother log-likelihood landscapes. The remaining distinct solutions can be interpreted as genuine scenarios that unfold as the time of the analysis flows, which can be compared directly via their likelihood ratio. Finally, we develop a multi-scale profile likelihood analysis to visualize the structure of the financial data at different scales (typically from 100 to 750 days). We test the methodology successfully on synthetic price time series and on three well-known historical financial bubbles.

Using trading strategies to detect phase transitions in financial markets

We show that the log-periodic power law singularity model (LPPLS), a mathematical embodiment of positive feedbacks between agents and of their hierarchical dynamical organization, has a significant predictive power in financial markets. We find that LPPLS-based strategies significantly outperform the randomized ones and that they are robust with respect to a large selection of assets and time periods. The dynamics of prices thus markedly deviate from randomness in certain pockets of predictability that can be associated with bubble market regimes. Our hybrid approach, marrying finance with the trading strategies, and critical phenomena with LPPLS, demonstrates that targeting information related to phase transitions enables the forecast of financial bubbles and crashes punctuating the dynamics of prices.