Johansen-Ledoit-Sornette Research Articles

Role of Diversification Risk in Financial Bubbles

We present an extension of the Johansen-Ledoit-Sornette (JLS) model to include an additional pricing factor called the factor'', which describes the diversification risk of the stock market portfolio. Keeping all the dynamical characteristics of a bubble described in the JLS model, the new model provides an additional information about the concentration of stock gains over time. This allows us to understand better the risk diversification and to explain the investors' behavior during the bubble generation. We apply this new model to two famous Chinese stock bubbles, from August 2006 to October 2007 (bubble 1) and from October 2008 to August 2009 (bubble 2). The Zipf factor is found highly significant for bubble 1, corresponding to the fact that valuation gains were more concentrated on the large firms of the Shanghai index. It is likely that the widespread acknowledgement of the 80-20 rule in the chinese media and discussion fora led many investors to discount the risk of a lack of diversification, therefore enhancing the role of the Zipf factor. For bubble 2, the Zipf factor is found marginally relevant, suggesting a larger weight of market gains on small firms. We interpret this result as the consequence of the response of the chinese economy to the very large stimulus provided by the Chinese government in the aftermath of the 2008 financial crisis.

The Role of Diversification Risk in Financial Bubbles

We present an extension of the Johansen-Ledoit-Sornette (JLS) model to include an additional pricing factor called the "Zipf factor", which describes the diversification risk of the stock market portfolio. Keeping all the dynamical characteristics of a bubble described in the JLS model, the new model provides additional information about the concentration of stock gains over time. This allows us to understand better the risk diversification and to explain the investors' behavior during the bubble generation. We apply this new model to two famous Chinese stock bubbles, from August 2006 to October 2007 (bubble 1) and from October 2008 to August 2009 (bubble 2). The Zipf factor is found highly significant for bubble 1, corresponding to the fact that valuation gains were more concentrated on the large firms of the Shanghai index. It is likely that the widespread acknowledgement of the 80-20 rule in the Chinese media and discussion forums led many investors to discount the risk of a lack of diversification, therefore enhancing the role of the Zipf factor. For bubble 2, the Zipf factor is found marginally relevant, suggesting a larger weight of market gains on small firms. We interpret this result as the consequence of the response of the Chinese economy to the very large stimulus provided by the Chinese government in the aftermath of the 2008 financial crisis.

Clarifications to Questions and Criticisms on the Johansen-Ledoit-Sornette Bubble Model

The Johansen-Ledoit-Sornette (JLS) model of rational expectation bubbles with finite-time singular crash hazard rates has been developed to describe the dynamics of financial bubbles and crashes. It has been applied successfully to a large variety of financial bubbles in many different markets. Having been developed for more than one decade, the JLS model has been studied, analyzed, used and criticized by several researchers. Much of this discussion is helpful for advancing the research. However, several serious misconceptions seem to be present within this collective conversation both on theoretical and empirical aspects. Several of these problems appear to stem from the fast evolution of the literature on the JLS model and related works. In the hope of removing possible misunderstanding and of catalyzing useful future developments, we summarize these common questions and criticisms concerning the JLS model and offer a synthesis of the existing state-of-the-art and best-practice advices.

Inferring Fundamental Value and Crash Nonlinearity from Bubble Calibration

Identifying unambiguously the presence of a bubble in an asset price remains an unsolved problem in standard econometric and financial economic approaches. A large part of the problem is that the fundamental value of an asset is, in general, not directly observable and it is poorly constrained to calculate. Further, it is not possible to distinguish between an exponentially growing fundamental price and an exponentially growing bubble price. In this paper, we present a series of new models based on the Johansen-Ledoit-Sornette (JLS) model, which is a flexible tool to detect bubbles and predict changes of regime in financial markets. Our new models identify the fundamental value of an asset price and a crash nonlinearity from a bubble calibration. In addition to forecasting the time of the end of a bubble, the new models can also estimate the fundamental value and the crash nonlinearity, meaning that identifying the presence of a bubble is enabled by these models. Besides, the crash nonlinearity obtained in the new models presents a new approach to possibly identify the dynamics of a crash after a bubble. We test the models using data from three historical bubbles ending in crashes from different markets. They are: the Hong Kong Hang Seng index 1997 crash, the S&P 500 index 1987 crash (black Monday) and the Shanghai Composite index 2009 crash. All results suggest that the new models perform very well in describing bubbles, forecasting their ending times and estimating fundamental value and the crash nonlinearity. The performance of the new models is tested under both the Gaussian residual assumption and non-Gaussian residual assumption. Under the Gaussian residual assumption, nested hypotheses with the Wilks statistics are used and the p-values suggest that models with more parameters are necessary. Under non-Gaussian residual assumption, we use a bootstrap method to get type I and II errors of the hypotheses. All tests confirm that the generalized JLS models provide useful improvements over the standard JLS model.

Diagnosis and Prediction of Market Rebounds in Financial Markets

We introduce the concept of “negative bubbles” as the mirror image of standard financial bubbles, in which positive feedback mechanisms may lead to transient accelerating price falls. To model these negative bubbles, we adapt the Johansen-Ledoit-Sornette (JLS) model of rational expectation bubbles with a hazard rate describing the collective buying pressure of noise traders. The price fall occurring during a transient negative bubble can be interpreted as an effective random downpayment that rational agents accept to pay in the hope of profiting from the expected occurrence of a possible rally. We validate the model by showing that it has significant predictive power in identifying the times of major market rebounds. This result is obtained by using a general pattern recognition method which combines the information obtained at multiple times from a dynamical calibration of the JLS model. Error diagrams, Bayesian inference and trading strategies suggest that one can extract genuine information and obtain real skill from the calibration of negative bubbles with the JLS model. We conclude that negative bubbles are in general predictably associated with large rebounds or rallies, which are the mirror images of the crashes terminating standard bubbles.

Detecting log-periodicity in a regime-switching model of stock returns

Log-periodic precursors have been identified before most and perhaps all financial crashes of the Twentieth Century, but efforts to statistically validate the leading model of log-periodicity, the Johansen–Ledoit–Sornette (JLS) model, have generally failed. The main feature of this model is that log-harmonic fluctuations in financial prices are driven by similar fluctuations in expected daily returns. Here we search more broadly for evidence of any log-periodic variation in expected daily returns by estimating a regime-switching model of stock returns in which the mean return fluctuates between a high and a low value. We find such evidence prior to the two largest drawdowns in the S&P 500 since 1950. However, if we estimate a log-harmonic specification for the stock index for the same time periods, fixing the frequency and critical time according to the results of the regime-switching model, the parameters do not satisfy restrictions imposed by the JLS model.

A Bayesian analysis of log-periodic precursors to financial crashes

A large number of papers have been written by physicists documenting an alleged signature of imminent financial crashes involving so-called log-periodic oscillations–oscillations which are periodic with respect to the logarithm of the time to the crash. In addition to the obvious practical implications of such a signature, log-periodicity has been taken as evidence that financial markets can be modelled as complex statistical-mechanics systems. However, while many log-periodic precursors have been identified, the statistical significance of these precursors and their predictive power remain controversial in part because log-periodicity is ill-suited for study with classical methods. This paper is the first effort to apply Bayesian methods in the testing of log-periodicity. Specifically, we focus on the Johansen–Ledoit–Sornette (JLS) model of log periodicity. Using data from the S&P 500 prior to the October 1987 stock market crash, we find that, if we do not consider crash probabilities, a null hypothesis model without log-periodicity outperforms the JLS model in terms of marginal likelihood. If we do account for crash probabilities, which has not been done in the previous literature, the JLS model outperforms the null hypothesis, but only if we ignore the information obtained by standard classical methods. If the JLS model is true, then parameter estimates obtained by curve fitting have small posterior probability. Furthermore, the data set contains negligible information about the oscillation parameters, such as the frequency parameter that has received the most attention in the previous literature.