A Consistent Model of 'Explosive' Financial Bubbles with Mean-Reversing Residuals

Abstract

We present a self-consistent model for explosive financial bubbles, which combines a mean-reverting volatility process and a stochastic conditional return which reflects nonlinear positive feedbacks and continuous updates of the investors' beliefs and sentiments. The conditional expected returns exhibit faster-than-exponential acceleration decorated by accelerating oscillations, called "log-periodic power law." Tests on residuals show a remarkable low rate (0.2%) of false positives when applied to a GARCH benchmark. When tested on the S&P500 US index from Jan. 3, 1950 to Nov. 21, 2008, the model correctly identifies the bubbles ending in Oct. 1987, in Oct. 1997, in Aug. 1998 and the ITC bubble ending on the first quarter of 2000. Different unit-root tests confirm the high relevance of the model specification. Our model also provides a diagnostic for the duration of bubbles: applied to the period before Oct. 1987 crash, there is clear evidence that the bubble started at least 4 years earlier. We confirm the validity and universality of the volatility-confined LPPL model on seven other major bubbles that have occurred in the World in the last two decades. Using Bayesian inference, we find a very strong statistical preference for our model compared with a standard benchmark, in contradiction with Chang and Feigenbaum [2006] which used a unit-root model for residuals.

Prediction accuracy and sloppiness of log-periodic functions

“…The associated eigenvectors confirm the wisdom that the stiffer a direction, the more likely that it is close to an axis, and yWhile the beginning of a bubble is supposed to be given by the lowest value since the previous crash, it may be moved from this minimum to some later date; in Johansen and Sornette (2001) the beginnings of half of the eight bubbles on the Hang Seng up to 1998 were thus moved (Bre´e and Joseph 2010). zLomb periodograms ( van Bothmer 2003) or the residuals (Gazola et al 2008, Lin et al 2009) have been proposed, with mixed results. reversely for sloppy eigenvalues (Gutenkunst et al 2007).…”

“…Of the recent progress, the residuals have been shown to be AR(1) (Gazola et al 2008, Lin et al 2009). It makes sense, therefore, to create artificial data with AR(1) noise.…”

“…The question is still unsettled as of yet (Laloux et al 1999, Feigenbaum 2001, van Bothmer 2003, Chang and Feigenbaum 2006, Gazola et al 2008. Problems are indeed numerous: what definition of a bubble and a crash to adopt (Jacobsen 2009, Lin et al 2009),? should the price in a bubble always be increasing (Bothmer and Meister 2003), should one impose constraints on the fitted parameters , where to start a fit of a bubble,y what test of goodness of fit to use,z why having different lengths of the data window greatly affects the parameters of the best fit of the LPPL to the data (Feigenbaum 2001), why leaving out a few data points can alter the parameters of the best fit sufficiently to change a no/bubble decision (Lin et al 2009), and why is the fitting error very sensitive to small (but not large!)…”

Dragon-Kings, Black Swans and the Prediction of Crises

“…Johansen and Sornette [31] have found that two-thirds of the dragon-kings identified in the distribution of drawdowns are actually stock market crashes, which were preceded by large bubbles. This has led them to build a theory [33][34][35] in which crashes are seen as the possible end of a bubble regime, associated with various positive feedback mechanisms that lead to faster-than-exponential unsustainable growth regime. The mechanisms responsible for positive feedbacks include portfolio insurance trading, option hedging, momentum investment and imitation-based herding.…”

“…At the time of writing, the last one on the Shanghai market is still running. It is important to stress that our methodology allows us to predict the end of bubbles, but not the crashes per se [33][34][35]. It is often the case that a bubble bursts into a crash but this is not always the case.…”