Speculative bubbles: what they are and how the LPPLS model detects them

Advanced12 min read · updated 29 September 2026 · 23 sources

A speculative bubble is a price rise that becomes detached from earnings and prospects and feeds on the expectation of further price gains. Whether bubbles can be identified before they end is one of the great debates in financial research. Here you will learn how economists define bubbles, why joining in can even be rational, how Didier Sornette's LPPLS model searches for bubbles in the price path, and what critics object.

What is a speculative bubble?

The history of financial markets is full of manias: rising prices attract new buyers, credit amplifies the rise, and the end is often a crash. Charles Kindleberger collected such episodes; his classic is now continued by Robert Aliber Aliber and Kindleberger (2015). Robert Shiller, author of *Irrational Exuberance* Shiller (2000), describes a bubble as a social epidemic: news of rising prices sparks enthusiasm that spreads from person to person and draws in more and more investors, partly out of envy of other people's gains and partly out of a gambler's excitement Shiller (2014). In his view, a sudden, final burst is not part of the definition: the booms before 1929, 2000 and 2007 were soon reflated.

Eugene Fama considers the word treacherous. It usually means an irrational strong price increase that is followed by a predictable strong decline. But there is no reliable evidence of predictable price declines, he argues Fama (2014). The big crashes in the US stock market coincided with recessions, and they were often quickly followed by new rallies. Fama therefore asks: which part of a bubble was irrational, the rise or the fall? More on his view in the article on the efficient market hypothesis.

Greenwood, Shleifer and You (2019) tested Fama's claim, using US industry returns from 1926 to 2014 and international sectors from 1985 to 2014. Fama is right in that a sharp price run-up in an industry does not, on average, signal unusually low subsequent returns. It does, however, markedly raise the probability of a crash, and features of the run-up (volatility, turnover, new issuance and the price path itself) help predict a later crash.

The physicist Didier Sornette (ETH Zurich) starts from the price path. With Peter Cauwels, he defines a bubble as a period of unsustainable growth in which the price rises faster and faster, in accelerating bursts of corrections and rebounds; at the end comes a regime change, a crash or a strong correction Sornette and Cauwels (2015).

Rational bubbles: why joining in can make sense

Bubbles do not require everyone to act irrationally. Blanchard and Watson (1982) show that bubbles are consistent with rational behavior in many markets, including runaway prices and crashes. The idea: the price consists of the fundamental value and a bubble component. Whoever buys the bubble knows it can burst. They stay in as long as it grows fast enough while it lasts. According to their findings, such bubbles can hardly be proven statistically; the tests provided circumstantial evidence at best.

The JLS model: herding and crash risk

The model of Johansen, Ledoit and Sornette (2000), called the JLS model after the authors' initials, links rational bubbles with herd behavior. It is a rational expectations model with one key assumption: a crash can arise from imitation among noise traders, that is, investors who do not trade on fundamentals. If their tendency to imitate their neighbors increases up to a critical point, they can all sell at the same time. How dangerous the situation is, is captured by the hazard rate: the probability per unit of time that the crash happens in the next instant, given that it has not happened yet.

Because the crash is not certain, it remains rational to stay invested, provided investors are compensated for the risk by faster growth of the bubble. The end of the bubble, the critical time tc, is therefore not the date of the crash: the crash can come at any time before, is most likely around tc, and the bubble can also end without a crash. The authors stress:

Simplified, without interest or a risk premium, this means: whatever the crash costs on average, the price has to make up in advance.

Superexponential growth and log-periodic oscillations

According to Sornette and Cauwels (2015), the typical shape arises from feedback loops, a hierarchical market structure and certain trading styles. For the log price, this yields the formula of the LPPLS model (log-periodic power law singularity):

The power-law term B × (tc − t)^m with 0 < m < 1 makes the growth rate rise without limit toward tc, even though the price itself stays finite. Hence the S for singularity. This is superexponential growth: with exponential growth the return is constant; here it rises every day.

The second component is log-periodic oscillations: the cosine is periodic not in time but in the logarithm of the remaining time tc − t. The swings therefore get faster and faster as tc approaches; the gaps between the peaks shrink each time by the same factor λ = e^(2π / ω). Physicists call this discrete scale invariance: the pattern looks the same when you stretch the remaining time by λ, λ², and so on. Sornette, Johansen and Bouchaud (1996) found such patterns in the S&P 500 before the crash of October 1987 and compared them with earthquakes; in the same year, Feigenbaum and Freund (1996) described crashes as critical points in a system with discrete scale invariance.

Advanced: calibration and confidence

The formula has seven parameters: tc, m, ω, φ and the three linear ones A, B and C. Filimonov and Sornette (2013) split the oscillation term into C₁ × cos(ω × ln(tc − t)) + C₂ × sin(ω × ln(tc − t)). The fit then depends on only three nonlinear parameters (tc, m, ω); A, B, C₁ and C₂ each follow from a linear regression. The cost function becomes smooth and, if the model fits the data, usually has only one minimum. This makes the estimate much more stable.

A single fit depends on where the window starts. You therefore fit the model to many windows that end on the same day and count how many of them deliver plausible parameters. Zhang et al. (2016) used quantile regressions over many time scales to build the DS LPPLS confidence indicator from this; in Gerlach et al. (2019), confidence is the share of qualified fits across many time windows. The exact bounds Portfolio Lab uses are given in the methodology.

With B > 0, the same formula describes an accelerating fall, a negative bubble: panic instead of euphoria, which can end in a strong rebound. The indicator counts positive and negative bubbles separately.

What research shows

• Johansen, Ledoit and Sornette (2000) found log-periodic patterns before the Wall Street crashes of 1929, 1962 and 1987 and before the 1997 crash in Hong Kong.
• Johansen, Sornette and Ledoit (1999) analyzed eight crashes from 1929 to 1998 in the stock markets of the US, Hong Kong and Russia and in currency markets. Their results suggest that over the preceding two decades, no major crash after a lengthy bubble came without such signatures.
• Sornette et al. (2015) document how the Financial Crisis Observatory at ETH Zurich diagnosed the bubble in the Chinese stock market publicly and in real time. It began in mid-2014 and burst from June 2015 on.
• Demirer et al. (2019) applied the confidence indicator to the S&P 500, FTSE and Nikkei. It flagged some major bubbles in advance, for example around Black Monday, the dot-com era and the subprime crisis. Short selling predicted negative bubbles, and market liquidity predicted both kinds, especially in the short term.
• Gerlach et al. (2019) broke Bitcoin's price history from 2012 to early 2018 down into three major and ten smaller peaks and analyzed the long bubbles with the confidence indicator.

It is striking that almost all of this evidence comes from Sornette and his coauthors. That does not count against it, but it does call for critical reading and a look at independent tests.

Criticism: how reliable are the signals?

• Significance: Feigenbaum (2001) re-examined the S&P 500 before the 1987 crash. If you leave out the last year before the crash, the log-periodic component is not statistically significant.
• Patterns created by the analysis: Huang et al. (2000), with Sornette as a coauthor, use aftershocks to show that log-periodic patterns can arise from the way the data are processed alone, for example through smoothing or cumulating. That is why enough oscillations are required within the window.
• Parameters outside the ranges: Brée and Joseph (2013) examined eleven crashes in the Hang Seng from 1970 to 2008. In only seven did the fitted parameters fall within the ranges that Johansen and Sornette had set after the fact. A fit could have predicted the sharp decline in the most recent global downturn, but the data do not support the mechanism assumed behind the model. Sornette et al. (2013) respond to such objections at length.
• Hindsight selection: after a crash, attention turns to those who predicted it. Fama urges that forecasting models be judged on all their predictions, not on individual hits Fama (2014).
• Many parameters: seven parameters can be fitted to almost any price path. This is the problem of overfitting. And tc shifts with every new price.

Alternatives: tests for explosive price behavior

Econometricians take a different route. Instead of fitting a formula to the price path, they test whether a price grows explosively for a time, that is, faster than a random walk allows. To do so, Phillips, Wu and Yu (2011) combined a right-tailed unit root test with a recursive supremum test. The unit root test is a Dickey-Fuller test following Dickey and Fuller (1979) that tests not for mean reversion but for explosive growth. For the Nasdaq in the 1990s, they dated the start of the exuberance to mid-1995, before Alan Greenspan's famous warning about “irrational exuberance” in December 1996.

Phillips, Shi and Yu (2015) generalized the procedure with flexible windows so that it can also detect and date several bubbles in one long series; after the authors, it is often called the PSY test. In the S&P 500 from 1871 to 2010, it found the well-known episodes of exuberance and collapse; the older procedure found far fewer.

None of the approaches predicts a crash with certainty. Valuations are explained in the article on the Shiller P/E.

What this means for investors

• A signal is not a date: a bubble signal describes a price pattern, not the day of a crash. Many signals end without a crash.
• Timing is doubly hard: the price rises fastest precisely in the superexponential phase. Get out too early and you miss it; get out too late and you ride the crash. More in the article on market timing.
• Rules without forecasts: broad diversification limits how much any single bubble matters, and rebalancing automatically sells part of whatever has risen sharply. A stock allocation at which you could sit through a drawdown of 50% needs no prediction.
• Check for concentration: if you hold a lot in an industry or theme that has risen sharply, you carry concentration risk.
• Not investment advice: models like LPPLS are research tools. They do not replace a decision that fits your goals and your capacity to bear risk.

How Portfolio Lab calculates

The Bubbles page runs the LPPLS model for 30 markets (stock indexes, bond ETFs, commodities and cryptocurrencies) or for a ticker of your choice, also as of a date in the past. For each end date, it fits the formula to 26 windows of 125 to 750 daily prices, with the Filimonov and Sornette calibration, and counts how many fits pass the filters, separately for positive (B < 0) and negative bubbles (B > 0). It also shows the median of the estimated tc with its range, the fit in the chart and the confidence history week by week over two years. All the details are in the methodology.

This article explains a method; it is not investment or tax advice. Worked examples are simplified and rounded. All terms in the glossary.