# The Log-Periodic Power Law: A Physicist's Bubble Model, Applied to the S&P 500 and Nasdaq 100

The log-periodic power law is a physicist's model that says a speculative bubble leaves a readable signature in the price itself. Here is how it fits the 1987 S&P 500 and the 2000 Nasdaq, and why its power to actually predict a crash stays contested.

Author: J.A. Watte
Published: July 6, 2026
Source: https://jwatte.com/blog/lppl-bubble-model-sp500-nasdaq/

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Most models of finance treat a market crash as a bolt from a clear sky, an event with no fingerprint until the moment it lands. A small group of physicists spent three decades arguing the opposite: that a speculative bubble leaves a specific mathematical signature in the price while it is still inflating, and that a trained eye can read it. The tool they built is called the log-periodic power law, or LPPL, and its most famous demonstrations sit on top of the two indices every American investor already watches, the S&P 500 and the Nasdaq. Whether it can actually forecast a crash, as opposed to describe one after the fact, is a genuinely open question, and worth walking through carefully.

## From earthquakes to markets

The physicist behind the idea is Didier Sornette, whose earlier career was in the physics of critical phenomena, the mechanics of material rupture, and the statistics of earthquakes ([Wikipedia](https://en.wikipedia.org/wiki/Didier_Sornette)). The move from cracking rock to crashing markets is less of a jump than it sounds. In each case the interesting question is whether a system racing toward a sudden failure gives off a warning in the way it accelerates.

The financial version is usually called the JLS model, after Johansen, Ledoit, and Sornette. It pictures a market containing two kinds of participants: rational traders who price assets sensibly, and imitative "noise" traders who buy mostly because other people are buying. When the herding among the second group intensifies, it drives up the hazard rate, the probability per unit of time that a crash will strike in the next instant, and a rational market has to compensate for that rising risk with faster price gains ([Johansen, Ledoit & Sornette](https://arxiv.org/abs/cond-mat/9810071)). The named model was set out in "Crashes as Critical Points," published in the International Journal of Theoretical and Applied Finance, volume 3, issue 2, in 2000 ([Johansen, Ledoit & Sornette](https://arxiv.org/abs/cond-mat/9810071)).

The empirical work came first. The earliest published log-periodic reading of a stock-index crash appeared in 1996, before Ledoit joined, when Sornette, Johansen, and Bouchaud fitted the S&P 500 around the October 1987 crash in "Stock Market Crashes, Precursors and Replicas" in the Journal de Physique I ([Sornette, Johansen & Bouchaud](https://hal.science/jpa-00247175v1)). Sornette later gathered the ideas in "Why Stock Markets Crash: Critical Events in Complex Financial Systems," published by Princeton University Press in 2003 ([Wikipedia](https://en.wikipedia.org/wiki/Didier_Sornette)).

## What the equation actually says

The heart of the model is a single formula for the logarithm of price as the bubble runs toward its critical moment. In the form used for the Nasdaq study it reads:

ln p(t) = A + B(tc - t)^m + C(tc - t)^m cos(ω ln(tc - t) - φ)

Each term earns its place ([Johansen & Sornette](https://ar5iv.labs.arxiv.org/html/cond-mat/0004263)). A is roughly the log-price the model expects at the critical time tc. B is negative during a bubble, so the middle term pushes the price up faster and faster as t approaches tc. The exponent m sits strictly between 0 and 1, which is the crucial detail: it keeps the price itself finite at tc, yet makes the growth rate diverge, producing the faster-than-exponential climb that is the model's headline signature ([Sornette et al.](https://arxiv.org/abs/1404.2140)). The final cosine term, governed by the angular frequency ω and the phase φ, adds the log-periodic wrinkles, oscillations that ride on top of the power-law climb and crowd closer and closer together as the peak nears.

Those tightening oscillations are the model's other claim to fame. They reflect what physicists call discrete scale invariance: the pattern repeats at geometrically shrinking intervals, so each wobble takes a fixed fraction of the time the previous one did, and the whole sequence compresses toward tc ([Cao et al.](https://arxiv.org/html/2510.10878v1)). It is worth being clear about what tc is and is not. It is not a guaranteed crash date. It is the end of the bubble regime, a change of behaviour that might be a crash or might be a large, slower correction ([Sornette et al.](https://arxiv.org/abs/1404.2140)).

Count the knobs on this machine and you get seven: three that enter the equation linearly (A, B, C) and four that do not (tc, m, ω, φ). In practice the fit exploits that split. For any trial of the four nonlinear parameters, the three linear ones can be solved directly, so the search only has to grind over four dimensions to minimise the root-mean-square error. A fit is treated as a genuine bubble regime only when it is physically sensible: m between 0 and 1, ω somewhere around 6 to 13, and B negative. The critical time is then read as the most likely interval for a change of regime, and the whole exercise is repeated across many overlapping windows to see whether that interval holds still or drifts ([Cao et al.](https://arxiv.org/html/2510.10878v1)). That number, seven, is the hinge of the argument that follows.

## The two crashes it made its name on

The 1987 fit is the model's origin story. Applied to the S&P 500 running into October 1987, the canonical calibration puts the critical time in late 1987, around 1987.7 to 1987.8 in decimal years, with an exponent m near 0.33 and a log-frequency ω near 7.4 ([Johansen, Ledoit & Sornette](https://arxiv.org/abs/cond-mat/9810071)). The crash itself landed on October 19, 1987, which in the same decimal-year units is 1987.78, comfortably inside that window.

The Nasdaq gave the group its most detailed public case. The Nasdaq Composite peaked at 5133 on March 10, 2000 (an intraday high of 5132.52), then broke on Friday, April 14, 2000, closing down close to 10% on the day at 3321, more than 35% below the peak ([Johansen & Sornette](https://arxiv.org/abs/cond-mat/0004263)). Fitting the run-up from a bubble they dated to the spring of 1997, Johansen and Sornette reported a best-fit critical time of about 2000.33, with the exponent z near 0.27, ω near 7.0, A near 9.5, B near -1.7, C near 0.06, and φ near -0.1. That best fit pointed to a crash around May 2, 2000 ([Johansen & Sornette](https://ar5iv.labs.arxiv.org/html/cond-mat/0004263)). The peak came a few weeks early, which is either an impressive near-miss or a reminder of how loosely "around" has to be read. Both fits were published, or refined, with the crash already in the rear-view mirror.

## Anti-bubbles, and an observatory that watches in real time

Turn the picture upside down and you get what Sornette and Johansen call an anti-bubble: a market that falls in a mirror image of the bubble, decaying along a log-periodic power law whose oscillations stretch out rather than compress. They first documented the pattern in the Nikkei after its peak at the end of 1989 and in gold after 1980 ([Johansen & Sornette](https://arxiv.org/abs/cond-mat/9901268)). A few years later, Zhou and Sornette reported the same downward structure across dozens of markets at once, finding a worldwide anti-bubble in the S&P 500 and other Western indices as the dot-com decline set in around the middle of 2000 ([Zhou & Sornette](https://arxiv.org/abs/cond-mat/0212010)).

To move from history to live forecasting, Sornette founded the Financial Crisis Observatory at ETH Zurich in 2008, a project that runs LPPL fits every day across tens of thousands of assets ([Financial Crisis Observatory](https://emeritus.er.ethz.ch/financial-crisis-observatory.html)). Its most disciplined test was the Financial Bubble Experiment, launched on November 2, 2009 ([Sornette et al.](https://arxiv.org/abs/0911.0454)). The design was meant to answer the obvious objection that anyone can fit a bubble after it pops. The team wrote up its forecasts, sealed them, and published only cryptographic fingerprints of the documents, an MD5 hash plus 256-bit and 512-bit SHA-2 hashes, using the arXiv submission timestamp as tamper-proof evidence of when the forecast was made. The plain-text forecasts and post-analysis were revealed on May 3, 2010 ([Sornette et al.](https://arxiv.org/abs/0911.0454)).

It is worth stressing what the experiment did not cover. The four assets it diagnosed were the Brazilian IBOVESPA index, a Merrill Lynch US corporate bond index, gold spot (all announced on November 2, 2009), and cotton futures (added on December 23, 2009). The S&P 500 and the Nasdaq 100 were not among them ([Financial Crisis Observatory](http://tasmania.ethz.ch/pubfco/fco.html)). The results were mixed rather than triumphant, which is part of why the method's predictive power stays contested.

The S&P 500 work did not stop at 1987. In a 2016 Physica A study, Zhang, Sornette, Balcilar, Gupta, Ozdemir, and Yetkiner ran the observatory's two headline diagnostics, the DS LPPLS Confidence and Trust indicators, across monthly S&P 500 data from August 1791 to August 2014, more than two centuries ([Zhang, Sornette et al.](https://www.sciencedirect.com/science/article/abs/pii/S0378437116301017)). The Confidence indicator is simply the fraction of fitting windows in which the calibration passes the model's filters, so a high reading means the bubble shape survives the choice of window rather than depending on it ([Shu and Zhu](https://arxiv.org/abs/1905.09647)). They report eight positive bubbles and two negative anti-bubbles from January 1814 to August 2014. It is a retrospective study, not a live call.

The observatory's most documented equity call landed not on a US index but on China. Sornette's team reported that its June 2015 FCO Cockpit flagged Shanghai and Shenzhen equities as a bubble before they turned: the Shanghai and Shenzhen Composites had climbed 60 percent and 122 percent between December 31, 2014 and June 12, 2015, then fell more than 43 percent and about 45 percent into the August 26, 2015 bottom ([Sornette et al.](https://www.risk.net/journal-of-investment-strategies/2427649/real-time-prediction-and-post-mortem-analysis-of-the-shanghai-2015-stock-market-bubble-and-crash)). That account is FCO self-reported. What sets it apart is an outside check: Min Shu and Wei Zhu of Stony Brook University calibrated the model to both indices and found the June 12 crash "can be well predicted by the LPPLS model as far back as two months before" ([Shu and Zhu](https://arxiv.org/abs/1905.09633)). Their test is retrospective, but independent of the FCO.

The observatory's better-known real-time call was on oil. Applying the model to the 2006 to 2008 run-up, the group posted a forecast to arXiv on June 6, 2008 diagnosing a speculative bubble in crude ([Sornette, Woodard & Zhou](https://ideas.repec.org/a/eee/phsmap/v388y2009i8p1571-1576.html)). Oil then closed at $145.29 on the NYMEX on July 3, 2008 and printed an intraday all-time high near $147 on July 11, 2008, before collapsing. The FCO also reports, in its own account, a real-time S&P 500 forecast issued on May 27, 2008 that bracketed a market peak inside an 80% window running from about May 17 to July 14, 2008. That particular claim rests on the observatory's self-reporting rather than an independently verified sealed record, and is best read as an FCO-reported result rather than an established fact ([Financial Crisis Observatory](https://emeritus.er.ethz.ch/financial-crisis-observatory.html)).

The work has since spun off a commercial service. The site sornette.finance says it monitors roughly 450 systemic assets and 850 single stocks across bonds, equities, commodities, currencies, and crypto, publishes daily rankings of the twelve strongest positive and negative bubbles, issues monthly Global Bubble Status Reports, and advertises an "89% success rate" drawn from the ETH Financial Bubble Experiment ([sornette.finance](https://sornette.finance/landing/)). Those are the vendor's own marketing figures, not independently peer-verified numbers, and should be read that way.

## The case against it

The criticism is as old as the model, and it starts with that count of seven parameters. Fitting a seven-parameter curve to noisy price data is exactly the sort of exercise that can produce a beautiful match to almost anything. Early critics including Laloux and co-authors warned that the model would suffer from severe over-fitting ([Brée & Joseph](https://ar5iv.labs.arxiv.org/html/1002.1010)).

The most careful takedown came from David Brée and Nathan Joseph. Testing the model across a set of historical crashes, they found that the predicted crash time was highly sensitive to the size and starting point of the data window, that the error function being minimised was extremely sensitive to tiny changes in the frequency ω, and, most damning, that the fitted curves often sloped downward during the supposed bubble. In 18 of 30 cases the fit had a negative slope, contradicting the model's own requirement that price rise toward the critical time ([Brée & Joseph](https://ar5iv.labs.arxiv.org/html/1002.1010)). In a companion analysis they checked whether real crashes fell inside Johansen and Sornette's own after-the-fact parameter ranges, and found that they did for only 7 of 11 crashes, concluding that the posited mechanism "does not hold" ([Brée & Joseph](https://www.researchgate.net/publication/45899349_Fitting_the_Log_Periodic_Power_Law_to_financial_crashes_a_critical_analysis)).

A separate strand of criticism questions whether the log-periodic wiggles are reliably there at all. James Feigenbaum, in work from 2001, and Chang and Feigenbaum in 2006, concluded that the log-periodic component is not statistically robust, small enough relative to market noise that it cannot be dependably detected, and that clean-looking fits are not confined to windows that happen to end near real crashes ([LPPL review](https://arxiv.org/pdf/2106.05116)). The exact quantitative wording of those findings is hard to pin down from the secondary literature, so it is fairer to describe them as serious statistical-significance objections than to attach a precise multiple to the size of the signal.

A recent survey frames the underlying problem crisply. Fitting an LPPL to a bubble after it has ended is easy and visually convincing; genuine out-of-sample prediction is much harder, dogged by false positives and by the fact that the results depend on subjective choices about which windows to use and which fits to accept ([Nature Humanities and Social Sciences Communications](https://www.nature.com/articles/s41599-025-05920-7)). That 2025 paper states plainly that the model's parameter interpretation and reliability are contingent on the researchers' own judgment, and proposes an AI-based reliability score precisely to get past ex-post fitting.

Sornette's group has not conceded the field. In a 2013 paper titled "Clarifications to questions and criticisms on the Johansen-Ledoit-Sornette financial bubble model," the authors argued that much of the criticism attacks a straw man, because tc was never meant to be a precise crash date. It marks a probabilistic change-of-regime window, and forecasts should be read as ensemble probabilities rather than single-point predictions ([Sornette, Woodard, Yan & Zhou](https://www.sciencedirect.com/science/article/abs/pii/S0378437113004342)). That reframing is reasonable, though it also makes the model harder to falsify, which is its own kind of problem.

## The bottom line

The log-periodic power law is not numerology, and it is not a crystal ball. It is a serious attempt to carry the mathematics of critical phenomena into finance, and on historical data its fits to the 1987 S&P 500 and the 2000 Nasdaq are genuinely striking. What stays unsettled, after thirty years, is whether those fits amount to prediction. The in-sample matches are easy to admire and easy to over-produce; the out-of-sample record is thin, contested, and partly self-reported. The most defensible reading is the one Sornette's own group retreated to: a bubble diagnosis is a probability over a window, not a date on a calendar. Treat any live bubble score on the S&P 500 or the Nasdaq the same way, as one contested indicator among many, and never as a reason to act. The current subscription-gated scores are not public, and nobody, including the people who built the model, can hand you the day the music stops.

## Related reading

- [Testing a trading idea against data](/blog/blog-claude-trading-lessons/): why an appealing market pattern means little until it survives real out-of-sample data.
- [The backtest that flattered a signal](/blog/blog-ai-portfolio-24-percent-backtest/): how a quantitative signal gets backtested, and the specific ways backtests mislead.
- [Compounding instead of timing](/blog/avenir-dodge-buffett-compounding/): the opposite temperament, growing money slowly rather than trying to date the crash.
- [The Fed, facts and myths](/blog/federal-reserve-facts-and-myths/): the institution whose policy turns often sit right where the model tries to place a bubble top.

## Fact-check notes and sources

- **Origins, the JLS model, and the book**: the physics-of-earthquakes background and the 2003 Princeton University Press book from [Wikipedia](https://en.wikipedia.org/wiki/Didier_Sornette); the two-agent structure, the crash hazard rate, and "Crashes as Critical Points" (IJTAF vol 3, issue 2, 2000) from [Johansen, Ledoit & Sornette](https://arxiv.org/abs/cond-mat/9810071); the first 1987 S&P 500 log-periodic fit (Journal de Physique I, 1996) from [Sornette, Johansen & Bouchaud](https://hal.science/jpa-00247175v1).
- **The equation, parameters, and fitting workflow**: the equation form and the m-between-0-and-1, ω-6-to-13, B-negative acceptance rules from [Cao et al.](https://arxiv.org/html/2510.10878v1) and [Johansen & Sornette](https://ar5iv.labs.arxiv.org/html/cond-mat/0004263); the finite-price-but-diverging-growth reading of m and the change-of-regime meaning of tc from [Sornette et al.](https://arxiv.org/abs/1404.2140).
- **The 1987 and 2000 fits**: tc around 1987.7 to 1987.8, m near 0.33, ω near 7.4 for the S&P 500 from [Johansen, Ledoit & Sornette](https://arxiv.org/abs/cond-mat/9810071); the Nasdaq Composite peak of 5133 on March 10, 2000, the April 14, 2000 close of 3321 (more than 35% below the peak, down close to 10% on the day), the 1997 bubble start, and the fit (tc about 2000.33, z near 0.27, ω near 7.0, A near 9.5, B near -1.7, C near 0.06, φ near -0.1, predicted crash around May 2, 2000) from [Johansen & Sornette](https://arxiv.org/abs/cond-mat/0004263).
- **Anti-bubbles and the observatory**: the Nikkei-after-1989 and gold-after-1980 anti-bubbles from [Johansen & Sornette](https://arxiv.org/abs/cond-mat/9901268); the worldwide S&P 500 anti-bubble from mid-2000 from [Zhou & Sornette](https://arxiv.org/abs/cond-mat/0212010); the 2008 founding and daily scanning of tens of thousands of assets from the [Financial Crisis Observatory](https://emeritus.er.ethz.ch/financial-crisis-observatory.html); the November 2, 2009 launch, the cryptographic-hash method, and the May 3, 2010 reveal from [Sornette et al.](https://arxiv.org/abs/0911.0454); the four diagnosed assets (IBOVESPA, a Merrill Lynch corporate bond index, gold spot, and cotton) from the [Financial Crisis Observatory results page](http://tasmania.ethz.ch/pubfco/fco.html).
- **The oil call, the S&P 500 self-report, and the commercial service**: the June 6, 2008 oil-bubble forecast and the July 2008 crude peak (a $145.29 NYMEX close on July 3, an intraday high near $147 on July 11) from [Sornette, Woodard & Zhou](https://ideas.repec.org/a/eee/phsmap/v388y2009i8p1571-1576.html); the May 27, 2008 S&P 500 forecast and its May 17 to July 14 window are FCO self-reported and are presented as such ([Financial Crisis Observatory](https://emeritus.er.ethz.ch/financial-crisis-observatory.html)); the "~450 systemic assets," "~850 single stocks," top-twelve rankings, monthly reports, and "89% success rate" are vendor marketing from [sornette.finance](https://sornette.finance/landing/).
- **The criticism and the reply**: the seven-parameter over-fitting worry, the window and ω sensitivity, and the 18-of-30 negative-slope finding from [Brée & Joseph](https://ar5iv.labs.arxiv.org/html/1002.1010); the "7 of 11 crashes" and "does not hold" conclusion from [Brée & Joseph](https://www.researchgate.net/publication/45899349_Fitting_the_Log_Periodic_Power_Law_to_financial_crashes_a_critical_analysis); the Feigenbaum (2001) and Chang & Feigenbaum (2006) statistical-significance objections, whose exact wording is characterized rather than quoted, from an [LPPL review](https://arxiv.org/pdf/2106.05116); the ex-post-versus-out-of-sample and researcher-subjectivity point from [Nature Humanities and Social Sciences Communications](https://www.nature.com/articles/s41599-025-05920-7); the probabilistic change-of-regime defence from [Sornette, Woodard, Yan & Zhou](https://www.sciencedirect.com/science/article/abs/pii/S0378437113004342).
- **What is not established**: an alternate 1987 fit (exponent near 0.68, ω near 8.9) circulates in secondary summaries but could not be tied to a primary source and is left out; the FCO's May 2008 S&P 500 forecast rests on self-reporting; the precise size of the log-periodic signal relative to noise is characterized, not quoted; and no current live bubble score for any index was retrievable, so none is stated.
- **A two-century S&P 500 test**: monthly S&P 500 data from August 1791 to August 2014, with eight positive bubbles and two negative anti-bubbles from January 1814 to August 2014, from [Zhang, Sornette et al.](https://www.sciencedirect.com/science/article/abs/pii/S0378437116301017); the DS LPPLS Confidence indicator defined as the fraction of fitting windows that pass the model's filters from [Shu and Zhu](https://arxiv.org/abs/1905.09647).
- **The 2015 China call and an independent check**: the June 2015 FCO Cockpit reading, the 60 percent and 122 percent run-ups from December 31, 2014 to June 12, 2015, and the later drops of more than 43 percent and about 45 percent into the August 26, 2015 bottom, all FCO self-reported, from [Sornette et al.](https://www.risk.net/journal-of-investment-strategies/2427649/real-time-prediction-and-post-mortem-analysis-of-the-shanghai-2015-stock-market-bubble-and-crash); the independent finding that the June 12, 2015 crash "can be well predicted by the LPPLS model as far back as two months before" from [Shu and Zhu](https://arxiv.org/abs/1905.09633).

*This post is informational and educational, not investment, financial, or trading advice. Figures, equations, and forecasts are reproduced from the cited academic papers, the researchers' own reports, and one vendor's marketing page, with self-reported and unverified items flagged as such. The people, institutions, and services named are discussed as a matter of public record under nominative fair use, with no affiliation, endorsement, or partnership implied.*


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