Why Financial Crises Keep Coming Back — Fat Tails, Black Swans, and Systemic Risk

Why do once-in-a-century crashes keep arriving so often? There is a deep gap between the "normal market" promised by textbooks and the market we actually experience. This article follows that gap, moving from the collapse of conventional wisdom, through contagion and networks, to attempts to foresee crises and honest reversals.

Let us unpack the terms in the title first. A fat tail refers to the shape of a distribution where extreme events happen more frequently than a normal distribution predicts. A black swan is an event that erupts from a place the model deemed nearly impossible. Systemic risk is the risk that a substantial portion of the entire system stops functioning and collapses. It arises when a shock from one place spreads.

One thing must be disclosed in advance. Nearly all the papers cited here are preprints (arXiv) released before undergoing peer review by other experts. This is because this field, where finance, physics, and economics intersect, often makes its findings public via preprint first. Therefore, a single line must not be read as a definitive conclusion.

Broken Consensus — Markets Do Not Follow a Normal Distribution

The oldest assumption underlying finance is that returns follow a normal distribution (the bell curve). Returns are a value expressed as a ratio of how much invested money has increased or decreased. The normal distribution is a shape where most values cluster near the average, and as you move away from the average, they rapidly become rarer. Since Bachelier in 1900 and Osborne in 1959, standard models have predicted stock price returns using a log-normal distribution (arXiv:cs/0304009).

This assumption was convenient. Risk could be calculated knowing only the average and variance (the degree to which values are scattered). Extreme events could effectively be treated as if they never happened.

The problem is that the data does not follow this assumption. Since Fama in 1965, empirical data has shown leptokurtosis, unlike the normal distribution. It has also shown fat tails (arXiv:cs/0304009). Kurtosis is the degree to which values are heavily clustered around the center and both ends. The higher it is, the more the peak tends to be sharp and the tails thick. Fat tails are a shape where probability remains thick at both ends of the distribution, namely toward the extreme values. Probability is more heavily clustered in the center and tails of the distribution, and the probability of extreme events is much greater than a normal distribution predicts.

The high-frequency return distribution has a sharp shape with thick tails. This directly shakes the Gaussian (normal distribution) assumptions of standard risk management and option pricing. There is no consensus on which family of distributions should be used to describe it. However, the compared distributions were fairly consistent with each other in the scaling properties of large price fluctuations (arXiv:2302.02769). Actual stock price returns do not fit the normal distribution and show high peaks, thick tails, and skewness (arXiv:2312.02472).

To these thick tails, a surprisingly consistent figure is attached. In mature markets—that is, large markets that have been traded for a long time—the tails are known to follow an inverse cubic law. This is true for both the profit side and the loss side. This means a power-law exponent of about 3. A power law is a law where as values get larger, the probability decreases, but much more slowly than in a normal distribution. The exponent is the number that determines that rate of decrease. The possibility was raised that this exponent is a universal constant independent of the market, industry, and market capitalization.

This number has been confirmed in multiple markets, alongside exceptions. In the Indian market, a fat-tail power law with an exponent of about 3 was confirmed (arXiv:physics/0605247). A power law of ζ≈3 was also confirmed across international market indices of 7 countries. However, in the same study, the Korean market followed an exponential distribution (β≈0.7) rather than a power law. An exponential distribution is a distribution where the tails decrease faster than a power law (arXiv:physics/0601126).

The same difference appeared in Chinese indices. When the time interval for measuring returns was short (Δt<240 minutes), the tails decreased according to a power law with an exponent of about 3. Beyond 240 minutes, they decreased exponentially, a difference not observed in mature markets. As the interval became longer, the distribution gradually approached a normal distribution (arXiv:2209.08521). The rank distribution of stocks ordered by stock price across 4 markets broadly matched power laws like Zipf's law. Yet in the same study, the probability density of returns compared after matching scale was closer to an exponential form than a power law (arXiv:cond-mat/0412014).

Research testing this universality in emerging markets yielded different results. For 1-minute returns of Chinese A-shares (mainland-listed stocks), the tail exponent decreased as the turnover rate was higher. The larger the market capitalization, the higher the tail exponent was. The conclusion was that the distribution is not universal in emerging markets (arXiv:1003.5984).

In a world promised by the normal distribution, a '20-sigma' event happens once in a time longer than the age of the universe. Sigma is a unit measuring how much a value deviates from the average, so this event is a value deviating twentyfold from the average. Yet "extremely implausible" crashes have been repeatedly observed. 'Black Monday,' when stock prices collapsed massively in a single day in 1987, can be cited as an example of a black swan. These losses could not be explained under a normal distribution environment (arXiv:1202.0142).

The Korean market adds an even more inconvenient truth. The more recent the data, and the smaller the stocks, the thicker the tails became. The GARCH filter is a correction that strips out the phenomenon where volatility—periods during which prices fluctuate wildly—clusters together. This fat tail did not disappear even after passing through the GARCH filter (arXiv:1904.02567).

Where Fat Tails Come From — Rules Repeated in the Market

The non-normality of return distributions is not a one-off quirk. There are statistical regularities in markets that repeat across assets, markets, and time zones. These are called stylized facts. The 11 stylized facts compiled by Rama Cont in 2001 are accepted as constraints that any model must reproduce.

However, whether those 11 still hold true today is a separate question. Intraday returns were re-examined using individual stocks from the Dow 30, a basket of representative blue-chip US stocks. Eight of the facts were confirmed in data from October 2018 to March 2019. The remaining three found no supporting evidence (arXiv:2311.07738). There is also a study that broadly checked the robustness of these regularities by mobilizing 150 years of data from Asia, Europe, and the US, as well as cryptocurrencies (arXiv:2504.08611).

At the core of these regularities lie two things. One is the fat tail, and the other is volatility clustering. Volatility clustering is a phenomenon where large fluctuations tend to cluster with large fluctuations, and small fluctuations with small fluctuations. Across roughly 100 years of Dow Jones daily returns, several regularities appeared together, including fat tails and long-term volatility correlation. Long-term volatility correlation is the property where the magnitude of fluctuations persists over a long period (arXiv:cond-mat/0401009).

Fat tails and volatility clustering may not be separate, but from the same root. In one analysis, when the order of daily returns was randomly shuffled, the fat tails at time scales longer than a day almost disappeared. This analysis interpreted that fat tails stem from known long-term volatility correlation. A control experiment where only the signs were shuffled while the absolute value order was left intact supports this interpretation (arXiv:cond-mat/0112484).

Another study sees it differently. It reported that the slow decay of absolute return autocorrelation is directly linked not to fat tails, but to the degree of clustering. The point that large losses generally cluster more severely than large gains was also quantified (arXiv:1002.0284).

There are also microscopic explanations. In one model, if liquidity is lacking in the order book so that "gaps" occur between orders, large price fluctuations occur. This creates fat tails (arXiv:1207.2946). An order book is a list where buy and sell orders accumulate by price. Another explanation has also been proposed. Markets move back and forth between "stress levels" and draw values from different normal distributions, which is why fat tails arise (arXiv:1310.4538). These explanations are in a competitive relationship. The Korean market study seen earlier reported that fat tails remain even after stripping away volatility clustering.

2008-Style Contagion — Networks Amplify Shocks

The fat tails of individual assets alone cannot explain the full picture of a crisis. What 2008 taught us is that crises spread along connections. When a crisis strikes, markets become strongly correlated and move almost as one. Analyzing correlation matrices of major world indices shows that markets synchronized during sharp crashes (arXiv:1102.1339).

A network is a web connecting firms or financial institutions as dots, and their correlations or trading relationships as lines. In the 2008 Korean market network, firms moved synchronously and correlations strengthened during the crisis. The degree distribution of the largest cluster during the crisis also became fatter than in other periods (arXiv:1307.6974). A brief 2009 empirical study depicted U.S. stock market losses as flowing like an epidemic or waterfall along stock correlation networks. It left whether that was a metaphor or a model as a matter of debate (arXiv:0901.1392).

This is where systemic risk becomes an issue. The financial system is represented as a giant complex network. Banks, hedge funds, and institutions are entangled by visible and invisible financial connections. Connections that are normally means of risk diversification can suddenly turn into conduits for risk propagation during a crisis (arXiv:1710.11512).

Types of systemic risk are divided into direct externalities and perception-feedback effects. The former type includes defaults (failing to repay debts), correlated portfolios, and fire sales. The latter type includes bank runs (depositors rushing to withdraw money all at once) and credit crunches (arXiv:2012.12702).

Mathematical models have explored how networks amplify shocks. Rigorous asymptotic results on the scale of contagion in large-scale financial networks have been derived. These results highlight the role of 'contagious links'. When a default occurs, the institutions that contribute most to network instability are those with many connections and a large proportion of contagious links (arXiv:1112.5687).

Let us look at a model from a dissertation. If capital requirements are not sufficient, a shock that is initially localized can spread to the entire system. As the 2007/08 crisis showed, it can be greatly amplified across multiple contagion channels (arXiv:1911.07313).

Amplification is not just a matter of direct exposure. Direct exposure refers to claims and liabilities directly established between institutions. Indirect exposure refers to connections created by holding similar assets. A multiplex network was analyzed where the two exposures were placed in different layers within the Mexican financial system. Consequently, focusing only on direct exposures underestimated total systemic risk by up to 50% (arXiv:1802.00311).

Analyzing four layers of the same system showed that looking at a single layer underestimated total systemic risk by up to 90%. In the same study, market-based systemic risk indicators systematically underestimated expected system losses. Those expected losses were up to 4 times higher than before the 2007–2008 crisis (arXiv:1505.04276).

Contrary to theory, empirical studies often find that direct exposure networks are not the primary channel of contagion. However, when simulated shocks (stress tests) were applied to the Austrian banking network, it was different. When counterparty losses and rollover risk were viewed separately, bankruptcies were fewer on average. When portfolio overlap was included as well, the direct trading network significantly increased systemic risk (arXiv:1306.3704).

Contagion also comes from outside the financial sector. Model estimations were performed using supply chain relationship data for all Hungarian firms. Incorporating supply chain contagion increased banks' expected loss by 4.3 times, VaR by 4.5 times, and ES by 3.2 times. VaR is the loss limit expected not to be exceeded at a given confidence level. ES is the average loss when that limit is exceeded. A small number of firms could impact up to 16% of the banking system's equity. Most of the estimated losses came from supply chain contagion, not financial channels (arXiv:2305.04865).

Can We See Crises in Advance — Early Warning Signals and Bubble Models

If the vulnerability of a system is measurable, can we warn of a crisis in advance? There have been two branches of attempts.

Early Warning Systems

Indicators that sound a warning signal before a crisis occur are called Early Warning Systems (EWS). The 2008 crisis highlighted the importance of early detection and prevention of financial crises. A bibliographic review of 616 papers from 1976 to 2023 reported that over 90% of the analyzed subjects were published after 2006. A bibliographic review is an analysis that gathers publication records of papers numerically to view the trends of a field. They also outlined that this field has evolved into bankruptcy prediction, banking crises, currency crises, and machine learning prediction (arXiv:2310.00490).

In the case of Greece, instability signals appeared in GDP growth long before the economic collapse. Rapidly soaring growth rates pushed GDP onto a pseudo-hyperbolic trajectory. It was a trajectory that, if left alone, would have diverged to infinity in 2017. It was something that could not be sustained. This is a single study looked back upon after the collapse (arXiv:1511.06992).

There are also attempts utilizing the spectral properties of market data. Correlation and covariance matrices are tables gathering the degree to which stocks move together. There is an indicator that checks whether the eigenvalue distribution of this table deviates from the "calm market" reference distribution. There is also an indicator that directly looks at the largest value among the eigenvalues. The latter relies on the intuition that large eigenvalues are a signal of dynamical instability (arXiv:1506.00806). There is also research incorporating these crisis indicators into trading rules (arXiv:1709.02701).

In the 2015 Chinese market crash, an illiquidity network became more densely connected on crisis days, accelerating the collapse. An illiquidity network is a connection network linking stocks whose degrees of blocked trading move together. There is an early signal that counts days with random failures among the preceding 5 days. The authors reported that this signal correctly identified over half of the crash days within the sample, particularly the days clustered in the early part of the crash (arXiv:2004.01917). Just before the 2008 crash, the UK benchmark index FTSE 100 showed precursors of a sharp surge in the collective behavior of the entire market (market mode) (arXiv:1111.4637).

Log-Periodic Bubble Model (LPPL)

The most ambitious attempt is the Log-Periodic Power Law (LPPL) model. A bubble is a phenomenon where asset prices inflate away from their fundamentals—the value the asset actually holds—and then burst. The core idea is that financial crashes resemble critical phenomena in physics. Critical phenomena are events where an entire system changes its state all at once at a certain limit (the critical point). Noise traders are investors who trade based on rumors or sentiment rather than solid information. This model assumes that they mimic each other. When that tendency strengthens up to the critical point, everyone places the same order (to sell) at the same time, and a crash occurs (arXiv:cond-mat/9810071).

This model predicts that log-periodic oscillations are imprinted on top of a power law in prices just before a crash. Log-periodic oscillations are fluctuations whose period shortens at a constant rate as they get closer to the time of the crash. Evidence matching this prediction was presented in early research (arXiv:cond-mat/9509033). The authors presented evidence that this pattern existed in prices just before the crashes on Wall Street in 1929, 1962, and 1987, and in Hong Kong in 1997. How statistically solid this evidence is is covered in the debate section later (arXiv:cond-mat/9810071).

Early research analyzing the precursors, aftershocks, and easing oscillations of the 1987 S&P 500 portrayed the crash as a 'dynamical critical point' (arXiv:cond-mat/9510036). The same authors saw the April 2000 Nasdaq crash as also fitting well into the LPPL framework they proposed (arXiv:cond-mat/0004263).

The perspective on bubbles themselves was also organized. A review came to a conclusion that bubbles can also be diagnosed by looking together at temporary 'super-exponential' price growths produced by positive feedback. Positive feedback is a loop where an upward trend calls forth another upward trend. Super-exponential growth is growth that accelerates more steeply than exponential growth (arXiv:0812.2449).

The 2008 crisis was diagnosed as the result of multiple bubbles accumulating and mutually reinforcing each other. Those bubbles created the illusion of a "perpetual money-making machine" (arXiv:0905.0220). There is also research that classified large crashes in the three major US indices (Dow, S&P 500, Nasdaq) over the past century. Drawdowns are the cumulative drop from the immediate peak to the trough. This study defined crashes as the top rankings among them. It saw that all crashes within the sample are linked to either an external shock or an LPPL bubble (arXiv:cond-mat/0401210).

The 2020 COVID-19 crash was also analyzed with the same framework. A study looking at 10 major indices found LPPL bubble signals in 7 of them. Those indices are the S&P 500, Dow, Nasdaq, DAX, CSI 300, BSESN, and BOVESPA. This study viewed that crash as originating endogenously from a bubble that had already formed. Endogenous means a cause accumulated from within the market. COVID-19 was merely the spark (arXiv:2101.00327). The US market was looked at by dividing it into market capitalization segments. This study viewed that the crash came from a bubble that started as early as September 2018. It reported that the start time of the bubble varied greatly across segments (arXiv:2101.03625).

The model itself was also refined. A methodology emerged that significantly increased calibration stability by reducing the LPPL formula to a function of three non-linear parameters (arXiv:1108.0099). The HLPPL model followed, which aims to capture positive and negative bubbles in a single framework by combining sentiment scores and hype indices (arXiv:2510.10878).

An Honest Twist — Vulnerabilities Are Visible, but the Timing Is Not

Up to this point, it sounds as though the crisis can be predicted imminently. However, the honest picture has a decisive limitation. System vulnerabilities can be measured and observed. However, whether the exact timing of a crash can be guessed in advance is not yet proven and remains under debate.

The sharpest debate has unfolded over LPPL itself. One critical study examined S&P 500 data right before the 1987 crash. As a result, it reported that if the final year of data immediately preceding the crash is excluded, the log-periodic component is not statistically significant (arXiv:cond-mat/0101031). The same study also examined another claim. It judged that the evidence explaining the drawdown of the S&P 500 through two distinct mechanisms (external shock versus bubble) was unconvincing.

In response, the LPPL camp countered. They argued that analyzing the data while excluding the most crucial data closest to the critical point is naive (arXiv:cond-mat/0106520). The critics counter-argued in turn (arXiv:cond-mat/0107445). This battle can be read as a scene showing the distance between ex-post identification and ex-ante prediction of a bubble.

The fundamental difficulty is this. One study noted that clearly discerning whether a bubble exists is still an unresolved problem in standard econometrics. The fundamental value of an asset is generally not directly observable. This study started from the following diagnosis: It is impossible to distinguish between a fundamentally growing price and a bubble price growing exponentially.

The authors extended the JLS model (a bubble model of the LPPL family). They presented a model that estimates fundamental value and crash non-linearity together. They reported that it fit well across three historical bubbles (1997 Hang Seng, 1987 S&P 500, 2009 Shanghai). However, the fact remains that all three are cases looked back upon after they had already ended (arXiv:1011.5343). Collapsing asset price bubbles have been studied for a long time. Yet, one review concluded that there is still almost no consensus on their causes and effects (arXiv:0812.2449).

It is worth emphasizing that even LPPL theory does not promise deterministic prediction. In this model, a crash is not a deterministic consequence of a bubble. Because of rational expectation constraints, the timing of the crash must be random. The theoretical end of a bubble is merely the 'most probable timing' of a crash, not a confirmed date. A finite probability that a bubble ends without a crash also exists (arXiv:cond-mat/9810071).

Power laws are also observed in the movements following a crash. There is an analysis showing that the mainshocks and aftershocks of the 1987, 2008, and 2020 crashes followed the Gutenberg-Richter power law of earthquakes (arXiv:2012.03012). It was also observed that the mitigating dynamics of the market immediately after a crash coincide with seismology's Omori's law (power-law decay). This observation is not explained by widely used simple stochastic volatility models (arXiv:cond-mat/0111257).

So, Why Do Crises Called Once-in-a-Century Come So Often?

Now we can answer the initial question. The intuition of "once in 100 years" is a calculation of a normal distribution. However, market returns have much thicker tails than a normal distribution. In mature markets, they commonly decay with a power law of an exponent around 3. That is why extreme crashes that a normal distribution considers astronomically rare are actually observed repeatedly (arXiv:1202.0142, arXiv:1003.5984).

The explanation from the studies followed in this article is as follows. Markets bundle large shocks and emit them through volatility clustering (arXiv:cond-mat/0401009). Networks amplify those shocks via direct and indirect channels (arXiv:1802.00311, arXiv:1505.04276), and speculative feedback inflates superexponential bubbles (arXiv:0812.2449).

The common perspective of these studies is that the reason "rare events" come often lies in the structure of the system. However, this is an explanation put forward by current research and not a finalized conclusion.

There is good news and bad news together. The good news is that tools exist to measure this vulnerability. There are network topology, spectral indicators, and even superexponential signatures of bubbles. One study calculated that in 10 quarters of data from the Austrian interbank lending network, rearranging the network optimally reduces systemic risk by roughly 3.5 times. The authors stated that this does not mean actually reorganizing the market, but rather showed the magnitude of the room for reduction (arXiv:1905.05931).

The bad news is that knowing the vulnerability does not mean knowing the date of the crash. In the LPPL model, the timing of the crash must be random. One critical study reported that if the final year right before the crash is removed, the log-periodic component is not significant. There have also been rebuttals and counter-rebuttals to this. There is also the diagnosis that fundamentally rising prices and exponentially rising bubble prices cannot be distinguished.

The conclusion is not simple and thus rather honest. We are getting better and better at seeing where things are unstable. But when they will collapse remains, at least for now, beyond the fog of the power law.

This article is for informational purposes and is not investment advice. Most of the cited evidence is preprints (arXiv) before peer review, and the details may be updated by subsequent research.

Source Papers (arXiv)

Collapse of Common Sense — Non-Gaussian, Fat-Tail, Power-Law

• Standard models are log-normal, empirical evidence shows kurtosis and fat tails (arXiv:cs/0304009)
• High-frequency returns show leptokurtic fat tails, Gaussian assumption collapses (arXiv:2302.02769)
• High peaks, fat tails, and skewness (arXiv:2312.02472)
• Mature markets show cubic inverse power law vs emerging markets being non-universal (indices depend on market cap and turnover) (arXiv:1003.5984)
• Indian market fat-tail power law α~3 (arXiv:physics/0605247)
• International markets in 7 countries follow power law ζ≈3 (arXiv:physics/0601126)
• Zipf's law and power laws in stock prices (arXiv:cond-mat/0412014)
• Chinese index tail power-law exponent is about 3 (arXiv:2209.08521)
• Heavy tails and Black Monday in economic data (arXiv:1202.0142)
• Korean market fat tails persist even after GARCH (arXiv:1904.02567)

Stylized Facts and Volatility Clustering

• 150 years and 11 stylized facts (arXiv:2504.08611)
• Re-examining Cont's 11 stylized facts (arXiv:2311.07738)
• 100 years of Dow Jones: fat tails, volatility clustering, and leverage (arXiv:cond-mat/0401009)
• Fat tails equal long-term volatility correlation (shuffle experiment) (arXiv:cond-mat/0112484)
• Volatility clustering and the bunching of large losses (arXiv:1002.0284)
• Order book gaps induce fat tails (arXiv:1207.2946)
• Origin of fat tails = mixture of normal distributions across stress levels (arXiv:1310.4538)

Contagion, Networks, and Systemic Risk

• Survey of systemic risk networks (arXiv:2012.12702)
• Review of systemic risk network models (arXiv:1710.11512)
• Contagious connectivity and contagion asymptotics (arXiv:1112.5687)
• Multi-layer networks, single layers underestimate risk by 90% (arXiv:1505.04276)
• Overlapping portfolios: looking only at direct exposures underestimates risk by 50% (arXiv:1802.00311)
• Interbank overlapping portfolios amplify contagion (Austria) (arXiv:1306.3704)
• Supply chain contagion amplifies losses 4.3 times (arXiv:2305.04865)
• 2007/08: Systemic propagation and amplification of local shocks (arXiv:1911.07313)
• Agent-based model of 183 European banks: interbank markets were extremely vulnerable until 2008, resilience in remaining capital increased afterwards, speed of collapse peaked in 2011 (arXiv:1611.04311)
• Market synchronization (surge in correlation) during crises (arXiv:1102.1339)
• 2008 Korean market network showed thick degree distribution during the crisis (arXiv:1307.6974)
• 2009: Epidemic/cascade-like flow of credit crisis losses following the US stock correlation network (whether metaphorical is debated) (arXiv:0901.1392)
• Optimal rearrangement of Austria's interbank lending network reduces systemic risk by about 3.5 times (computational baseline) (arXiv:1905.05931)

Early Warning Indicators

• Systematic review of financial EWS (616 papers) (arXiv:2310.00490)
• Warning signals before the Greek GDP collapse (arXiv:1511.06992)
• Random matrix-based crisis indicators (arXiv:1506.00806)
• Spectral crisis indicator trading strategies (arXiv:1709.02701)
• 2015 Chinese illiquidity network predicts half of crash days (arXiv:2004.01917)
• Collective behavior precursors before the 2008 crash (FTSE100) (arXiv:1111.4637)

Log-Periodic Power Law (LPPL) Bubble Model

• Crashes as critical points, timing is random (arXiv:cond-mat/9810071)
• Discrete scaling and log-periodicity before crashes (arXiv:cond-mat/9509033)
• 1987 S&P 500 precursors and aftershocks (arXiv:cond-mat/9510036)
• April 2000 Nasdaq LPPL (arXiv:cond-mat/0004263)
• Review of bubbles and crashes, super-exponential growth (arXiv:0812.2449)
• 2008 = accumulation of bubbles and the illusion of a "money-printing machine that works forever" (arXiv:0905.0220)
• In-sample crashes in US major three indices = external shocks or LPPL bubbles (arXiv:cond-mat/0401210)
• 2020 crash endogeneity (LPPL) (arXiv:2101.00327, arXiv:2101.03625)
• Stable LPPL calibration methods (arXiv:1108.0099)
• HLPPL bubble identification (arXiv:2510.10878)

Honest Reversal — Limits of Prediction and Debate

• Removing the final year makes log-periodicity meaningless (criticism) (arXiv:cond-mat/0101031)
• Excluding critical point data is naive (rebuttal) (arXiv:cond-mat/0106520)
• Counter-rebuttal on log-periodic precursor debate (arXiv:cond-mat/0107445)
• Bubble identification unsolved, fundamental vs bubble indistinguishable (arXiv:1011.5343)
• Gutenberg-Richter power law for mainshocks and aftershocks (arXiv:2012.03012)
• Relaxation of Omori's law after crashes (arXiv:cond-mat/0111257)