The Vault

THE FRACTALS OF FINANCE | Research Series Phase 2 | Episode 1 of 9

Zero Doesn't Mean Nothing

How we looked beneath the most quoted statistic in finance and found a machine that should not exist.

Phase 1 of this series proved that markets have memory. Across sixty-eight futures contracts spanning eight asset classes and four decades of daily data, the evidence was unanimous. Every market carried a statistical fingerprint that the random walk forbids. Feedback was universal. The efficient market hypothesis did not survive contact with the data.

But Phase 1 was a photograph. A cross-sectional exposure of a moving system. It told us what was there. It did not tell us what it was doing.

Phase 2 asks the deeper question. Not whether feedback exists, but what it does over time. Not whether the machine is running, but whether it is speeding up or slowing down. To answer that question, we begin with the single most important number in quantitative finance.

Zero.

Specifically, the near-zero autocorrelation that appears when you compute the serial dependence of daily returns over a long sample. It is the number that launched a thousand assumptions. It is the number that persuaded generations of researchers and practitioners that markets are efficient, that the past is irrelevant, that structure does not exist.

It is also a lie.

The Number Everyone Trusts

Take the S&P 500. Compute the first-order autocorrelation of daily returns from 1984 to 2026. Forty-two years. More than ten thousand trading days. The answer is negative 0.07.

That number is indistinguishable from zero. If you published it, no journal would find it remarkable. If you presented it to a risk committee, they would nod and move on. It confirms what everyone already believes: the S&P 500 has no exploitable serial structure. Returns are essentially independent. The random walk holds.

Across all sixty-eight markets in our dataset, the picture is the same. The mean full-sample autocorrelation is negative 0.001. Sixty-eight markets, eight asset classes, four decades of data, and the answer averages to zero.

This is the foundation on which modern finance was built.

We did not dispute the number. We simply asked a different question. Instead of measuring what the autocorrelation is, we measured what it does. Instead of compressing forty years into a single statistic, we watched the statistic evolve.

The Rolling Window

We took the same S&P 500 return series and computed the autocorrelation not once, but hundreds of times. Each calculation used a two-year window of daily returns. The window advanced one month at a time, producing a time series of autocorrelation values stretching from 1986 to 2026.

The full-sample statistic said zero. The rolling window told a completely different story.

Figure 1.1: Three panels for the S&P 500. Panel A shows the full-sample autocorrelation function at lags one through twenty. It is flat and unremarkable. Panel B shows the rolling two-year autocorrelation over time. The trace oscillates between positive values (positive feedback, trending) and negative values (negative feedback, oscillating), with crisis periods visible as sharp regime shifts. Panel C compares the histogram of rolling values to the theoretical distribution under independence. The actual distribution is 1.45 times wider than the random null.

Panel A is what finance sees. The full-sample autocorrelation function for the S&P 500, measured at lags one through twenty. Flat. Unremarkable. Consistent with the random walk.

Panel B is what finance has been ignoring.

The rolling two-year autocorrelation oscillates between positive 0.06 and negative 0.25. Periods of positive feedback, where moves tend to follow through, give way to periods of negative feedback, where moves tend to reverse, in a structured pattern that persists across four decades. Black Monday appears as a violent negative spike. The dot-com collapse. The Global Financial Crisis. COVID. Each crisis marks a sharp regime shift, the system slamming from one end of the spectrum to the other.

This is not a market wandering aimlessly. This is a system driven by two opposing forces: positive feedback that makes moves persist and negative feedback that makes them oscillate. When you measure it over forty years, the forces cancel. The average is zero. But the average conceals a system that never stops cycling between them.

The zero is not the absence of signal. It is the exhaust of a machine.

The Three States

Phase 2 introduces a framework that will structure every episode that follows. There are three possible states for a financial market, and the data can distinguish between them.

The first state is true equilibrium. Zero autocorrelation at every scale, in every window, at all times. This is what the Efficient Market Hypothesis predicts. If it were true, the rolling autocorrelation would be a tight band of noise centred on zero, never straying far from the theoretical boundaries of randomness.

The second state is near equilibrium. Oscillations exist, but they are damping. The system is converging on efficiency. Each decade, the swings get smaller. The engine is winding down. In this state, the zero is the future, not just the average.

The third state is far from equilibrium. Oscillations maintain or increase their amplitude. The engine is self-sustaining. The zero is not a destination. It is the centre of gravity for a system that swings permanently between trending and oscillating behaviour.

The burden of proof for this series is not merely ruling out the first state. It is distinguishing the second from the third. That distinction will be drawn in Episodes 4 and 5. But first, we must establish the foundation.

The first state is dead.

Eighty-Two Percent

The S&P 500 is not special. It is representative.

We ran the identical analysis on all sixty-eight contracts. For each market, we computed the rolling two-year autocorrelation, assembled the distribution of values, and tested whether that distribution was consistent with independence. We used the Jarque-Bera test, which examines whether the shape of the distribution matches what randomness would produce.

Eighty-two percent rejected independence. Fifty-six out of sixty-eight markets produced rolling autocorrelation distributions that are statistically impossible under the random walk. Not marginal. Not borderline. Impossible.

Figure 1.2: The dispersion ratio for all sixty-eight futures markets, ranked from lowest to highest. Each bar shows the ratio of actual variability to the theoretical variability under independence. A value of 1.0 means consistent with random. The mean across all sixty-eight markets is 1.39 times wider than random. Every asset class sits above the line.

The chart ranks all sixty-eight markets by how much wider their rolling autocorrelation is than what randomness allows. The red dashed line at 1.0 marks the boundary. Everything above it is wider than random. The mean across the entire universe is 1.39 times the random null. The median is 1.33. Reality runs nearly forty percent wider than theory permits.

And it is universal. Equities average 1.50 times. Energy runs at 1.50. Grains at 1.49. Metals at 1.42. Even the quietest asset class, foreign exchange, runs at 1.14 times the random boundary. Not a single asset class falls below.

Figure 1.3 Box plots of the dispersion ratio by asset class. Every asset class median sits above the random boundary. Equities, energy, grains, and metals show the widest dispersion. Fixed income and FX are narrower but still above it.

The box plots make the universality visible at a glance. Eight asset classes. Eight medians above the random null. The range of behaviour varies, equities and energy oscillate more violently than bonds and currencies, but the direction is the same everywhere. The machine runs in every market.

What the Full Sample Hides

This is the central insight of Episode 1. It deserves to be stated plainly.

Figure 1.4: A side-by-side comparison for all sixty-eight markets. Left: full-sample autocorrelation values, all clustered near zero. Right: the range of rolling autocorrelation values for the same markets. The mean range is 0.319, spanning more than thirty percentage points of autocorrelation that the full-sample statistic completely conceals.

On the left: the full-sample autocorrelation for every market. A cluster of small numbers hugging zero. Mean: negative 0.001. This is what the textbooks report. This is the evidence for efficiency.

On the right: the range of rolling autocorrelation values for the same sixty-eight markets. Each bar stretches from the most negative two-year window to the most positive. The mean range is 0.319. That is more than thirty percentage points of autocorrelation swing, compressed into a full-sample value that rounds to zero. But an average can conceal as much as it reveals.

The full sample does not show the absence of structure. It shows the cancellation of structure. Two opposing forces, positive feedback and negative feedback, operating in succession, averaging out over decades, producing a zero that looks like silence but is actually the sound of two forces colliding.

The Volatility Signature

There is a second confirmation hiding in the same data.

Figure 1.5: Two panels for the S&P 500. Top: rolling autocorrelation of signed returns, oscillating between positive feedback (trending) and negative feedback (oscillating). Bottom: rolling autocorrelation of absolute returns, persistently positive, confirming volatility clustering. Both are layers of memory that independence forbids.

The top panel shows the rolling autocorrelation of returns. This is the machine we have already described. Oscillating. Structured. Two opposing forces.

The bottom panel shows the rolling autocorrelation of absolute returns. It is persistently and massively positive. This is volatility clustering: the well-documented tendency for large moves to follow large moves and quiet periods to follow quiet periods. But notice that it is not constant. It surges around crises and recedes during calm. It, too, has structure.

If returns were truly independent, both panels would be flat noise near zero. The fact that absolute returns carry deep positive memory while signed returns oscillate between positive and negative tells us the underlying process has two layers of memory. One in direction. One in magnitude. Neither is consistent with independence. Together, they form the fingerprint of a complex system driven by feedback.

The Universal Distribution

To see the machine at full scale, we pooled every rolling autocorrelation observation from all sixty-eight contracts. Twenty-nine thousand data points spanning every asset class, every geography, four decades of trading.

Figure 1.6: The pooled distribution of all 29,253 rolling autocorrelation values from sixty-eight markets, overlaid with the theoretical distribution under independence. The actual distribution is 1.65 times wider than the random null. Eighty-two percent of individual markets reject independence.

The pooled dispersion is 1.65 times the independence null. The actual distribution is nearly two-thirds wider than what random data could produce, with heavier tails and more time spent at extreme values than any independent process allows.

This is not a single anomaly in a single market. This is a universal property of financial returns. The machine is everywhere.

What This Means

If you manage money, allocate capital, or construct portfolios, here is what this episode has established.

The near-zero autocorrelation in full-sample statistics is an artefact. It does not mean the market lacks structure. It means the structure oscillates between two opposing forces, positive feedback and negative feedback, and the forces cancel when you compress time. The zero is real, but it is an average, not a description.

Eighty-two percent of futures markets are statistically inconsistent with independence. The random walk is not a reasonable model for the vast majority of traded markets.

The structure is universal. It appears in every asset class, across every geography, spanning four decades. It is not an anomaly confined to equities or to liquid markets or to developed economies. It is a property of markets themselves.

But we have only begun. Ruling out the first state, true randomness, is necessary but not sufficient. The critical question is whether the engine is winding down or maintaining its force.

That question will be answered. But first, Episode 2 maps the system across all sixty-eight markets simultaneously and reveals the permanent coupling that connects Soybeans to the S&P 500.

Episode 1 established that the near-zero autocorrelation in financial markets is not evidence of randomness. It is the centre of gravity for a system driven by two opposing forces, positive feedback and negative feedback, that cancel when you average across time. Episode 2 maps that system across all sixty-eight markets simultaneously, revealing that these markets are permanently coupled along a spectrum from trending to oscillating behaviour, and that the transition between states follows an earthquake-like pattern visible in four decades of data.

The machine runs in every market. Episode 2 shows you what it looks like from above.

Endnotes

References

Full-sample first-order autocorrelation (ACF at lag 1) of daily log returns for S&P 500 E-mini futures (ES), September 1984 to January 2026. Computed as the Pearson correlation between r_t and r_t-1 across 10,441 observations. The value of -0.0715 is within the 95% confidence interval for a white noise process of this length (±0.019), though marginally outside at the 99% level. The mean full-sample ACF(1) across all sixty-eight contracts is -0.001.
Under the null hypothesis that returns are independent and identically distributed (IID), the sampling distribution of the sample autocorrelation at lag 1 is approximately Normal with mean zero and standard deviation 1/√N, where N is the window length. For our two-year (504-day) rolling windows, this gives a theoretical standard deviation of 0.0445. Values exceeding ±0.089 (two standard deviations) would occur in fewer than 5% of windows under the null.

Methodology

Jarque-Bera test applied to the empirical distribution of rolling ACF(1) values for each contract. The test assesses whether skewness and kurtosis are consistent with a Gaussian distribution. Under the IID null, the rolling ACF distribution should be approximately N(0, 1/√N). Rejection at the 5% level indicates the rolling autocorrelation distribution is inconsistent with independence. Fifty-six of sixty-eight contracts (82%) reject. When evaluated by the Kolmogorov-Smirnov test against the theoretical N(0, 0.0445) distribution, sixty-five of sixty-eight (96%) reject at the 5% level.
The persistent positive autocorrelation of absolute returns is one of the most robust stylised facts in empirical finance, first documented by Mandelbrot in 1963 and systematically catalogued by Rama Cont in “Empirical Properties of Asset Returns: Stylized Facts and Statistical Issues,” Quantitative Finance, 2001. The coexistence of flat directional autocorrelation and persistent magnitude autocorrelation is inconsistent with any IID process and implies at minimum a conditionally heteroscedastic data-generating process.
Pooled dispersion ratio computed as the standard deviation of all 29,253 rolling ACF(1) observations across sixty-eight contracts (σ = 0.0736) divided by the IID theoretical standard deviation (0.0445), giving a ratio of 1.65. Excess kurtosis of the pooled distribution is 1.93 (relative to Normal = 0), indicating heavier tails. The pooling treats each rolling window as an independent draw, which overstates independence across overlapping windows; the per-contract Jarque-Bera tests (endnote 3) provide the formal contract-level rejection rates.

Data

All data sourced from Commodity Systems Incorporated (CSI). Contracts are ratio (proportional) back-adjusted continuous futures, which preserves percentage returns across contract rolls. The universe comprises sixty-eight contracts: sixteen equity indices, eight fixed income, eight currencies, seven energy, six metals, seven grains, twelve softs, and four livestock/dairy. Sample period: September 1984 to January 2026. Returns computed as log returns: r_t = ln(Close_t / Close_t-1). Rolling windows: 504 trading days (approximately two years), advanced in 21-day (approximately one month) steps.

Figures

Figure 1.1: Three-panel chart for S&P 500 E-mini (ES). Panel A: full-sample ACF at lags 1–20. Panel B: rolling 504-day ACF(1) with monthly steps, 1986–2026. Shading: blue = positive feedback (trending), orange = negative feedback (oscillating). Dotted lines = ±2 sigma IID bounds. Crisis periods annotated. Panel C: histogram of 474 rolling ACF(1) values vs N(0, 0.0445) IID null. Dispersion ratio = 1.45×. Panel D: summary statistics.
Figure 1.2: Dispersion ratio (actual sigma of rolling ACF / IID theoretical sigma) for all sixty-eight contracts, sorted ascending. Coloured by asset class. Red dashed line at 1.0 = IID boundary.
Figure 1.3: Box plots of dispersion ratio by asset class. Medians, interquartile ranges, and outliers shown. Red dashed line at 1.0.
Figure 1.4: Left panel: full-sample ACF(1) for all sixty-eight contracts (mean = -0.001). Right panel: range of rolling ACF(1) for the same contracts, displayed as horizontal bars. White dots mark full-sample values. Mean range = 0.319.
Figure 1.5: S&P 500 rolling 504-day ACF(1) for signed returns (top) and absolute returns (bottom), 1986–2026. Signed returns oscillate around zero. Absolute returns are persistently positive with surges during crisis periods.
Figure 1.6: Pooled distribution of 29,253 rolling ACF(1) values from all sixty-eight contracts vs IID null N(0, 0.0445). Actual: μ = 0.006, σ = 0.074. Dispersion ratio = 1.65×. Excess kurtosis = 1.93.

This research series is drawn from The Fractals of Finance: Determinism, Adaptation and the Geometry of Markets

The book explores the full architecture of feedback, fat tails, and fractal structure in financial markets, and what it means for how we trade, invest, and understand risk.

Available now on Amazon in paperback, hardcover, and Kindle.

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