The Stylized Facts a Generator Must Reproduce

Suppose you build a synthetic-data generator, point it at a decade of S&P 500 returns, and it dutifully learns that the mean daily return is about 0.03% and the daily standard deviation is about 1%. You let it spit out a fresh decade of “market” returns that match those two numbers exactly. Congratulations — you have produced garbage that no trader would recognise as a market.

Why? Because the mean and the variance are only the first two moments of a distribution, and markets live in the higher moments and in the time structure. A series can have the right average and the right wobble while having no crashes, no calm-then-storm rhythm, no memory, and a perfectly symmetric up/down profile. It would price options wrong, size risk wrong, and backtest strategies into a fantasy. A real market is not “random noise with the right volume knob” — it has a fingerprint.

That fingerprint is a short list of stylized facts: empirical regularities that show up in every liquid market — equities, FX, commodities, crypto — across decades and across continents, no matter the asset. They were catalogued by Mandelbrot, Fama, Cont and others, and they are remarkably universal. For us they are not trivia: they are the rubric. Every generator you meet later in this course — the classical Geometric Brownian Motion of lesson 3, and the GANs, VAEs and diffusion models of lessons 4–6 — gets graded against exactly this checklist. So learn the rubric before you meet the contestants.

Notice the scoreboard: a plain Gaussian random walk passes essentially one of the six, and even that one only by accident. Hold that thought — we’ll return to the full scorecard at the end. First, one section per fact, each with the test you’d actually run to grade a generator on it.

Fat tails (excess kurtosis)

Analogy. A normal distribution is a cautious airline that promises a delay “almost never.” Real markets are an airline that promises the same — and then strands you on the tarmac twice a year. The average delay can match the brochure while the catastrophes are wildly more frequent.

Definition. Kurtosis is the standardized fourth moment, $\kappa = \mathbb{E}\!\left[\left(\frac{r-\mu}{\sigma}\right)^4\right]$. A normal distribution has $\kappa = 3$, so analysts quote excess kurtosis $\kappa - 3$ (zero for a Gaussian). Daily equity returns routinely show excess kurtosis of 5 to 30 — the tails are not a little heavy, they are heavy.

Worked example — the 25,000-year storm. Under a normal model with daily $\sigma = 1\%$, a $-5\sigma$ day means a $-5\%$ move. The probability of a $5\sigma$ event in a Gaussian is about $2.87 \times 10^{-7}$. So the expected wait between such days is

$\frac{1}{2.87 \times 10^{-7}} \approx 3{,}480{,}000 \text{ days} \approx 13{,}800 \text{ years (one tail)},$

and roughly 25,000 years if you only care about the down tail at that exact threshold across the calendar. Yet markets deliver $\pm 5\sigma$ days every few years — October 1987 was a $-20\sigma$+ event under the Gaussian, which the normal model rates at “once per many times the age of the universe.” When your model says “geological epoch” and reality says “Tuesday,” your tails are too thin.

When to use it

To test a generator for fat tails: compute the sample excess kurtosis of its returns and compare it to the real series (real should be well above 0; a Gaussian generator sits near 0). Better, compare the empirical tail quantiles — e.g. the 0.1% worst day — or fit a generalized Pareto / power-law to the tail and check the tail index. A QQ-plot of synthetic vs. real returns makes thin tails jump out: the points bend away from the diagonal at the extremes.

Volatility clustering

Analogy. Earthquakes don’t arrive on a metronome — they come in swarms. One big tremor and the ground keeps shaking with aftershocks for days, then it goes quiet for a long stretch. Volatility is exactly that: storms cluster, calm clusters.

Definition. Big moves tend to follow big moves and small moves follow small moves, regardless of sign. Formally: the autocorrelation of raw returns $r_t$ is approximately zero at all lags (markets are roughly unpredictable in direction), but the autocorrelation of squared or absolute returns is significantly positive for many lags:

$\rho_{|r|}(k) = \operatorname{Corr}\!\left(|r_t|,\, |r_{t-k}|\right) > 0 \quad\text{for } k \ge 1.$

This is precisely the regularity that GARCH (from Time-Series Finance) was built to capture: today’s variance depends on yesterday’s squared shock and yesterday’s variance, so shocks persist.

Worked example — an ACF table. Compute autocorrelations at lags 1, 5 and 10 for a real daily series:

The raw column hovers around 0 (no directional predictability). The squared column is clearly positive and only fades slowly — that gap is volatility clustering. A series whose squared-return ACF is also ~0 has no clustering: its big days are sprinkled at random instead of bunching into storms.

When to use it

To test for clustering: plot the ACF of $r_t$ and of $r_t^2$ (or $|r_t|$) side by side, with confidence bands. Real and good-generator series show a flat raw ACF and a clearly positive squared ACF. Quantitatively, run a Ljung–Box test on squared returns (or an ARCH-LM test): a small p-value confirms clustering. Run it on both the real and synthetic series and compare.

Slow decay of absolute-return autocorrelation (long memory)

Analogy. A struck church bell doesn’t go silent the instant the next pew is empty; it rings on, the hum fading over a long, slow tail. Market volatility rings like that bell — its echo in absolute returns persists far longer than any short-memory model expects.

Definition. Beyond merely being positive, the autocorrelation of absolute (and squared) returns decays slowly with lag — empirically close to a power law, $\rho_{|r|}(k) \sim k^{-\gamma}$ with a small $\gamma$ (often $0.2$–$0.4$), rather than the fast geometric decay $\rho(k) \sim \phi^{k}$ of a short-memory process. This is the long-memory or long-range dependence of volatility: the past influences the present over very long horizons.

Worked example — raw vs. absolute ACF across lags. Contrast the two series at increasing lags:

The raw-return ACF is statistical noise around zero at every lag — flat from the start. The absolute-return ACF starts at 0.30 and is still about 0.20 at lag 20 and 0.14 at lag 50. A short-memory (geometric) decay from 0.30 with $\phi = 0.8$ would already be down to $0.30 \times 0.8^{20} \approx 0.0035$ by lag 20 — effectively gone. Reality stays loud; that persistence is the long memory.

When to use it

To test for long memory: plot $\rho_{|r|}(k)$ out to large lags (50–100) on a log–log scale; a roughly straight line of small slope signals power-law decay. Estimate the long-memory exponent directly — e.g. the Hurst exponent $H$ of absolute returns (real volatility gives $H > 0.5$), via R/S analysis or a GPH/log-periodogram estimator. Compare the synthetic series’ exponent and full ACF curve to the real one.

The leverage effect

Analogy. Good news lifts a stock the way a calm tide rises — smoothly. Bad news drops it the way a fire alarm clears a building: shoving, shouting, chaos. The panic on the way down stirs far more turbulence than the cheer on the way up.

Definition. Volatility tends to increase more following price declines than following price increases of equal magnitude — there is a negative correlation between current returns and future volatility, $\operatorname{Corr}(r_t,\, \sigma_{t+1}) < 0$. Classically attributed to financial leverage (a price drop raises the firm’s debt-to-equity ratio, making equity riskier), hence the name — though the effect is larger than leverage alone explains.

Worked example — post-up vs. post-down realized vol. Split days by the sign of the prior day’s return and measure the next day’s realized volatility (annualized):

Same-sized shocks, very different aftermath: a down day is followed by volatility about $21/14 \approx 1.5\times$ higher than after an up day. A symmetric model (GARCH, GBM) would give roughly equal numbers in both rows; the gap is the leverage effect. Asymmetric models like EGARCH or GJR-GARCH were invented precisely to capture this.

When to use it

To test for the leverage effect: estimate the cross-correlation $\operatorname{Corr}(r_t,\, |r_{t+k}|)$ or $\operatorname{Corr}(r_t,\, r_{t+k}^2)$ for $k \ge 1$ — real markets give significantly negative values, a generator without the effect gives ~0. Or simply bin next-day realized vol by the sign of today’s return (as in the table) and check for a gap. In option language, the effect shows up as a negative volatility skew.

Aggregational Gaussianity

Analogy. Up close, a coastline is jagged with violent cliffs and inlets. Zoom out to a map of the whole continent and the wild detail smooths into a gentle curve. Aggregate returns over longer windows and the fat-tailed jaggedness of daily moves smooths toward the tidy bell curve.

Definition. As the sampling interval lengthens — daily → weekly → monthly — the distribution of returns approaches a normal distribution, and the excess kurtosis decays toward 0. This is a Central-Limit-Theorem-flavoured effect: a monthly return is a sum of ~21 daily returns, and sums of (weakly dependent) shocks drift toward Gaussian. Crucially, this is the one fact a Gaussian random walk trivially “passes” — because it was Gaussian at every horizon to begin with.

Worked example — kurtosis falling with horizon. Take a daily series with excess kurtosis 8. If daily returns were independent, summing $n$ of them divides excess kurtosis by $n$: a 21-day month gives $8 / 21 \approx 0.38$. Real returns aren’t fully independent (volatility clustering slows the convergence), so monthly excess kurtosis lands a bit higher — empirically around 1 — but the direction is unmistakable:

From 8 to about 1 as you zoom out — the tails deflate. A generator that’s Gaussian everywhere matches the monthly shape fine; the test is whether it also has the fat daily tails it’s supposed to converge from.

When to use it

To test for aggregational Gaussianity: compute excess kurtosis (and run a normality test such as Jarque–Bera) at several horizons — daily, weekly, monthly — and confirm the value trends downward toward 0 as the window grows. A good generator reproduces the whole curve: fat at daily, near-normal at monthly. Matching one horizon is not enough; it’s the shape of the convergence that must agree.

Gain/loss asymmetry

Analogy. The market climbs the stairs and rides the elevator down. Gains are a patient hike — one step at a time over weeks. Losses are a trapdoor: a cliff that opens in days. Symmetric in size, wildly asymmetric in speed and shape.

Definition. Large drawdowns tend to be sharper and faster than run-ups of equal magnitude, producing negative skewness in the return distribution, $\operatorname{skew}(r) < 0$. Equivalently (the “gain/loss asymmetry” of Johansen–Sornette): the typical waiting time to reach a given negative cumulative return is shorter than the waiting time to reach the same positive return.

Worked example — time to ±20%. Track how long an index typically takes to move a cumulative 20% from a starting point:

Same 20% magnitude, but the drop gets there in roughly a third of the time. That speed asymmetry leaves a fingerprint of negative skew (here, say, $\operatorname{skew} \approx -0.6$ on daily returns): the left tail is longer and more abrupt than the right.

When to use it

To test for gain/loss asymmetry: compute the sample skewness of returns (real equities give a small negative value) and compare to the generator’s. More directly, measure the first-passage time distribution to $+\theta$ versus $-\theta$ for a threshold $\theta$ (e.g. 20%) and check that the down-side hits sooner. Drawdown statistics — maximum drawdown, drawdown duration — are the practitioner’s version of the same audit.

The whole scorecard

Now total it up. Line the six facts against a plain Gaussian random walk — the lesson-3 baseline (GBM with constant volatility) — and the grade is brutal:

One out of six, and the one it passes it passes for the wrong reason. A Gaussian random walk is not a market — it’s the null hypothesis a market is defined against. This is exactly why lesson 3’s classical models (GBM, and the patched-up jump-diffusion / stochastic-vol families that try to bolt some facts back on) are judged against this table, and why the learned generators of lessons 4–6 (GANs, VAEs, diffusion) live or die on the same six rows. When someone shows you a shiny new synthetic-data model, you now know the six questions to ask before you believe a word of it.

Recap

Stylized facts are the universal fingerprint of liquid markets — empirical regularities every asset class shares, sitting in the higher moments and the time structure that mean and variance ignore. The big six: fat tails (excess kurtosis 5–30, crashes far more frequent than Gaussian), volatility clustering (positive squared-return ACF while raw-return ACF ≈ 0), slow decay / long memory (absolute-return ACF fading like a power law, still positive at lag 20+), the leverage effect (volatility rises more after drops — negative return/future-vol correlation), aggregational Gaussianity (tails deflate toward normal as you sum over longer horizons), and gain/loss asymmetry (negative skew — markets fall faster than they rise). Each comes with a concrete test — kurtosis and QQ-plots, Ljung–Box on squared returns, log–log ACF and Hurst exponents, return/future-vol cross-correlation, kurtosis-vs-horizon curves, and skew / first-passage times. A plain Gaussian random walk flunks five and a half of the six. This table is the rubric every generator in this course will be graded against — memorize it.