Stylized Facts of Financial Returns¶

Recall (see § Stock Data and Return Computation): log return series and their sample moments have already been defined. We now catalogue the empirical regularities that appear robustly across assets, markets, and time periods. These regularities — called stylized facts — are the empirical constraints any credible model must respect. The implications of each fact for modelling are stated as constraints only; the question of which mathematical framework satisfies them is taken up in § Why Deterministic Models Fail and § Bridge to Stochastic Differential Equations.

Concept Definition¶

A stylized fact is an empirical regularity of financial returns that:

1. appears consistently across many assets, markets, and sample periods,
2. is robust to the precise measurement method used,
3. cannot be explained by an i.i.d. Gaussian model, and
4. constrains the class of plausible mathematical models.

The term was popularised in finance by Rama Cont (2001) in a systematic survey of equity and FX return data. The five facts documented below are the ones most directly relevant to the SDE models developed in this book.

Stylized Fact 1: Heavy Tails (Leptokurtosis)¶

Observation¶

The marginal distribution of daily log returns has fatter tails than the Gaussian distribution with the same mean and variance. This is measured by excess kurtosis:

For a Gaussian distribution \(\kappa = 0\) by definition. Empirical estimates for major equity indices at daily frequency:

As a consequence, extreme events occur far more frequently than Gaussian models predict:

Mathematical Characterisation¶

Tail behaviour is well approximated by a power law:

with tail index \(\alpha \approx 3\)–\(5\) for daily equity returns. A Gaussian has exponentially thin tails; any finite \(\alpha\) implies dramatically heavier tails.

A tractable parametric family capturing this is the Student-\(t\) distribution with \(\nu\) degrees of freedom:

Empirical fits for equity indices typically give \(\nu \approx 4\)–\(6\).

Constraint on Models¶

Any acceptable model must produce a marginal return distribution with excess kurtosis in the empirical range above. A Gaussian innovation process (\(\kappa = 0\)) cannot satisfy this constraint.

Stylized Fact 2: Volatility Clustering¶

Observation¶

Mandelbrot (1963) noted: large changes tend to be followed by large changes, of either sign, and small changes tend to be followed by small changes.

Quantitatively:

• Log returns \(r_t\) show little or no autocorrelation: \(\operatorname{Corr}(r_t, r_{t-k}) \approx 0\) for \(k \geq 1\).
• Squared returns \(r_t^2\) (a proxy for volatility) show strong positive autocorrelation that decays slowly: \(\operatorname{Corr}(r_t^2, r_{t-k}^2) > 0\) for lags \(k\) up to 100 days or more.
• Absolute returns \(|r_t|\) show similarly persistent autocorrelation.

The combination — unpredictable direction, predictable magnitude — is the defining signature of volatility clustering.

Testing¶

Engle's ARCH test regresses \(r_t^2\) on its own lags and tests whether all slope coefficients are zero. For virtually every equity index at daily frequency, the null is rejected at \(p \ll 0.001\).

```python from statsmodels.stats.diagnostic import het_arch

def test_arch(returns, nlags=10): stat, pval, _, _ = het_arch(returns, nlags=nlags) print(f"ARCH({nlags}): stat = {stat:.2f}, p-value = {pval:.2e}") conclusion = "ARCH effects present" if pval < 0.05 else "No ARCH effects" print(f"Conclusion: {conclusion}") ```

Constraint on Models¶

Any acceptable model must reproduce positive, persistent autocorrelation of \(r_t^2\) (and \(|r_t|\)) while keeping \(\operatorname{Corr}(r_t, r_{t-k})\) negligible. A constant-volatility model cannot satisfy this constraint.

Stylized Fact 3: The Leverage Effect¶

Observation¶

There is a negative correlation between today's signed return and tomorrow's variance — specifically, \(\operatorname{Corr}(r_t, \sigma_{t+1}^2)\) — whose magnitude for large-cap equities is typically in the range \(-0.5\) to \(-0.8\). The relationship is asymmetric:

• A large negative return today substantially increases tomorrow's volatility.
• A large positive return today has a smaller, sometimes negligible, effect.

The effect was first documented by Black (1976), who attributed it to the increase in financial leverage following a price decline (equity falls, debt/equity ratio rises, making the firm riskier). An alternative explanation is the volatility feedback hypothesis: anticipated higher future volatility raises the risk premium, requiring a price fall today.

Testing¶

The standard non-parametric diagnostic computes the cross-correlation between the signed return \(r_t\) and the future squared return \(r_{t+k}^2\) at lags \(k = 1, 2, \ldots\). Negative values confirm the direction of the leverage effect.

```python import numpy as np

def leverage_test(returns, max_lag=20): """ Compute Corr(r_t, r_{t+k}^2) for k = 1, ..., max_lag.

Negative values at positive lags indicate the leverage effect:
a negative return today predicts higher variance tomorrow.

Parameters
----------
returns  : pd.Series of signed log returns
max_lag  : int, number of forward lags to compute

Returns
-------
list of float, one correlation per lag
"""
corrs = []
for k in range(1, max_lag + 1):
    r_signed  = returns.iloc[:-k].values        # r_t  (signed)
    r_sq_lead = (returns.iloc[k:].values) ** 2  # r_{t+k}^2 (variance proxy)
    corrs.append(float(np.corrcoef(r_signed, r_sq_lead)[0, 1]))
return corrs

```

Constraint on Models¶

Any acceptable model must produce a negative correlation between the signed return at time \(t\) and the conditional variance at time \(t+1\). A model in which volatility is independent of the price process cannot satisfy this constraint.

Stylized Fact 4: Absence of Return Autocorrelation¶

Observation¶

For daily and lower frequencies, log returns show no statistically significant autocorrelation:

This is the empirical content of the Efficient Market Hypothesis in its weak form: past prices contain no exploitable information about future returns.

Minor exceptions exist at very short horizons (intraday) due to bid-ask bounce and market microstructure, but these disappear at daily and lower frequencies.

The Ljung–Box test provides a formal check:

```python try: # statsmodels >= 0.13 from statsmodels.stats.diagnostic import acorr_ljungbox except ImportError: from statsmodels.tsa.stattools import acorr_ljungbox

def test_return_autocorrelation(returns, lags=20): result = acorr_ljungbox(returns, lags=lags, return_df=True) n_sig = (result["lb_pvalue"] < 0.05).sum() print(f"Ljung-Box: {n_sig}/{lags} lags significant at 5 %") ```

For most equity indices fewer than 1–2 lags out of 20 will be significant, consistent with no systematic autocorrelation.

Constraint on Models¶

Any acceptable model must produce returns with no exploitable serial correlation at daily or lower frequencies.

Stylized Fact 5: Aggregational Gaussianity¶

Observation¶

Daily log returns are non-Gaussian (heavy tails, negative skewness). As we aggregate over longer horizons, the distribution becomes progressively more Gaussian. The kurtosis table in Fact 1 shows that the full S&P 500 history reaches excess kurtosis ≈ 20; at shorter post-2000 samples the daily figure is ≈ 9, and it decreases steadily with horizon:

This convergence is predicted by the Central Limit Theorem: the sum of \(n\) weakly-dependent random variables with finite variance converges to a Gaussian as \(n \to \infty\).

Constraint on Models¶

Any acceptable model must reproduce the systematic decrease of excess kurtosis with aggregation horizon, while remaining accurate at the daily scale where non-Gaussianity is most pronounced.

Summary of Empirical Constraints¶

These are the five empirical bars that the modelling framework introduced in later sections must clear.

Conclusion¶

The stylized facts reveal that financial returns are:

• non-Gaussian (heavy tails, negative skewness),
• conditionally heteroskedastic (volatility clusters),
• asymmetrically responsive to shocks (leverage effect),
• unpredictable in direction (no return autocorrelation), and
• approximately Gaussian at long horizons (aggregational Gaussianity).

A model that ignores these facts will produce incorrect risk measures, mispriced options, and unreliable simulations. These five constraints are the empirical bar that any candidate framework must clear.

Next: § Why Deterministic Models Fail. \(\square\)

Exercises¶

Exercise 1. The excess kurtosis of a Student-\(t\) distribution with \(\nu > 4\) degrees of freedom is \(6/(\nu - 4)\). If an empirical estimate of excess kurtosis for daily equity returns is \(\hat{\kappa} = 6\), what value of \(\nu\) does this imply? For this \(\nu\), does the fourth moment of the Student-\(t\) distribution exist? What happens to the theoretical kurtosis as \(\nu \to 4^+\)?

Exercise 2. Suppose daily log returns \(\{r_t\}\) satisfy \(\operatorname{Corr}(r_t, r_{t-k}) = 0\) for all \(k \geq 1\), but \(\operatorname{Corr}(r_t^2, r_{t-1}^2) = 0.25\). Explain why these two conditions are not contradictory. Write down a simple GARCH(1,1) model

and verify that \(\mathbb{E}[r_t \mid \mathcal{F}_{t-1}] = 0\) while \(\operatorname{Var}(r_t \mid \mathcal{F}_{t-1}) = \sigma_t^2\) varies over time.

Exercise 3. The tail of a distribution follows a power law \(P(|r| > x) \sim C x^{-\alpha}\) with \(\alpha = 4\). A Gaussian distribution with the same variance \(\sigma^2\) has \(P(|r| > x) \approx \frac{\sigma}{x\sqrt{2\pi}} e^{-x^2/(2\sigma^2)}\) for large \(x\). Taking \(\sigma = 0.01\) (daily), compute both probabilities at \(x = 5\sigma = 0.05\) (choose \(C\) so that the power-law probability at \(x = 3\sigma\) matches the Gaussian probability at \(x = 3\sigma\)). How many times more likely is a \(5\sigma\) event under the power-law tail than under the Gaussian?

Exercise 4. The leverage effect is characterised by \(\operatorname{Corr}(r_t, \sigma_{t+1}^2) < 0\). In the Heston model the correlation between the Brownian motions driving the price and variance processes is \(\rho\). Explain qualitatively why \(\rho < 0\) generates the leverage effect. Specifically, if \(dS_t = \mu S_t\,dt + \sqrt{V_t}\,S_t\,dW_t^S\) and \(dV_t = \kappa(\theta - V_t)\,dt + \xi\sqrt{V_t}\,dW_t^V\) with \(\operatorname{Corr}(dW_t^S, dW_t^V) = \rho\,dt\), trace through what happens to \(V_t\) when \(S_t\) experiences a large negative shock (i.e., \(dW_t^S \ll 0\)) and \(\rho = -0.7\).

Exercise 5. Aggregational Gaussianity states that excess kurtosis decreases with the return horizon. Under i.i.d. assumptions, if a daily return has excess kurtosis \(\kappa_1\), show that the \(n\)-day return (sum of \(n\) daily returns) has excess kurtosis

Using this formula, if \(\kappa_1 = 8\) at daily frequency, compute the excess kurtosis at weekly (\(n = 5\)), monthly (\(n = 21\)), and quarterly (\(n = 63\)) horizons. At what horizon does the excess kurtosis drop below 0.5?

Exercise 6. You are given a time series of 2520 daily log returns (approximately 10 years). Describe step-by-step how you would test for each of the five stylized facts documented in this section. For each fact, state: (a) the specific quantity you would compute or the test you would run, (b) the null hypothesis, and (c) what outcome would confirm the presence of the stylized fact. You do not need to perform the computations; outline the methodology only.