2.5 The stylized facts

2.5.1Cont's eleven facts#

Cont's list (Quantitative Finance 1(2):223–236): (1) absence of linear autocorrelation, except at intraday scales; (2) heavy tails, with tail index between 2 and 5; (3) gain/loss asymmetry (large drawdowns without equally large upward moves); (4) aggregational Gaussianity (the distribution approaches normal as the horizon lengthens); (5) intermittency (bursts of large changes at all scales); (6) volatility clustering; (7) conditional heavy tails (residuals remain heavy-tailed after a GARCH-type correction); (8) slow decay of the autocorrelation of absolute returns, roughly a power law with exponent between 0.2 and 0.4; (9) the leverage effect (volatility is negatively correlated with past returns); (10) volume/volatility correlation; (11) asymmetry in time scales (coarse-grained volatility predicts fine-scale volatility better than the reverse).

2.5.2The facts in Indian data#

Evidence on the Nifty sample (daily, 2007–2026)

Fact	Statistic	Nifty value	Under iid normal
(1) no autocorrelation	$\hat\rho_1, \dots, \hat\rho_5$ of $r$	0.037, −0.005, −0.025, −0.024, 0.004	$\pm 0.029$ (2 s.e.)
(2) heavy tails	excess kurtosis	15.2	0 ± 0.14
(2) heavy tails	days with $\lvert z\rvert > 4$	38	0.30 expected
(2) heavy tails	days with $\lvert z\rvert > 5$	12	0.003 expected
(2) heavy tails	Hill $\hat\alpha$, $k = 100$, both tails	3.05	$\to \infty$
(3) asymmetry	skewness; Hill left vs right ($k = 100$)	−0.28; 2.61 vs 2.86	0
(6), (8) clustering	$\hat\rho$ of $\lvert r\rvert$ at lags 1, 5, 10, 20, 50, 100	0.28, 0.27, 0.28, 0.22, 0.15, 0.14	0 ± 0.03
(6) clustering	annual volatility by year	9.0% (2017) to 44.7% (2008)	constant

Two features deserve emphasis. The autocorrelation of $|r|$ is still 0.14 at a lag of 100 trading days, which is five months: volatility has long memory. And the largest standardised moves were $-10.8\sigma$ (23 Mar 2020), $-10.1\sigma$ (24 Oct 2008) and $+12.6\sigma$ (18 May 2009, the first session after the 2009 general-election results, when the index hit its upper circuit), measured with the full-sample standard deviation. Under a normal model a $10\sigma$ day has probability about $10^{-23}$.

The existing exercise file exercises/modules/M02-statistics-of-returns/exercises/m02_returns_stats.py (functions excess_kurtosis, autocorr, vol_clustering, tail_ratio, looks_like_real_market) implements these diagnostics; its tests check that they separate GARCH-with-jumps synthetic data from GBM. Run them on a cached Nifty series (toolkit.data.load("NIFTY", source="cache"), from NSE's index files) to reproduce the table.