Stylized Facts

The empirical phenomenon where large price changes tend to be followed by large changes and small changes by small changes, creating persistent periods of high and low turbulence. This contradicts the assumption of constant variance in simple random walk models.

• First documented by Mandelbrot (1963) in cotton prices
• Measured by autocorrelation of squared returns
• Implies volatility is predictable in the short term
• Captured by GARCH and stochastic volatility models
• Synthetic data must exhibit clustering to pass validation

Financial returns exhibit leptokurtosis — a higher probability of extreme events than predicted by a normal distribution. The tails follow a power law decay, meaning 3-sigma events occur far more frequently than Gaussian models suggest.

• Daily S&P 500 returns show kurtosis of ~10-30 vs. 3 for normal
• Tail index α typically between 3 and 5 for equity returns
• Critical for Value at Risk (VaR) and CVaR calculations
• Synthetic generators must reproduce tail behavior exactly
• Related: Mandelbrot's stable Paretian hypothesis

Linear autocorrelation of raw returns is effectively zero for liquid instruments beyond very short horizons, consistent with the efficient market hypothesis. However, this does not imply independence — nonlinear dependencies abound.

• Daily return autocorrelation typically < 0.05
• Violated at very high frequencies due to bid-ask bounce
• Does NOT contradict volatility clustering
• Implies linear forecasting of returns is futile
• Synthetic data must show this decorrelation property

A negative correlation between asset returns and subsequent volatility: volatility tends to rise when prices fall and decline when prices rise. First identified by Black (1976), this asymmetry is a universal feature of equity markets.

• Correlation between returns and volatility changes: typically -0.3 to -0.5
• Explained by financial leverage increasing equity risk when firm value drops
• Also attributed to volatility feedback effects
• Captured by asymmetric GARCH models (EGARCH, GJR-GARCH)
• Less pronounced in commodities and FX markets

The autocorrelation function of absolute or squared returns decays hyperbolically rather than exponentially, indicating long-range dependence in volatility. This persistence spans weeks to months, far longer than short-memory models predict.

• Hurst exponent for volatility typically 0.6-0.8 (vs. 0.5 for random walk)
• Implies volatility shocks have very long half-lives
• Modeled by fractionally integrated GARCH (FIGARCH)
• Contrasts with rapid decay of return autocorrelation
• Synthetic generators must reproduce this slow decay

Large positive returns and large negative returns exhibit different statistical properties. Drawdowns (cumulative losses) tend to be faster and steeper than recoveries, creating an asymmetry in the temporal structure of extreme moves.

• Market crashes are typically sharper than rallies
• Drawdown distributions follow different scaling laws than drawups
• Related to the leverage effect but distinct in mechanism
• Observed across all major asset classes
• Critical for tail risk hedging strategy design

The empirical observation that large price changes tend to be followed by large changes, and small changes by small ones, creating persistent periods of high and low turbulence. This autocorrelation of absolute returns contradicts the assumption of constant variance in simple random walk models.

• Measured by autocorrelation functions on squared or absolute returns
• Driven by information arrival clustering and leverage effects
• A critical stylized fact that any realistic synthetic market generator must reproduce
• Failure to capture clustering leads to underestimation of sequential risk

A probability distribution where extreme events have significantly higher likelihood than predicted by a Gaussian (normal) distribution. Financial returns exhibit leptokurtosis, meaning the fourth moment (kurtosis) exceeds 3.

• Captured by distributions like Student's t, Generalized Pareto, or α-stable
• Responsible for the frequency of Black Swan events in markets
• Synthetic data must match the tail index of real returns to be useful for tail risk hedging
• Underestimating tails leads to catastrophic strategy failure in live trading

The negative correlation between an asset's returns and its subsequent volatility. When prices fall, volatility tends to rise more than it would after an equivalent price increase.

• Originates from the mechanical increase in financial leverage as equity values decline
• Also driven by the volatility feedback effect where anticipated higher volatility depresses prices
• Captured in models like EGARCH and GJR-GARCH with asymmetric terms
• Essential for generating realistic synthetic paths where crashes amplify turbulence

A modeling paradigm where volatility paths exhibit Hölder regularity significantly less than 0.5, meaning they are rougher than Brownian motion. Empirical studies show the log-volatility of financial assets behaves like a fractional Brownian motion with a Hurst exponent around 0.1.

• Explains the observed behavior of the ATM volatility skew
• Provides superior fit to VIX dynamics and option surfaces
• Synthetic market generators must incorporate roughness to produce realistic volatility-of-volatility patterns
• Challenges traditional diffusion-based stochastic volatility models

A self-exciting point process where the arrival of an event increases the probability of subsequent events in the near future. The intensity function depends on the history of past events through a triggering kernel.

• Models clustered trade arrivals and order flow autocorrelation
• Captures the empirical observation that trades and quote updates arrive in bursts
• Used to simulate realistic limit order book dynamics in synthetic environments
• The branching ratio parameter controls the degree of endogeneity vs. exogenous excitation