Why These Properties Matter
Model Selection
The empirical properties of returns—volatility clustering, fat tails, and negative skewness—determine which stochastic models are appropriate. GBM assumes none of these, while Heston captures clustering, jump-diffusion models capture fat tails, and Lévy processes offer flexible tail behavior.
Risk Management
The tail quantiles (1% and 99%) show the magnitude of extreme daily moves. Fat tails mean these extremes occur more frequently than a normal distribution predicts—critical for VaR calculations and portfolio risk assessment.
Volatility Dynamics
The autocorrelation of squared returns reveals volatility clustering: periods of high volatility tend to persist. This motivates stochastic volatility models like Heston that explicitly model time-varying volatility.
Option Pricing
The implied volatility smile observed in options markets reflects these empirical properties. Models that capture skewness and kurtosis produce more realistic option prices than Black-Scholes.
Explore how different stochastic models capture these properties: View all models →