October 15, 2023

In the intricate world of finance, a profound understanding of asset behavior is crucial for investors, traders, and economists. Financial assets, ranging from stocks and bonds to commodities, demonstrate unique patterns and characteristics often referred to as “stylized facts.” These stylized facts offer invaluable insights into the intricate nature of asset dynamics and play an instrumental role in guiding investment decisions. In this article, we will delve into these key stylized facts, reinforced by mathematical equations, to unveil the fascinating universe of financial markets in greater detail. Returns Distribution The distribution of asset returns serves as the foundation for comprehending the dynamics of financial markets. Contrary to the expectations set by classical finance theory, empirical observations frequently reveal that asset returns do not adhere to a normal distribution. Instead, they often exhibit fat-tailed distributions, signifying that extreme events occur more frequently than predicted. To model these non-normal distributions, the Student’s t-distribution is frequently employed, introducing the degrees of freedom (ν) parameter: Volatility Clustering Volatility clustering is a phenomenon where periods of heightened volatility tend to cluster together, followed by periods of relative calm. This pattern is accurately captured by the Autoregressive Conditional Heteroskedasticity (ARCH) model, pioneered by Robert Engle: Here, Leverage Effect The leverage effect portrays a negative correlation between asset returns and changes in volatility. When asset prices decline, volatility tends to rise. This phenomenon is aptly described by the GARCH (Generalized Autoregressive Conditional Heteroskedasticity) model: In this context, γ embodies the leverage effect. Serial Correlation Serial correlation, or autocorrelation, is the tendency of an asset’s returns to exhibit persistence over time. Serial correlation can be measured through the autocorrelation function (ACF) or the Ljung-Box Q-statistic: Here, Tail Dependence Tail dependence quantifies the likelihood of extreme events occurring simultaneously. This concept is of paramount importance in portfolio risk management. Copula functions, such as the Clayton or Gumbel copulas, are utilized to estimate the tail dependence coefficient (TDC): For the Clayton copula: For the Gumbel copula: Mean Reversion Mean reversion is the tendency of asset prices to revert to a long-term average or equilibrium level over time. This phenomenon suggests that when an asset’s price deviates significantly from its historical average, it is likely to move back toward that average. The Ornstein-Uhlenbeck process is a mathematical model that describes mean reversion: Where: Volatility Smile and Skew The volatility smile and skew refer to the implied volatility of options across different strike prices. In practice, options markets often exhibit a smile or skew in implied volatility. This means that options with different strike prices have different implied volatilities. The Black-Scholes model, when extended to handle such scenarios, introduces the concept of volatility smile, and skew. Long Memory Long memory, also known as long-range dependence, describes the persistence of past price changes in asset returns. This suggests that asset returns exhibit memory of past price movements over extended time horizons. The Hurst exponent (H) is often used to measure long memory in asset returns, with 0.5<H<1 indicating a positive long memory. Jumps and Leptokurtosis Asset returns frequently exhibit jumps or sudden large price movements. These jumps can lead to leptokurtic distributions, where the tails of the return distribution are thicker than a normal distribution. The Merton Jump-Diffusion model is used to capture this behavior, adding jumps to the standard geometric Brownian motion model: Where: