Stylized Facts of Assets: A Comprehensive Analysis

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:

f(x|μ,σ²,ν) = \frac{Γ((ν+1)/2)}{Γ(ν/2)√(πνσ²)} * (1 + \frac{(x - μ)²}{νσ²})^(-(ν+1)/2)

import numpy as np
from scipy.stats import t

# Simulated or historical returns data
returns = np.random.standard_t(df=3, size=1000)

# Fit a Student's t-distribution
params = t.fit(returns)

# Calculate the excess kurtosis (a measure of fat tails)
excess_kurtosis = t.stats(params[0], moments='k') - 3

if excess_kurtosis > 0:
    print("The distribution has fat tails.")
else:
    print("The distribution has normal tails.")

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:

σ²_t = ω + αε²_(t-1) + βσ²_(t-1)

Here,

σt² signifies the conditional variance of returns at time t.
ω represents a constant.
α indicates the short-term memory of the process.
β symbolizes the long-term memory of the process.
ε_(t-1) represents the innovation (residual) at time t-1.

import pandas as pd
from arch import arch_model

# Simulated or historical returns data
returns = pd.Series(np.random.randn(1000))

# Estimate GARCH(1,1) model
model = arch_model(returns, vol='Garch', p=1, q=1)
results = model.fit()

# Print model summary to check for volatility clustering
print(results.summary())

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:

σ²_t = ω + (α + γε_(t-1)²)σ²_(t-1) + βσ²_(t-1)

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:

Q(k) = n(n + 2) ∑(j=1)^k (\frac{r²(j)}{n - j})

import numpy as np
from statsmodels.tsa.stattools import acf, q_stat

# Simulated or historical returns data
returns = np.random.randn(1000)

# Calculate ACF
acf_values = acf(returns)

# Perform Ljung-Box test
lb_stat, lb_p_value = q_stat(acf_values[1:], len(returns))

if lb_p_value < 0.05:
    print("Serial correlation is present.")
else:
    print("No significant serial correlation.")

Here,

Q(k) stands for the Ljung-Box statistic.
n represents the sample size.
r(j) signifies the ACF at lag j.
k denotes the number of lags under consideration.

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:

TDC = 1 + \frac{1}{α}

For the Gumbel copula:

TDC = \frac{6}{1 - θ}

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:

dX_t = θ(μ - X_t)dt + σdW_t

Where:

X_trepresents the asset’s price.
θ is the speed of mean reversion.
μ is the long-term mean.
σ is the volatility.
dW_t is a Wiener process (Brownian motion).

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:

dX_t = μX_tdt + σX_tdW_t + JdN_t