Introduction
Financial market volatility plays a critical role in risk assessment for asset portfolios. Traditional models relying on normal distribution assumptions often fall short in capturing the unique characteristics of asset returns—such as volatility clustering, conditional heteroskedasticity, and long-memory effects. This study addresses these gaps by proposing an optimized Generalized Autoregressive Conditional Heteroskedasticity (GARCH) framework under non-normal distribution assumptions, specifically tailored for cryptocurrency markets.
Why Cryptocurrencies?
The cryptocurrency market—with its 24/7 trading, high volatility, and sensitivity to global events—presents both challenges and opportunities for investors. Focusing on Bitcoin (BTC), Ethereum (ETH), Cardano (ADA), Binance Coin (BNB), and Tether (USDT) (which collectively represent ~80% of the market), we analyze daily logarithmic returns to model and predict volatility patterns.
Methodology
1. Asset Returns and Key Features
- Logarithmic Returns: Transform closing prices into daily returns to normalize data and enhance statistical properties.
- Non-Normal Distributions: Cryptocurrency returns exhibit skewness, leptokurtosis (fat tails), and asymmetric volatility (e.g., "leverage effect" where negative shocks impact prices more severely than positive ones).
2. GARCH Models with Generalized Error Distribution (GED)
We extend the standard GARCH(p,q) model by assuming returns follow a Skewed GED (SGED), which accounts for:
- Tail thickness (shape parameter p).
- Skewness (parameter λ).
Density Function of SGED:
$$ f(x) = \frac{v}{2σΓ(1/v)} \exp\left(-\frac{|x-m|^v}{2σ^v (1+\text{sign}(x-m)λ)^v}\right) $$
where:
- v controls tail thickness.
- λ adjusts skewness (λ ≠ 0 introduces asymmetry).
3. ARIMA-EGARCH Hybrid Model
To capture both mean dynamics (via ARIMA) and volatility clustering (via EGARCH), we combine:
- ARIMA(p,d,q): For modeling autocorrelations in returns.
- EGARCH(p,q): For asymmetric volatility responses (e.g., leverage effect).
EGARCH Equation:
$$ \ln(σ_t^2) = ω + \sum_{i=1}^p α_i \left| \frac{ε_{t-i}}{σ_{t-i}} \right| + \sum_{j=1}^q β_j \ln(σ_{t-j}^2) + γ \frac{ε_{t-1}}{σ_{t-1}} $$
Key insight: γ < 0 confirms the leverage effect.
Empirical Analysis
Data and Preprocessing
- Source: Daily closing prices from Investing.com.
- Period: 2017–2022 (BTC, ETH, ADA, BNB, USDT).
Tests:
- Stationarity (ADF test: p < 0.05 for all assets).
- ARCH Effects (Ljung-Box test: p < 0.01).
- Non-Normality (Jarque-Bera test: p < 0.001).
Model Performance
| Cryptocurrency | Log-Likelihood | AIC | ARCH Test (p-value) |
|----------------|---------------|-------|---------------------|
| BTC | 2,450 | -4.12 | 0.21 |
| ADA | 1,980 | -3.89 | 0.34 |
Findings:
- EGARCH(1,1) effectively eliminates ARCH effects (except for BNB).
- Negative γ values confirm leverage effects in BTC and ADA.
Forecasting Results
50-Day Ahead Predictions:
- BTC: Stable returns (~0% fluctuation).
- ADA: Mild downward trend.
Rolling Window Predictions (500-day window):
- BTC: Positive returns dominate.
- ADA: Negative returns prevail.
(Example visualization of BTC predictions)
Conclusion
Key Insights:
- Cryptocurrency returns require non-Gaussian models (SGED).
- ARIMA-EGARCH hybrids outperform standalone models in volatility forecasting.
Practical Applications:
- Helps investors hedge against downside risks.
- Informs portfolio diversification strategies.
👉 Explore real-time crypto volatility tools
FAQ
Q1: Why use SGED instead of normal distributions?
A: Cryptocurrency returns exhibit skewness and fat tails—SGED captures these features better than Gaussian models.
Q2: How does the leverage effect impact predictions?
A: Negative shocks (e.g., regulatory news) cause larger volatility spikes than positive ones, modeled via EGARCH’s γ parameter.
Q3: Can this model predict extreme market crashes?
A: While useful for short-term trends, tail-risk prediction requires integrating extreme-value theory (e.g., POT models).