Understanding different variants of GARCH Models in Volatility Modelling

Volatility is a fundamental aspect of financial time series data, influencing risk management, option pricing, and portfolio optimization. Generalized Autoregressive Conditional Heteroskedasticity (GARCH) models provide a robust framework for modeling and forecasting volatility. These models build on the assumption that volatility is time-varying and can be predicted using past information. In this

comprehensive guide, we will explore different variants of GARCH models, their mathematical formulations, and implementation guidelines, and discuss their limitations and advancements.

Underlying Assumption

The underlying assumption in GARCH models is that volatility is conditional on past observations. Specifically, it assumes that the conditional variance σt²​ of a financial time series at time t depends on past squared returns and past conditional variances.

GARCH(1,1) Model

The GARCH(1,1) model is one of the most widely used variants and is expressed as follows:

\sigma_t^2 = \alpha_0 + \alpha_1 r_{t-1}^2 + \beta_1 \sigma_{t-1}^2

• σt^2​ is the conditional variance at time t.
• α_0​ is a constant term representing the baseline level of conditional variance.
• α₁​ and β₁​ are parameters governing the impact of past squared returns and past conditional variances on σt²​.
• r_t−12​ is the squared return at time t−1.
• σ_t−12​ is the conditional variance at time t−1.

GARCH(p, q) Model

The GARCH(p, q) model is a more general version allowing for more lags in both the squared returns and conditional variances. It is expressed as:

\sigma_t^2 = \alpha_0 + \sum_{i=1}^{p} \alpha_i r_{t-i}^2 + \sum_{j=1}^{q} \beta_j \sigma_{t-j}^2

• α_i​ (1 ≤ i ≤ p) and β_j​ (1 ≤ j ≤ q) are parameters controlling the impact of past squared returns and past conditional variances.
• r_t−i2​ represents squared returns at time t−i.
• σ_t−j2​ represents conditional variances at time t−j.

Implementation Guidelines

1. Data Preprocessing: Ensure that your time series data is stationary. If not, apply differencing until stationarity is achieved.
2. Model Specification: Choose the appropriate GARCH variant (e.g., GARCH(1,1), GARCH(p, q)) based on the nature of your data and the complexity of the volatility patterns you want to capture.
3. Parameter Estimation: Estimate the model parameters (e.g., α₀​, α_i​, β_j​) using maximum likelihood estimation (MLE) or alternative methods.
4. Model Validation: Assess the goodness-of-fit and conduct diagnostic tests for residual autocorrelation and normality.
5. Forecasting: Utilize the GARCH model for volatility forecasting, which can be crucial for risk management and financial decision-making.

Limitations and Drawbacks

1. Stationarity Assumption: GARCH models assume that the time series data is stationary, which may not hold in practice. Non-stationary data may lead to unreliable results.
2. Model Complexity: Higher-order GARCH models (e.g., GARCH(p, q) with large values of p and q) can become computationally intensive and may overfit the data.
3. Sensitivity to Initial Values: GARCH models can be sensitive to the choice of initial parameter values, making estimation less robust.

Advancements and Improvements

1. Integrated GARCH (IGARCH): IGARCH models allow for non-stationary data by introducing differencing operators.
2. Fractional GARCH (F-GARCH): F-GARCH models capture long-memory processes by including fractional differencing.
3. Regime-Switching GARCH: These models account for changing volatility regimes, enhancing their applicability in real-world scenarios.
4. Bayesian GARCH: Incorporating Bayesian techniques for parameter estimation provides more robust results, particularly when dealing with limited data.

Certainly, there are several variants and extensions of the Generalized Autoregressive Conditional Heteroskedasticity (GARCH) model, each designed to address specific characteristics or complexities of financial time series data. Let’s explore some of these variants and extensions along with their explanations:

Integrated GARCH (IGARCH):

Explanation: IGARCH models are used when the financial time series data is non-stationary. They introduce differencing operators to make the data stationary before modeling volatility.

Mathematical Formulation: The conditional variance in IGARCH is defined as follows:

\sigma_t^2 = \alpha_0 + \sum_{i=1}^{p} \alpha_i \left(r_{t-i}^2 - \mu\right) + \sum_{j=1}^{q} \beta_j \sigma_{t-j}^2

Where μ is the mean of the squared returns.

Usage: IGARCH models are suitable for financial data with trends or non-stationarity, allowing for more accurate modeling of volatility.

GJR-GARCH (Glosten-Jagannathan-Runkle GARCH):

Explanation: GJR-GARCH extends the traditional GARCH model by incorporating an additional parameter that allows for asymmetric effects of past returns on volatility. It captures the phenomenon where positive and negative shocks have different impacts on volatility.

Mathematical Formulation: The GJR-GARCH(1,1) model is expressed as:

\sigma_t^2 = \alpha_0 + \alpha_1 r_{t-1}^2 + \gamma_1 I_{t-1} r_{t-1}^2 + \beta_1 \sigma_{t-1}^2

Where It−1​ is an indicator variable that takes the value 1 if rt−1​<0 and 0 otherwise.

Usage: GJR-GARCH models are useful for capturing the asymmetric effects of market shocks, which are often observed in financial data.

EGARCH (Exponential GARCH):

Explanation: EGARCH models are designed to capture the leverage effect, where negative returns have a stronger impact on future volatility than positive returns. Unlike GARCH, EGARCH allows for the conditional variance to be a nonlinear function of past returns.

Mathematical Formulation: The EGARCH(1,1) model can be expressed as:

\log(\sigma_t^2) = \alpha_0 + \sum_{i=1}^{p} \alpha_i \log(r_{t-i}^2) + \sum_{j=1}^{q} \beta_j \log(\sigma_{t-j}^2)

Usage: EGARCH models are particularly useful for capturing the asymmetric and nonlinear dynamics of financial volatility, especially in the presence of leverage effects.

TARCH (Threshold ARCH):

Explanation: TARCH models extend the GARCH framework by incorporating a threshold or regime-switching component. They are used to model volatility dynamics that change based on certain conditions or regimes.

Mathematical Formulation: The TARCH(1,1) model is expressed as:

\sigma_t^2 = \alpha_0 + \sum_{i=1}^{p} \alpha_i \varepsilon_{t-i}^2 + \sum_{j=1}^{q} \beta_j \sigma_{t-j}^2 + \sum_{k=1}^{r} \gamma_k I_{t-k} \sigma_{t-k}^2

Where I_t−k​ is an indicator variable that captures the regime switch.

Usage: TARCH models are valuable for capturing changing volatility regimes in financial markets, such as during financial crises or market shocks.

Long Memory GARCH (LM-GARCH):

Explanation: LM-GARCH models are designed to capture long memory or fractional integration in financial time series. They extend GARCH to account for persistent, autocorrelated shocks over extended periods.

Mathematical Formulation: The LM-GARCH(1,1) model can be expressed as:

\sigma_t^2 = \alpha_0 + \sum_{i=1}^{p} \alpha_i \varepsilon_{t-i}^2 + \sum_{j=1}^{q} \beta_j \sigma_{t-j}^2 + \sum_{k=1}^{s} \delta_k \varepsilon_{t-k}^2

Where δk​ captures the long memory component.

Usage: LM-GARCH models are suitable for capturing the slow decay in volatility correlations over time, which is observed in long-term financial data.

Limitations and Advancements:

Limitations:

• GARCH models assume constant parameters, which may not hold in practice.
• They are sensitive to the choice of initial parameter values, affecting estimation results.
• GARCH models are known for their inability to capture sudden regime changes effectively.

Advancements:

• Bayesian GARCH models incorporate Bayesian techniques for robust parameter estimation.
• Regime-switching GARCH models combine GARCH with state-switching models to capture regime changes.
• Realized volatility models use intraday data to provide more accurate volatility forecasts.
• High-frequency GARCH models address the challenges of modeling volatility at high-frequency intervals.

In conclusion, GARCH models and their variants offer a versatile toolbox for modeling volatility in financial time series data. Depending on the specific characteristics of the data and the phenomena to be captured, practitioners can choose from various GARCH variants and extensions. These models have evolved to address limitations and provide more accurate representations of financial market dynamics.