GARCH Model

GARCH models are designed to capture volatility clustering, the empirical observation that large changes in asset prices (high volatility) tend to be followed by large changes, and small changes (low volatility) by small changes. This is modeled by making today's conditional variance depend on past squared errors and past variances.

• Example: A day with a large price drop increases the model's forecast for volatility in the subsequent days.

The core mechanism addresses conditional heteroskedasticity, meaning the variance of the error term is not constant over time but depends on past information. The model specifies that the conditional variance (h_t) is a function of:

• Past shocks: Lagged squared residuals (\epsilon_{t-i}^2) (the ARCH component).
• Past variances: Lagged conditional variances (h_{t-j}) (the GARCH component). This structure allows volatility to evolve dynamically.

A key advantage of the GARCH(1,1) model is its parsimony. It often provides a good fit to financial data with just three parameters: a constant term, one lag for past shocks, and one lag for past volatility. This makes it efficient to estimate and less prone to overfitting compared to higher-order ARCH models, which require many parameters to capture persistent volatility.

GARCH models are primarily used for volatility forecasting. By estimating the model on historical data, one can generate multi-step ahead forecasts for conditional variance. These forecasts are crucial for:

• Risk management: Calculating Value at Risk (VaR).
• Derivative pricing: Informing option pricing models like Black-Scholes.
• Portfolio optimization: Adjusting for time-varying risk.

The basic GARCH framework has been extended to address specific financial phenomena:

• EGARCH: Models asymmetric effects (leverage effect), where negative shocks increase volatility more than positive shocks.
• GJR-GARCH: Another model capturing asymmetry via an indicator function for negative shocks.
• IGARCH: For integrated processes where volatility shocks are persistent.
• Multivariate GARCH: Models time-varying covariances between multiple assets (e.g., DCC-GARCH).

GARCH models are typically estimated using Maximum Likelihood Estimation (MLE). This involves:

• Assuming a conditional distribution for the errors (often Normal or Student's t).
• Constructing the likelihood function based on the model's conditional variance equation.
• Using numerical optimization algorithms to find the parameter values that maximize the likelihood of observing the historical data.

In traditional finance, GARCH models are foundational for forecasting the volatility used in options pricing models like Black-Scholes. By modeling how volatility clusters over time (e.g., high-volatility periods follow high-volatility periods), traders can derive more accurate implied volatility surfaces and price options contracts. This is essential for market makers and risk desks managing large portfolios of derivatives.

Financial institutions use GARCH to estimate Value at Risk (VaR), a key metric for quantifying potential portfolio losses. By forecasting conditional volatility, GARCH provides dynamic VaR estimates that adjust to current market conditions, offering a more realistic risk assessment than models assuming constant volatility. This is crucial for banks and hedge funds to meet regulatory capital requirements and manage trading desk limits.

In DeFi, protocols like Aave or Compound can use GARCH-inspired models to dynamically adjust risk parameters. By analyzing the volatility of collateral assets (e.g., ETH, WBTC), a protocol could automatically modify loan-to-value (LTV) ratios or liquidation thresholds. This creates a more responsive system where risk controls tighten during high-volatility market regimes to protect the protocol's solvency.

Some algorithmic stablecoin designs could incorporate volatility signals from GARCH models to inform their monetary policy. For example, if the model forecasts high volatility for the stablecoin's paired assets, the protocol's expansion or contraction mechanisms could be triggered more aggressively to maintain the peg. This applies a quantitative finance tool to the decentralized governance of a currency's stability.

Both traditional and crypto asset managers use volatility forecasts from GARCH for portfolio optimization (e.g., Modern Portfolio Theory) and dynamic hedging strategies. By predicting which assets will become more or less volatile, managers can adjust portfolio weights or hedge ratios (like delta-hedging for options) to target a specific risk-return profile, minimizing unexpected losses from volatility spikes.

Quantitative analysts (quants) rely on GARCH models to create more realistic market simulations for backtesting. By generating synthetic price paths that replicate the volatility clustering and leverage effect (where negative returns increase future volatility more than positive returns) observed in real markets, traders can more accurately assess the historical performance and robustness of algorithmic trading strategies before deploying capital.

A key empirical observation in financial time series where large changes in asset prices tend to be followed by more large changes (of either sign), and small changes tend to be followed by small changes. This phenomenon, also known as conditional heteroskedasticity, is the primary stylized fact that the GARCH model was designed to capture and quantify. It contradicts the assumption of constant variance in simpler models.

The Autoregressive Conditional Heteroskedasticity (ARCH) model, introduced by Robert Engle in 1982, is the direct predecessor to GARCH. It models volatility as a function of past squared error terms. The key limitation GARCH addressed is that ARCH models often require many parameters to capture persistent volatility, leading to the more parsimonious Generalized ARCH (GARCH) model by Tim Bollerslev in 1986, which includes lagged conditional variances.

A critical risk management metric estimating the maximum potential loss of a portfolio over a specified time horizon at a given confidence level (e.g., 95%). GARCH models are extensively used to forecast the volatility input for VaR calculations, particularly for conditional VaR, which adapts to changing market conditions. This provides a more responsive and accurate risk measure than using a simple historical average volatility.

An alternative class of models to GARCH for modeling time-varying volatility. While GARCH models volatility as a deterministic function of past observations, Stochastic Volatility (SV) models treat it as an unobserved latent variable following its own stochastic process (often modeled with a separate error term). SV models are typically estimated using more complex methods like Markov Chain Monte Carlo (MCMC) but can offer different flexibility in capturing volatility dynamics.

The observed negative correlation between an asset's return and its subsequent volatility changes (i.e., price drops lead to increased future volatility more than price rises). Standard GARCH models cannot capture this asymmetry. This led to the development of asymmetric GARCH extensions like:

• EGARCH (Exponential GARCH)
• GJR-GARCH (Glosten-Jagannathan-Runkle GARCH)
• TGARCH (Threshold GARCH) These models include terms that allow bad news to have a larger impact on volatility than good news.

A non-parametric, model-free measure of volatility calculated from high-frequency intraday price data (e.g., 5-minute returns). It serves as a highly accurate ex-post observation of daily volatility. In modern econometrics, Realized Volatility is often used as the observable target variable in GARCH-X models, where it is included as an exogenous regressor to improve the in-sample fit and forecasting power of the conditional variance equation.