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Calibration of GARCH(1,1) Model from Historical Prices

The Generalized Autoregressive Conditional Heteroskedasticity (GARCH) model, typically GARCH(1,1), is calibrated using Maximum Likelihood Estimation (MLE) on the asset's log returns ($r_{t}$).

Key Calibration Steps:

• Step 1: Data Transformation
  Historical prices ($P_{t}$) are converted to log returns ($r_{t}$) using the formula: $$r_{t}=ln(\frac{P_{t}}{P_{t-1}})$$.
• Step 2: Model Specification
  The model is defined by a Mean Equation, $r_{t}=\mu+\epsilon_{t}$, and a Variance Equation. The conditional variance ($\sigma_{t}^{2}$) is modeled as a function of the long-run variance ($\omega$), the previous shock squared ($\epsilon_{t-1}^{2}$), and the previous conditional variance ($\sigma_{t-1}^{2}$): $$\sigma_{t}^{2}=\omega+\alpha\epsilon_{t-1}^{2}+\beta\sigma_{t-1}^{2}$$.
• Step 3: Parameter Estimation (MLE)
  The parameters $\Theta=\{\mu,\omega,\alpha,\beta,(\nu)\}$ are estimated by maximizing the log-likelihood function ($\mathcal{L}$). For a Normal Distribution, the log-likelihood contribution at time t ($l_{t}$) is: $$l_{t}(\Theta)=-\frac{1}{2}[ln(2\pi)+ln(\sigma_{t}^{2})+\frac{(r_{t}-\mu)^{2}}{\sigma_{t}^{2}}]$$.
• Step 4: Diagnostics and Validation
  Validation involves checking the standardized residuals ($\hat{z}_{t}$) for autocorrelation and ensuring the parameters satisfy stationarity ($\alpha+\beta<1$) and positivity ($\omega>0, \alpha\ge0, \beta\ge0$) constraints. The standardized residual ($\hat{z}_{t}$) is calculated as: $$\hat{z}_{t}=\frac{r_{t}-\hat{\mu}}{\hat{\sigma}_{t}}$$.

A step-by-step calibration algorithm and an illustrative example with recursive conditional variance calculation are included in Calibration of GARCH(1,1) Model from Historical Prices.