Approximating GARCH from a Mean-Reverting Stochastic Variance Process

Summary

The document asks how to approximate a GARCH(1,1) variance recursion from a discrete mean-reverting stochastic variance process associated with the Heston model. Returns are modeled as the square root of variance times a unit-variance shock, while variance moves toward a long-run level with a random innovation. The proposed approximation expresses next-period variance as a constant plus a weighted squared return and a weighted lagged variance.

The author derives how the mean of the variance process reverts and uses that relation to constrain the GARCH coefficients. The unresolved question is how to determine the relative contributions of squared returns and lagged variance from the original recursion, especially when the shocks may be correlated. The document presents the setup and an ad hoc parameterization, but no solution, calibration procedure, evidence, or conditions under which the approximation is reliable.

Key ideas

• The variance process combines mean reversion toward a long-run level with stochastic innovations.
• Returns are modeled as a variance-scaled shock.
• The proposed GARCH approximation links next-period variance to squared returns and prior variance.
• Mean reversion constrains the coefficient sum, but the document does not determine the individual coefficients.

Tags

# Determine GARCH(1,1) from a mean reverting time series recursion


# Determine GARCH(1,1) from a mean reverting time series recursion












Let $(v_t)$ be a discrete time series of variance obeying a mean-reverting variance process $v_t$, which is actually the discrete version of the Heston model in finance. \begin{align} x_t &= \sqrt{v_t} z_{1,t} \tag1\\ v_{t}-v_{t-1} &= -\lambda(v_{t-1}-v_\infty)+\eta\sqrt[]{v_{t-1}}z_{2,t-1} \tag2 \end{align} where $z_{1,t}$ and $z_{2,t}$ is two not necessarily independent unit variance random walks with correlation and $\eta$ is some positive constant.

I would like to make an approximate GARCH(1,1) model for the variance out of the above time series in the form of $$v_t = \alpha_0+\alpha x_{t-1}^2+\beta v_{t-1}$$ where $\alpha_0,\,\alpha,\,\beta$ are positive and $\alpha+\beta<1$.

Here is my rough ad hoc attempt. Take expectation of Equation (2) and arrange the terms we have $$u_t = \lambda u_\infty+(1-\lambda)u_{t-1}$$ where $u_t:=\mathbf E[v_t]$. Now set $$\tilde u_t = \lambda \tilde u_\infty+(1-\lambda)(ax_{t-1}^2+b\tilde u_{t-1})$$ for some positive $a$ and $b$ where $a+b=1$. Now $\alpha_0 = \lambda u_\infty,\,\alpha=(1-\lambda)a,\,\beta=(1-\lambda)b$. How would one determine $a,\,b$ from the time series recursion Equation (1) and (2)?

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.