Reconsidering the continuous time limit of the GARCH(1,1) process
Let the cumulative returns \(Y_k\) and the volatility process \(\sigma_k^2\) follow GARCH(1,1) process \[ Y_k-Y_{k-1}=\sigma_{k- 1} \varepsilon_k,\quad\sigma_k^2=\omega_0+\omega_1\sigma_{k-1}^2 +\omega_2\sigma_{k-1}^2\varepsilon_k^2 \] with \(\omega_0, \omega_1, \omega_2>0\), \(\omega_1+\omega_2<1\), and \(\varepsilon_k\sim\) iid \(N(0,1)\). For \(h>0\), an approximation scheme is \[ Y_{kh}- Y_{(k-1)h}=\sigma_{(k-1)h}\varepsilon_{kh},\quad\sigma_{kh}^2 -\sigma_{(k-1)h}^2=\omega_{0h}+(\omega_{1h}-1)\sigma_{(k-1)h}^2 +h^{-1}\omega_{2h}\sigma_{(k-1)h}^2\varepsilon_{kh}^2. \] This scheme can be parametrized either by the condition \(\lim_{h\to 0} h^{-\delta}\omega_{2h}=0\), \(\forall\delta<1\), or by \(\lim_{h\to 0} 2h^{-1}\omega_{2h}^2=\alpha^2\). It is shown that it leads either to a degenerate or to a non-degenerate diffusion limit. Further it is shown that there exists a Euler approximation to a degenerate diffusion that leads to a GARCH(1,1) process and that any Euler approximation of a non-degenerate diffusion leads to a stochastic volatility process.
scientific article; zbMATH DE number 1437076
scientific article