Perturbative Solution of GARCH(1,1) model within the Many-Body Theory

In this paper, we derive an approximate unconditional distribution for normal GARCH(1, 1) model by application of perturbation theory. We compute the characteristic function and distribution up to the fourth order approximation. Additionally, we derive an integral equation for the characteristic function. By application of Dyson equation, we develop a new method for calculating distribution from the characteristic function. Utilizing self-similarity transformation in parametric space, we improve the accuracy of our results, especially in the neighborhood of α + β = 1. Kullback-Leibler divergence, computed against simulated data shows good performance of the approximation.