On the choice of GARCH parameters for efficient modelling of real stock price dynamics

• Two self-consistent methods to determine GARCH(1,1) parameters are proposed.
• Fitting higher-order moments can lead to inefficient GARCH(1,1) parameter estimation.
• Higher-order moment analysis produces similar results to MLM-based software.
• Fitting Fourier spectrum leads to a more stable GARCH(1,1) stochastic process.
• Fitting spectrum produces a shorter characteristic autocovariance time.

We propose two different methods for optimal choice of GARCH(1,1) parameters for the efficient modelling of stock prices by using a particular return series. Using (as an example) stock return data for Intel Corporation, we vary parameters to fit the average volatility as well as fourth (linked to kurtosis of data) and eighth statistical moments and observe pure convergence of our simulated eighth moment to the stock data. Results indicate that fitting higher-order moments of a return series might not be an optimal approach for choosing GARCH parameters. In contrast, the simulated exponent of the Fourier spectrum decay is much less noisy and can easily fit the corresponding decay of the empirical Fourier spectrum of the used return series of Intel stock, allowing us to efficiently define all GARCH parameters. We compare the estimates of GARCH parameters obtained by fitting price data Fourier spectra with the ones obtained from standard software packages and conclude that the obtained estimates here are deeper in the stability region of parameters. Thus, the proposed method of using Fourier spectra of stock data to estimate GARCH parameters results in a more robust and stable stochastic process but with a shorter characteristic autocovariance time.