  In this paper we implement a combination of data-science and fuzzy theory to
improve the classical Barndorff-Nielsen and Shephard model, and implement this
to analyze the S&P 500 index. We pre-process the index data based on fuzzy
theory. After that, S&P 500 stock index data for the past ten years are
analyzed, and a deterministic parameter is extracted using various machine and
deep learning methods. The results show that the new model, where fuzzy
parameters are incorporated, can incorporate the long-term dependence in the
classical Barndorff-Nielsen and Shephard model. The modification is based on
only a few changes compared to the classical model. At the same time, the
resulting analysis effectively captures the stochastic dynamics of the stock
index time series.
