Are capital market prices the product of a stochastic processes or are they determined by deterministic chaos? Is there a non-linear causality between them? We are addressing these questions by studying two basic time series: the closing prices of the S&P 500 index which is one of the main indicators of the US capital market and the volatility index (VIX) of the New York Stock Exchange. With the help of Takens theorem (Takens 1981), Singular Spectrum Analysis and the Convergent Cross Mapping (CCM) algorithm as described in (Huffaker et al. 2017) we deal with these questions in the present framework. Implementing those relative new tools in the financial sector, we try to gain a new perspective on the financial indexes and try to take an alternative approach in the field of understanding the financial time series. An effort is made also to show that a prediction on the basis of next day values is feasible.
In order to achive that we examine the daily prices of the indexes for the period 2000 to 2019. The selection of the specific period is based on the fact that it contains the Financial Crisis of 2007-2008, an event that caused a major collapse in the financial sector. This perspective that could provide another insight in the financial data from the point of view of complexity theory, is relatively new and it is getting much attention among financial analysts, as there are many major successive predictions. The CCM algorithm, despite the fact that is computational costly and its results are highly sensitive to several parameters, can provide via lagged embedding time series, another causality measure avoiding the stochastic process procedure which is implemented in the Granger Causality test. The distinctive characteristic of the financial time series, which is noise, can affect the analysis and conclude in controversial results. This matter will be addressed in the conclusion sector of the present paper.