Financial markets are based on the general trading rule: buy with low price and sell with high price. The aim is the development of strategies with low risk and succeeds this general rule. Pure Arbitrage is a category of strategies with zero risk. As an example we can refer the case of buying and selling a stock at the same time with a different value in two different exchanges. The profit results from the difference in prices, breaking the law of one price.

Another category is the statistical arbitrage which is not risk free at all. Strategies of this type are aimed at the expected gain which is greater than the risk. The profit results from the mispricing of the stocks. To achieve this, one needs to assess whether the price of a stock is overvalued or undervalued relative to the actual value which is really hard to determine. The fundamentals of the stock, the demand of each period and the general economic environment are some of the factors that make the fair value evaluation difficult. Part of this category is the Pair Trading and is presented in several references as in [1] -[10] . The Pairs trading is a strategy which is based on the relative pricing of stocks without actually interested in the true value of them. The relative pricing is based on the idea that two assets with the same features can be priced about the same price as the basis of the law of one price. When at a given time the stock prices are different then one is overvalued and the other undervalued relative to the actual price. The classic trading Pairs strategy derives from the characteristics of this incorrect pricing (mispricing) between the two stocks.

In the current work instead of the two stocks the mispricing between the index S&P500 and a sub-set of stocks belonging to this is considered. This forms a portfolio consisting of one unit of the Index in long (or short) and the corresponding number γ (Hedge ratio) of subset’s stocks in the opposite position. Three methods adopted to determine the number γ (Hedge ratio): a) the ordinary, b) the rolling ordinary least squares (OLS) regression and c) the Kalman filter process as it is presented below. According to the strategy, the spread of the index S&P500 and the stocks prices’ combination are computed and when this deviates from its historical average value then the investor bets on the return to the historical with selling and buying respectively the stocks of the portfolio. In practice to construct the synthetic asset is necessary to investigate the appropriate stock exhibiting long relationship between them with the index. The technique used for this purpose is the Cointegration as this presented in [1] according to which two time series and are cointegrated if for and notation I(d) means integrated order d.

In [1] concluded that pairs who cointegrate in sample period behave better in the out-of-sample period than those not cointegrate in the sample period. Based on the previous, the spread of the linear combination of the cointegrated stocks and the S&P500 index have to be stationary process. To analyze this relationship the augmented Dickey-Fuller test (ADF) was used as presented in [2] . According to the strategy the stocks are not restricted to cointegrate in the out-of-sample period but it is an indication that they will present mean-reverting behavior.

Additionally the log of prices used is also instead the prices as in [3] . The main reason is that log-returns are time additive. So, in order to calculate the return over n periods using real returns we need to calculate the product of n numbers:.

If defined as:

And

Consequently the profit of spread over a period is equal to:

The classic application of the trading pair strategy has published in many works in the past and has been applied in practice. In [4] the author went a further step and presented an algorithm for the pair trading of a stock basket cointegrated with the S&P500 using constant cointegrating coefficient with a single value for all stocks belonging to the basket. To capture the real behavior of the markets this work presents new Time Adaptive algorithms using rolling ordinary least squares (OLS) regression and Multivariate Kalman Filter process where the time dependent hedge ratio is computed separately for each of the stocks forming the synthetic asset creating thereby statistical arbitrage conditions in index S&P500 and increasing the strategy performance.

The proposed methodology and the trading algorithm designed based on that divided in two different spaces. The first refers to the in sample period which is used to make all the appropriate test and construct the synthetic asset and the other to the out of sample period (Trading Period) where the synthetic asset trading based on the specific rules (Figure 1).

In reality the system of trading is dynamic and updated as new information get to in and the cointegration coefecient (or the hedge ratio) cannot stay constant during the trading period. For that reason time adaptive algorithms are developed to capture the real conditions of the markets.

The algorithm of the pair trading strategy is based on the distance of the spread from its historical mean value and its mean-reverting behavior. To measure this distance a normalized variable called z-score introduced as:

where:

: is the mean value of the spread over a lookback period.

: is the standard deviation of the spread over the same period.

The trading it takes place when this variable exceeds some limits based on the spread mean reverting behavior. Thus:

Open-long position if.

Open-short position if.

Exit-long position if.

Exit-short position if.

When a long (short) position is opened we buy (sell) one unit of S&P500 and sell (buy) the following amount

of stocks from synthetic asset as: in the case of constant hedge ratio, when the rolling OLS regression is applied, and if the Kalman Filter algorithm applied. The performance

of strategy evaluated using the following metrics: 1) Cumulative return, 2) Annualized Return, 3) Sharpe Ratio, 4) Maximum drawdown, 5) beta.

Five different time periods was studied. In each case one year data collection was used in order to make the entire test and construct the synthetic asset. After that the algorithm starts trading with ending day for all cases the 30/12/2015. All the sample periods started at the first day of the year and ending at the last of the same year. The trading started on the first day the next year. In Table 1 the back testing periods are presented. In Figure 2 and Figure 3 the results of cumulative return of Multivariate Kalman Filter algorithm against S&P500 and its maximum drawdown are shown. In Figure 4 and Figure 5 the cumulative return of rolling OLS regression algorithm against S&P500 and the cumulative return of rolling OLS regression algorithm against constant hedge ratio are illustrated. In t Table 2 the sets of synthetic assets are presented while in Table 3 the name of symbols is given. In Tables 4-8 all the metrics of the algorithms are displayed.

It can be shown that the dimension of the set and the constituents are different from period to period. This is an evidence of the non constant cointegration behavior of the stocks with the index but as we can see from the graphs the synthetic asset of each period continues to trade with profit and good metrics results.