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A stationary time series makes effective and precise predictions. Also, a stationary time series means that the pair of stocks is co-integrated and can be traded together by generating trading signals. Hence, stocks are needed to be selected for performing the pairs trading. Assumption: n, the hedge ratio is constant. Hence, we regress the stock prices to calculate the hedge ratio.

Theory: In regression , we get a term called the residuals which represents the distance of observed value from the curve fitting line or estimated value. These residuals are studied so that we understand whether or not they form a trend.

If they do not form a trend, that means the spread moves around 0 randomly and is stationary. The Dickey Fuller test is a hypothesis test which gives a p-value as the result. If this value is less than 0. So far, we have discussed the challenges and statistics involved in selecting a pair of stocks for statistical arbitrage. By using the cointegration tests, we can say within a certain level of a confidence interval that the spread between the two stocks is a stationary signal.

In other words, this signal is mean-reverting. Having already established that the equation above is mean reverting, we now need to identify the extreme points or threshold levels that when crossed by this signal, trigger trading orders for pairs trading. To be able to identify these threshold levels, a statistical construct called z-score is widely used in pairs trading.

Define threshold as anything between 1. This parameter will change as per the backtesting results without risking overfitting data. We have now understood entry points in pairs trading. Now we will move on to the other end, exit points. Defining Exit points Stop loss Stop loss is defined for scenarios when the expected outcome does not occur. For instance, if we chose entry signals at 2-sigma, we are expecting that the spread will revert back to the mean from this threshold.

However, it is possible that the spread continues to blow up. Say it reaches 2. To prevent further losses, you place stop loss at say 3-sigma. In addition to placing a predefined stop-loss criterion such as 3-sigma or extreme variation from the mean, you can check on the cointegration value. If the cointegration is broken while the pair is ON, the strategy warrants cutting the positions since the basic hypothesis is nullified. Take profit It is defined as scenarios where you take profit before the prices move in the other direction.

For instance, say you are LONG on the spread, that is, you have bought stock A and sold stock B as per the definition of spread in the article. The expectation is that spread will revert back to the mean or 0. In a profitable situation, the mean would be approaching zero or very close to it. You can keep the take profit scenario as when the mean crosses zero for the first time after reverting from the threshold levels.

There can be many ways of defining take profits depending on your risk appetite and backtesting results. What often works is your experience and a broad range of potent skill sets that allow you to grasp a hold of the complete scenario before jumping to conclusions. As we mentioned, your appetite for risk and backtesting results will work for you. Automation and practical applications are the keys here. Let us take a recap of what we have understood so far. Pairs Trading can be called a mean reversion strategy where we bet that the prices will revert to their historical trends.

For performing the pairs trading strategy, we have the following: Assumptions For simplification purposes, we ignore bid-ask spreads. Prices are available at 5 minutes intervals and we trade at the 5-minute closing price only. Since this is discrete data, squaring off of the position happens at the end of the candle i. Only the regular session T is traded.

Input parameters Please note that all the values for the input parameters are configurable. An average of 10 candles one candle is equal to every 5-minute price is considered. It also helps in the mitigation of risks as the pairs strategy involves dealing with two securities so if one is underperforming then there are chances that the other absorbs the losses. Good returns Pairs trading strategy helps the trader to get good returns regardless of the conditions of the market.

Hedging The best advantage of pairs trading is that the trader is completely hedged. Hedging is done in this strategy as the trader sells the overvalued security and purchases the undervalued security, thereby, limiting the chances of loss. Disadvantages of pairs trading The disadvantages of pairs trading are: Reliance of the High Statistical Correlation Pairs trading relies on the securities having a high statistical correlation.

Most of the traders require a correlation of at least 0. This implies that we can artificially construct a mean reverting time series through the appropriate combination of non-stationary time series. For example, we can construct a portfolio of assets whose market value is a stationary time series and thus amenable to profitable exploitation through mean-reversion techniques, even through the price series of the constituent assets are not themselves mean reverting.

A pairs trading strategy, where we buy one asset and short another with an appropriate allocation of capital to each, is an example of this method for exploiting the concept of cointegration, but we can also create more complex portfolios of three or more assets. We can test whether a given combination of assets forms a stationary process using the stationarity tests described in the previous post.

However, it is impossible to know a priori the coefficients or hedge ratios that form a stationary portfolio. How then does one test for cointegration? The example below illustrates this concept using the currencies of Australia and New Zealand since they seem likely to cointegrate given that the economies of both countries are commodity-based and are affected by similar geopolitical forces.

In this example, we allow for a flexible hedge ratio and attempt its optimization. In order to achieve this, we need to introduce a common quote currency, the more liquid the better. It makes sense to choose the US dollar. However, the negative value of the test statistic indicates that the portfolio is not trending.

One shortcoming of the ordinary least squares approach is that it is asymmetric: switching the dependent and independent variables in the regression results in a different hedge ratio. Good practice would dictate that both options be tested and the arrangement that results in the more negative test statistic be selected.

Another approach is to use total least squares regression, which can be used to derive a symmetric hedge ratio. In a geometrical sense, total least squares minimizes the orthogonal distance to the regression line as opposed to the vertical distance in the case of ordinary least squares and thus takes into account variance of both the dependent and independent variables.

Johansen test The Johansen test allows us to test for cointegration of more than two variables. The urca package contains an implementation of the Johansen test that provides critical values that we can use to test whether we can reject the null hypothesis that there exist 0, 1, 2, …, n-1 cointegrating relationships, where n is the number of constituent time series. Conveniently, the eigenvectors can be used as the hedge ratios of individual price series to form a stationary portfolio.

That is, it is unlikely that we can form a stationary portfolio from the price history used in this example. However, it may still be worth pursuing a mean reverting strategy if the half-life of mean reversion is sufficiently low see the previous post for more details. As stated above, the eigenvectors form the optimal hedge ratio. In this case, unfortunately, the resulting portfolio does not look any more stationary than that constructed using the ordinary least squares and total least squares regression approaches: Mean reversion of a portfolio of more than two instruments We can add a third asset and use the Johansen test to determine the probability that there exists a mean reverting portfolio along with the hedge ratios of such a portfolio.

Therefore, we will retain the first eigenvector to form a portfolio of the three instruments for further investigation. Linear mean reversion on a cointegrated time series Below is the equity curve of the linear mean reversion strategy from the previous post on the three-instrument portfolio with the value of the portfolio overlaid on the equity curve: The strategy suffers significant drawdown and only returns a profit factor of 1. Obviously, the linear mean reversion strategy presented above and detailed in the previous post would not be suitable for live trading even if the example shown here had generated an impressive backtest.

Applied to equities, it would require buying and selling an infinitesimal number of shares when price moves an infinitesimal amount. This is less of a problem when applied to currencies since we can buy and sell in units as small as one-hundredth of a lot.

Having said that, there is still much value in testing a mean reversion idea with this linear strategy as it shows whether we can extract profits without any data snooping bias as there are no parameters to optimize.

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What is the logic behind pairs trading? In the case of a pairs trading strategy, the two stocks or the financial instruments need to be trending at a similar mean price and remain close to each other. But, on certain occasions, one of the instruments may go through a short period of deviation from another in terms of price. In this short period, the trader can take the opportunity to go long on one of the financial instruments while shorting the other.

The positions are based on the current market price of both the stocks and their standard deviation. The correlation coefficient indicates the degree of correlation between the two variables. A perfect positive correlation is when one variable moves in either an upward or downward direction and the other variable also moves in the same direction with the same magnitude. Whereas a perfect negative correlation is when one variable moves in the upward direction and the other variable moves in the downward i.

This high number represents a strong relationship between the two stocks. So if A goes up, the chances of B going up are also quite high. Based on this assumption a market neutral strategy is played where A is bought and B is sold; bought and sold decisions are made based on their individual patterns. Just looking at correlation might give you spurious results. For instance, if your pairs trading strategy is based on the spread between the prices of the two stocks, it is possible that the prices of the two stocks keep on increasing without ever mean-reverting.

This will result in a loss since stock A is increasing at a rate lower than stock B and you are short on stock B. Thus, one should be careful of using only correlation for determining the pairs of the stocks while performing the pairs trading strategy. Cointegration is a statistical property of two or more time-series variables which indicates if a linear combination of the variables is stationary.

Let us understand the statement above. The two time series variables, in this case, are the log of prices of stocks A and B. For each stock of A bought, you have sold n stocks of B. If A and B are cointegrated, the equation above is stationary.

A stationary process has very valuable features which are required to model pairs trading strategies. For instance, in this case, if the equation above is stationary, that suggests that the mean and variance of this equation remain constant over time. Any deviation from this expected value is a case for statistical abnormality, hence a case for pairs trading! Z-score Given a normal distribution of raw data points, the z-score is calculated so that the new distribution is a normal distribution with a mean of 0 and a standard deviation of 1.

For instance, in pairs trading, we have a distribution of spread between the prices of stocks A and B. We can convert these raw scores of spread into z-scores as explained below. This new distribution will have a mean of 0 and a standard deviation of 1. It is easy to create threshold levels for this distribution such as 1. Augmented Dickey Fuller Test The augmented Dickey-Fuller test is an extension of the standard Dickey-Fuller test , which also checks for both stationarity and non-stationarity in the time series.

The main difference from the Dickey Fuller Test is that the Augmented Dickey Fuller test can also be applied to a large sized set of time series models. Also, the ADF Test works on the data with missing values. Steps for pairs trading Select stocks for pairs trading For the pair of stocks to be traded in a pairs trading strategy, it is required that the time series is stationary.

A stationary time series makes effective and precise predictions. Also, a stationary time series means that the pair of stocks is co-integrated and can be traded together by generating trading signals. Hence, stocks are needed to be selected for performing the pairs trading.

Assumption: n, the hedge ratio is constant. Hence, we regress the stock prices to calculate the hedge ratio. Theory: In regression , we get a term called the residuals which represents the distance of observed value from the curve fitting line or estimated value.

These residuals are studied so that we understand whether or not they form a trend. If they do not form a trend, that means the spread moves around 0 randomly and is stationary. The Dickey Fuller test is a hypothesis test which gives a p-value as the result. If this value is less than 0. So far, we have discussed the challenges and statistics involved in selecting a pair of stocks for statistical arbitrage. By using the cointegration tests, we can say within a certain level of a confidence interval that the spread between the two stocks is a stationary signal.

In other words, this signal is mean-reverting. Having already established that the equation above is mean reverting, we now need to identify the extreme points or threshold levels that when crossed by this signal, trigger trading orders for pairs trading. To be able to identify these threshold levels, a statistical construct called z-score is widely used in pairs trading. Define threshold as anything between 1. This parameter will change as per the backtesting results without risking overfitting data.

Lets calculate the mean and the standard deviation of the ratio from to , and test the strategy using these parameters on data from to If the ratio is truly mean reverting, then the values calculated from to should also be profitable. We can see that the stocks have a pretty high correlation of 0.

Now, we will write VBA code to enter the trades from to Print row Sheets "Trades". You will see the reason for its poor performance when you plot a graph of the ratio with time — The ratio of the stocks is not really mean reverting in the long run.

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