Pairs trading is supposedly one of the most popular types of trading strategy. In this strategy, usually a pair of stocks are traded in a market-neutral strategy, i.e. it doesn’t matter whether the market is trending upwards or downwards, the two open positions for each stock hedge against each other. The key challenges in pairs trading are to:

- Choose a pair which will give you good statistical arbitrage opportunities over time
- Choose the entry/exit points

In this article on pairs trading, we will cover the following topics:

- Correlation
- Cointegration
- How to choose stocks for pairs trading?
- What is z-score?
- Defining Entry points
- Defining Exit points
- A simple Pairs trading strategy in Excel
- Explanation of the model

Statistics play a crucial role in the first challenge of deciding the pair to trade. The pair is commonly chosen from the same basket of stocks, for instance, Microsoft and Google (technology domain) or ICICI & Axis (Indian Banking) or Nifty Index and MSCI index (market indices). Among each domain, there are thousands of pairs are possible. The best ones are those which are based on mathematical or statistical tests. We will learn about two statistical methods in the next section of pairs trading.

## Correlation

Though not common, a few Pairs Trading strategies look at correlation to find a suitable pair to trade.

Correlation is quantified by the correlation coefficient ρ, which ranges from -1 to +1. The correlation coefficient indicates the degree of correlation between the two variables. The value of +1 means there exists a perfect positive correlation between the two variables, -1 means there is a perfect negative correlation and 0 means there is no correlation.

A perfect positive correlation is when one variable moves in either up or down direction, the other variable also moves in the same direction with the same magnitude while a perfect negative correlation is when one variable moves in the upward direction, the other variable moves in the downward (i.e. opposite) direction with the same magnitude.

The correlation coefficient for the two variables is given by

Correlation(X,Y) = ρ = COV(X,Y) / SD(X).SD(Y)

where, cov (X, Y) is the covariance between X & Y while SD (X) and SD(Y) denotes the standard deviation of the respective variables.

If the correlation is high, say 0.8, traders may choose that pair for pairs trading. 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.

Spread = log(a) – nlog(b), where ‘a’ and ‘b’ are prices of stocks A and B respectively.

For each stock of A bought, you have sold n stocks of B.

Now, both ‘a’ and ‘b’ increases in such a way that the value of spread decreases. 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 pairs trading.

Let us now move to the next section in pairs trading basics, ie Cointegration.

## Cointegration

The most common test for Pairs Trading is the cointegration test. 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 this statement above. The two-time series variables, in this case, are the log of prices of stocks A and B. Linear combination of these variables can be a linear equation defining the spread:

As you know, Spread = log(a) – nlog(b), where ‘a’ and ‘b’ are prices of stocks A and B respectively.

For each stock of A bought, you have sold n stocks of B.

If A and B are cointegrated then it implies that this 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 remains constant over time. So if we start with ‘n’, which is called the hedge ratio, so that spread = 0, the property of stationary implies that the expected value of spread will remain as 0. Any deviation from this expected value is a case for statistical abnormality, hence a case for pairs trading!

With the theory in mind, let us try to answer the question which you might be thinking of, in the next section of Pairs trading basics.

## How to choose stocks for pairs trading?

For any pair of stocks, define the spread as below:

Spread = log(a) – nlog(b), where ‘a’ and ‘b’ are prices of stocks A and B respectively.

Assumption: n, the hedge ratio is constant.

Calculate ‘n’ using regression so that spread is as close to 0 as possible. 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 tell us how much the actual value of ‘spread’ deviates from 0 for the calculated ‘n’. 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.

Run the Dicky Fuller test on the spread (more complicated and popular version is called Augmented Dicky Fuller Test or ADF) values inserting the value of ‘n’.

Dickey Fuller test is a hypothesis test which gives pValue as the result. If this value is less than 0.05 or 0.01, we can say with 95% or 99% confidence that the signal is stationary and we can choose this pair.

So far, we have discussed the challenges and statistics involved in selecting a pair of stocks for statistical arbitrage. We understood that by using the cointegration tests, we can say within a certain level of confidence interval that the spread between the two stocks is a stationary signal. In other words, this signal is mean-reverting. The spread is defined as:

Spread = log(a) – nlog(b), where ‘a’ and ‘b’ are prices of stocks A and B respectively. For each stock of A bought, you have sold n stocks of B. n is calculated by regressing prices of stocks A and B.

Having already established that the equation above is mean reverting, we now need to identify the extreme points or threshold levels which when crossed by this signal, we 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. In the next section, along with the z-score, we will also do a brief dive in Moving averages which is another important component in Pairs trading.

## What is z-score?

Simply put, given a normal distribution of raw data points z-score is calculated so that the new distribution is a normal distribution with mean 0 and standard deviation of 1. Having such a distribution ~ N(0, 1) is very useful for creating threshold levels. For example, 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 mean 0 and standard deviation of 1. It is easy to create threshold levels for this distribution such as 1.5 sigma, 2 sigma, 2.5 sigma, and so on.

### How to calculate z-score?

z = (x – mean) / standard deviation, where x is a raw data point and z is the z-score.

Mean and standard deviation can be rolling statistics for a period of ‘t’ days or minutes or time intervals.

### Moving average

We divide the data into subsets of size ‘t’, where,‘t’ specifies a fixed time period for which the average is to be calculated. For example, to calculate the moving average of the prices of stock A where ‘t’ is 10 days, we start by calculating average after the first 10 days in the dataset. So we calculate moving average at 10th day, 11th day, 12th day and so on. The average is moving or rolling. Moving average and the standard deviation is calculated for ‘t’ as 10 days in the table below.

The moving average for 1-08-2001 or 11th entry would not take into account the first data point, that is, stock A prices on 18-07-2001.

Using these concepts of moving averages and z-score we create the entry points for Pairs Trading.

## Defining Entry points

Let us denote the Spread as s. Thus,

Spread = s = log(a) – nlog(b)

Calculate z-score of ‘s’, using rolling mean and standard deviation for a time period of ‘t’ intervals. Save this as z.

Define threshold as anything 1.5-sigma, 2-sigma. This parameter will change as per the backtesting results without risking overfitting to data.

When Z-score crosses upper threshold, go SHORT:

Sell stock A

Buy stock B

When z-score crosses the lower threshold, go LONG:

Buy stock A

Sell stock B

Maintain the hedge ratio to calculate the stock quantity

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 do not happen. For example, if we chose entry signals at 2-sigma, we are expecting that the spread will revert back to mean from this threshold. However, it is possible that spread continues to blow up. Say it reaches 2.5-sigma and you incurred losses. To prevent further losses, you place Stop Loss at say 3-sigma.

In addition to placing a pre-defined stop-loss criterion such as 3-sigma or extreme variation from the mean, you can check on the co-integration value. If the co-integration is broken during 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 brought stock A and sold stock B as per the definition of spread in the article. The expectation is that spread will revert back to mean or 0. In a profitable situation, the mean would be approaching to zero or very close to it. You can keep Take Profit scenario as when the mean crosses zero for the first time after reverting from threshold levels.

There can be many ways of defining take profits depending on your risk appetite and backtesting results.

Let us try to recap what we have understood so far. Pairs Trading is a trading strategy that matches a long position in one stock/asset with an offsetting position in another stock/asset that is statistically related. Pairs Trading can be called a mean reversion strategy where we bet that the prices will revert to their historical trends.

So far, we have gone through the concepts and now let us try to create a simple Pairs Trading strategy in Excel.

## A simple Pairs trading strategy in Excel

This excel model will help you to:

- Learn the application of mean reversion
- Understand of Pairs Trading
- Optimize trading parameters
- Understand significant returns of statistical arbitrage

### Why should you download the trading model?

As the trading logic is coded in the cells of the sheet, you can improve the understanding by downloading and analyzing the files at your own convenience. Not just that, you can play around the numbers to obtain better results. You might find suitable parameters that provide higher profits than specified in the article.

## Explanation of the model

In this example, we consider the MSCI and Nifty pair as both of them are stock market indexes. We implement mean reversion strategy on this pair. Mean reversion is a property of stationary time series. Since we claim that the pair we have chosen is mean reverting we should test whether it follows stationarity.

Plotting of the logarithmic ratio of Nifty to MSCI makes it appear to be mean reverting with a mean value of 2.088 but we use Dicky Fuller Test to test whether it is stationary with a statistical significance. The results under Cointegration output table shows that the price series is stationary and hence mean-reverting. Dicky Fuller Test statistic and a significantly low p-value (<0.05) confirms our assumption. Having determined that the mean reversion holds true for the chosen pair we proceed with specifying assumptions and input parameters.

### Assumptions

- For simplification purpose, 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.e. at the price available at the end of 5 minutes.
- Only the regular session (T) is traded
- Transaction costs are $0.375 for Nifty and $1.10 for MSCI.
- The margin for each trade is $990 (approximated to $1000).

### Input parameters

Please note that all the values for the input parameters mentioned below are configurable.

- Average of 10 candles (one candle is equal to every 5-minute price) is considered
- A “z” score of +2 is considered for buy and -2 for selling
- A stop loss of $100 and profit limit of $200 is set
- The order size for trading MSCI is 50 (1 lot) and for Nifty is 6 (3 lots)

The market data and trading parameters are included in the spreadsheet from the 12th row onwards. So when the reference is made to column D, it should be obvious that the reference commences from D12 onwards.

### Explanation of the columns in the Excel Model

Column C represents the price for MSCI.

Column D represents Nifty price.

Column E is the logarithmic ratio of Nifty to MSCI.

Column F calculates 10 candle average. Since 10 values are needed for average calculations, there are no values from F12 to F22.

The formula =IF(A23>$C$3, AVERAGE(INDEX($E$13:$E$1358, A23-$C$3):E22), "") means that the average should be calculated only if the data sample available is more than 10 (i.e. the value specified in cell C3), otherwise the cell should be blank.

Consider cell F22. Its corresponding cell A22 has a value of 10. Since A22>$C$3 fails, the entry in that cell is blank. The next cell F23 has a value since A23>$C$3 is true. Let’s move to the next column.

In column G, the formula, AVERAGE(INDEX($E$13:$E$1358, A23-$C$3):E22) calculates the average value of last 10 (as mentioned in cell C3) candles of column E data. Similar logic holds for column G where the standard deviation is calculated.

The “z” score is calculated in the column H. Formula for calculating “z” score is z= (x-μ)/(σ). Here x is the sample (Column E), μ is the mean value (Column F) and σ is the standard deviation (Column G).

Column I represents the trading signal. As mentioned in the input parameters, if “z” score goes below -2 we buy and if it goes above +2 we sell. When we say buy, we have a long position in 3 lots of Nifty and have a short position in 1 lot of MSCI. Similarly, when we say sell, we have a long position in 1 lot of MSCI and have a short position in 3 lots of Nifty thus squaring off the position. We have one open position all the time.

To understand what this means, consider two trading signals “buy” and “sell”. For the “buy” signal, as explained before, we buy 3 lots of Nifty future and short 1 lot of MSCI future. Once the position is taken, we track the position using the Status column, i.e. column M. In each new row while the position is continuing, we check whether the stop loss (as mentioned in cell C6) or take profit (as mentioned in cell C7) is hit. The stop loss is given the value of USD -100, i.e. loss of USD 100 and take profit is given the value of USD 200 in the cells C6 and C7 respectively.

While the position does not hit either stop loss or take profit, we continue with that trade and ignore all signals that are appearing in column I. Once the trade hits either the stop loss or take profit, we again start looking at the signals in column I and open a new trading position as soon as we have a Buy or Sell signal in column I.

Column M represents the trading signals based on the input parameters specified. Column I already has trading signals and M tells us about the status of our trading position i.e. are we long or short or booked the profits or exited at the stop loss. If the trade is not exited, we carry forward the position to the next candle by repeating the value of the status column in the previous candle. If the price movement occurs in such a way that it breaches the given TP or SL then we square off our position thus denoting it by “TP” and “SL” respectively.

Column L represents Mark to Market. It specifies the portfolio position at the end of time period. As specified in the input parameters we trade 1 lot of MSCI and 3 lots of Nifty. So when we trade our position is the appropriate price difference (depending on whether we are bought or sold) multiplied by the number of lots.

Column N represents the profit/loss status of the trade. P/L is calculated only when we have squared off our position. Column O calculates the cumulative profit.

### Outputs

The output table has some performance metrics tabulated. Loss from all loss-making trades is $3699 and profit from trades that hit TP is $9280. So the total P/L is $9280-$3699=$5581. Loss trades are the trades that resulted in losing money on the trading positions. Profitable trades are the successful trades ending in gaining cause. Average profit is the ratio of total profit to the total number of trades. Net average profit is calculated after subtracting the transaction costs which amounts to $91.77.

### Now it is your turn!

- First, download the model
- Modify the parameters and study the backtesting results
- Run the model for other historical prices
- Modify the formula and strategy to add new parameters and indicators! Play with logic! Explore and study!

Comment below with your results and suggestions

## Summary

Thus, we have understood the concept behind Pairs trading strategy, including correlation and cointegration. We also took a look at Z-score and defined the entry and exit points when we are executing a pairs trading strategy. We also created an Excel model for our Pairs Trading strategy!

Learn how to implement pairs trading/statistical arbitrage strategy in FX markets through a project work including live examples. If you want to dig deeper and try to find suitable pairs to apply the strategy, you can go through the blog on K-Means algorithm.

If you want to learn various aspects of Algorithmic trading then check out our Executive Programme in Algorithmic Trading (EPAT®). The course covers training modules like Statistics & Econometrics, Financial Computing & Technology, and Algorithmic & Quantitative Trading. EPAT® is designed to equip you with the right skill sets to be a successful trader. Enroll now!

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