The concept of equity pairs trading is relatively straight-forward:

• Identify pairs whose spread historically fluctuates around the same level
• Estimate this average historical spread
• Investigate how the spread deviates from this average value at any point in time
• Take a long position on the stock that trades below the average and a short position on the one that trades above

The underlying assumption is that the spread follows a mean reverting process- which means that eventually it will move back towards its historical average.

Key Concepts

The following items are crucial in our analysis:

• Co-Integration:   The metric that shows if the spread historically fluctuates around   the same levels.
• Correlation:   A measure of co-movement, and dependency. It is essentially a   statistical technique that can show whether and how pairs of   variables are related.
• Back-testing:   The process of applying a strategy to historical data in order to   see how good it performs and extract meaningful statistics.

Co-integration

• If two or more series are individually integrated but some linear combination of them has a lower order of integration, then the series are said to be co-integrated.
• In time series, the order of integration reflects the minimum number of differences required to obtain a covariance stationary series.
• A stochastic process is strictly stationary when its joint probability distribution does not change when shifted in time. Consequently, parameters such as the mean and variance, also do not change over time.
• Hence, if the spread of a pair of stocks is co-integrated it means that it does not change over time.
• Since, it is impossible to identify strictly stationary spreads, we are seeking for weak stationarity. Practically, their spread is expected to fluctuate around a tight range over time rather than remain unchanged.
• The mathematics essentially show that if you go long stock A and short stock B with some appropriate hedging factor to cancel out the drift/growth terms in the Brownian motion then you are left with a stationary signal which is the spread between the two stocks.
• The math also show that in expectation the daily change in the spread is zero and hence any deviation from this presents the opportunity for a trade.
• It is assumed that the stock growth terms are constant (or drifting slowly over time both at the same rate) or in other words the hedge ratio is constant.

spread < historical average + n * historical st. deviation

OR   spread > historical average – n * historical st. deviation

Correlation

Problem:

• The implementation of the Dickey Fuller test required for the co-integration test requires a regression analysis of the log price returns of each pair.
• When investigating a large number of stocks you end up with an extremely large number of combinations.
• The model is significantly slow.

Solution:

• Introduce another filter before testing for co-integration.
• We only test pairs that are strongly correlated: their correlation is above a certain threshold.
• As a result, the number of pairs investigated is decreased significantly whereas the filtered pairs are more likely to be co-integrated.