**Pairs Trading Overview**

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.

Pairs Trading Trigger:

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.