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Statistical Arbitrage / Pairs Trading: Distance, Cointegration, and Kalman Filter Methods

Cointegration and the Common-Trends Model

Two random walks can never be traded against each other in isolation — each price series wanders without a stable mean to revert to. But two companies exposed to the same underlying economic driver (the same commodity input, the same regulatory regime, the same consumer trend) have log-price series that share the same random-walk component. That shared component cancels in a linear combination, leaving a stationary residual with a well-defined equilibrium. This is cointegration, and the equity pairs trade is a bet that the residual will revert to zero.
This note develops the formal machinery behind that bet: the common-trends representation, the error-correction model, and the simple trading strategy that Vidyamurthy constructs from them. The hypothesis-testing procedures that certify whether a candidate pair actually passes the cointegration criterion are in the companion derivation note.
Learning objectives. By the end of this note you should be able to:
  • state the definition of cointegration for two I(1) series and identify the cointegrating vector γ\gamma;
  • write down the common-trends representation and explain why it implies stationarity of the spread;
  • interpret the error-correction coefficients αA\alpha_A and αB\alpha_B and explain what sign constraints must hold for the system to error-correct;
  • apply the cointegration model to log stock prices and derive the trading strategy from the spread dynamics;
  • compute the portfolio return as the increment to the spread and describe the Δ\Delta-band trading rule.

In this lesson

    1. The intuition: a shared random walk
    1. The common-trends representation
    1. The error-correction representation
    1. Applying the model to log stock prices
    1. A simple Δ\Delta-band trading strategy
    1. Estimating the cointegrating coefficient

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