Two stocks follow random walks with 2% daily volatility each and their daily increments have a correlation of 0.9. How far apart does the spread typically drift over a year, and what does that say about selecting pairs on correlation?

Two stocks follow random walks with 2% daily volatility each and their daily increments have a correlation of 0.9. How far apart does the spread typically drift over a year, and what does that say about selecting pairs on correlation?

Approach: Compute the volatility of the difference of the two increments, then note that a difference of random walks is itself a random walk and scale it over the year.

14.2%. The spread has daily volatility sigma*sqrt(2 - 2*rho) = 2%*sqrt(0.2) = 0.894%, and the difference of two random walks is a random walk, so its dispersion grows with the square root of time: 0.894%*sqrt(252) = 14.2% over a year and 20.1% over two. A correlation of 0.9 constrains the co-movement of the daily changes and says nothing about the level, so a pair chosen on correlation has no mechanism pulling the spread back and any stop loss is reached eventually with probability one. Cointegration is the separate property that some linear combination of the levels is stationary, which requires an error correction term in the dynamics. Correlation describes first differences and cointegration describes levels, so neither one implies the other.

Follow-up: How would you build two series that are cointegrated while their returns are almost uncorrelated?

Key concepts: cointegration, correlation, random walk, stationarity.