A spread follows dX = -theta*(X - mu)*dt + sigma*dW with theta = 0.05 per day and sigma = 0.8 per square root day. What is its stationary standard deviation, and what does a two sigma entry rule imply about the trade?

A spread follows dX = -theta*(X - mu)*dt + sigma*dW with theta = 0.05 per day and sigma = 0.8 per square root day. What is its stationary standard deviation, and what does a two sigma entry rule imply about the trade?

Approach: The stationary variance of a mean reverting diffusion balances the diffusion input against the pull toward the mean. Then read the half life off theta and combine the two into a trading rule.

2.53. The stationary variance of an Ornstein-Uhlenbeck process is sigma^2/(2*theta) = 0.64/0.10 = 6.4, so the stationary standard deviation is sqrt(6.4) = 2.53 and a two sigma entry sits 5.06 units from mu. The half life is ln(2)/theta = 13.9 days, so a position entered there is expected to give back half its distance in a fortnight and the expected gross profit per trade is on the order of 5.06 units against a holding period of several weeks. Two consequences follow for the rule. A wider entry earns more per trade and fires far less often, so the threshold that maximises profit per unit of time is well inside the one that maximises profit per trade. Estimates of theta from short samples are biased toward faster mean reversion, so the realised half life is usually longer than the fitted one and the capital is tied up longer than the backtest assumed.

Follow-up: What entry threshold maximises expected profit per unit time for this process with a cost of 0.1 per round trip?

Key concepts: ornstein-uhlenbeck, mean reversion, half life, stationary variance.