Problem 330 of 333MediumStochastic CalculusP330

Two half-lives into a pairs trade

You model the spread between two stocks as an Ornstein-Uhlenbeck process
$dX_t = -\lambda X_t\,dt + \sigma\,dW_t,$
with time measured in trading days. Estimation gives a half-life of 5 days and a stationary standard deviation of 1 spread point. The spread is at $+2$ . You sell it and plan to buy it back in 10 trading days, by which time you expect most of the gap to have closed.
(a) Find $\lambda$ and $\sigma$ .
(b) Find the mean and standard deviation of the spread 5 days and 10 days from now.
(c) What is the probability that the spread is still above 0 when you buy it back? What is the probability that the trade is losing money after 5 days?
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