The following questions have to do with mean-reversion trading of a single pair of cointegrated stocks.

= 1. Suppose that Zt is an OU process with dZt -KZ dt + odWt. Use Ito's lemma to confirm that Zt Zoe-*t to e-kt-s) dW.. What is the long-term mean and variance of Zt? 2. There exists another Brownian motion Bt such that Ztekt Zo = By(t). What is 7(t)? 3. Consider the trade signal where we open a position if |Z0| >0/V2K and then close at time t = inf{t > 0\2+ = 0}. Use the reflection principle? and By(t) from part 2 to show that P(7 0. 4. If Zo o/v2k, what is the probability that the pair trade is closed out by time t=l/k for l 1, 2, 3? 5. Suppose two stocks S and S are co-integrated as Z+ = log(St) Blog(S?), where Zt is a mean-reverting process. In terms of positions in Si and S2, what is a static pair trade to make if = 1? Why is a pair trade non-static if # 1? For Brownian motion Bt and stopping T = - inf{t > 0|B= a} where a > 0, the distribution of 7 is P(T 0/V2K and then close at time t = inf{t > 0\2+ = 0}. Use the reflection principle? and By(t) from part 2 to show that P(7 0. 4. If Zo o/v2k, what is the probability that the pair trade is closed out by time t=l/k for l 1, 2, 3? 5. Suppose two stocks S and S are co-integrated as Z+ = log(St) Blog(S?), where Zt is a mean-reverting process. In terms of positions in Si and S2, what is a static pair trade to make if = 1? Why is a pair trade non-static if # 1? For Brownian motion Bt and stopping T = - inf{t > 0|B= a} where a > 0, the distribution of 7 is P(T