What is the sample space Ω

    Problem Set 1 Problem (An extension of Example 1 of Lecture 1) Suppose the economy in Example 1, Lecture 1 lasts for three quarters. Similar to Example 5 of Lecture 1, consider a security that pays dt = $1 if the economy state in quarter t is G and dt = $0 if the economy state in quarter t is B. 1. What is the sample space Ω? 2. Following Example 1, find the filtration that corresponds to the σ-algebras Ft at t = 0, 1, 2. (If the answer is too long, a short description in words will suffice) 3. Calculate the probability measure P that is associated with each σ−algebra Ft above for t = 0,1,2,3. (If the answer is too long, a short description in words will suffice) 4. Consider a security X with date-3 payoff defined as X = d1 + d2 + d3 Let Y be the payoff to a put option on X with a strike price of K=2 and maturity of T = 3. Recall that the payoff for this call option is Y = max (K − X, 0). (a) Describe Y as a map: Y: Ω→R. (b) Find the smallest possible σ−algebra that makes Y a random variable.