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 ?

2. Following what we did in Example 1, find the filtration that corresponds to the -algebras Ft at t = 0,1,2,3. (If the answer is too long, a short description in words will suffice)

3. Calculate the probability measure Pt that is associated with each 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 = $2.5 and maturity of T = 3. Recall that the payoff for this put option is Y = max(K X, 0).

(a) DescribeY asamap: Y :R. (b) Is Y a random variable of the probability space (,F2,P2)? Why or why not?

(c) Find the smallest possible algebra that makes Y a random variable.