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Suppose the economy in Example 1, Lecture 1 lasts for three quarters. Similar to Example 5 of Lecture 1, consider a security that pays de = $1 if the economy state in quarter t is G and dı = $0 if the economy state in quarter t is B. 1. What is the sample space ? 2. Following Example 1, find the filtration that corresponds to the o-algebras Fi at t = 0,1,2, 3. (If the answer is too long, a short description in words will suffice) 3. Calculate the probability measure P that is associated with each o-algebra F, 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 = dı + 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:N + R. (b) Find the smallest possible o-algebra that makes Y a random variable.

Example 1 Suppose the state of the economy in each quarter is either good (G) with probability q and bad (B) with probability 1 4. Suppose further that these probabilities are identical and independent over time. Consider the following probability space of the state of the economy in the next two quarters: (1) 2 = {GG, GB, BG, BB) (2) Even in this simple example, there are several o-algebras. Let's consider the following class of o-algebras called a filtration which will play an important role later:

Example 1 The simplest (coarsest) a-algebra is 5o = {0,2}. 1 at quarter 0, no information about the economy's states in the next two quarters. A finer one is F1 = {1,{GG, GB), {BG, BB), 2}, when we arrive at quarter 1. we know what the state of the cconomy in this quarter is. Therefore, we know that either event {GG, GB} or event {BG, BB) is realized. The finest o-algebra is F= 2.2 = all possible subsets of 12. At date 2, we know which events have happened. Note F CFCF. Such a collection of a-algebras is called a filtration. Ft represents information known up to t. (3) The probability measures associated with Fo and Fi are: For F. P() = 0, P(12) = 1. For $1. P(0) = 0, P(12) = 1, P({GG, GB}) = 4, P{{BG, BB})=1-4.

Example 5 In Example 1, consider a security that pays de = $1 if the economy state in quarter t is G and de = $0 if the economy state in quarter t is B. Then X = dı + d2 is a random variable with 0(X) ={0,2, {BB}, {GG}, {GB, BG}, {BB, GG}, {GB, BG, BB}, {GB, BG, GG}} 2? Note that o(X) represents information known by observing X.