1. Let Ω = [0, 1] with the σ-algebra B of Borel sets and let µ0  the Lebesgue measure on [0, 1]. Find the E(X|Y) if

X(ω) = 2ω and Y (ω) = ω 2

X(ω) = 2ω and Y (ω) = ω if ω ∈ [0, 1/2) and Y (ω) = 1/2 if ω ∈ [1/2, 1] X(ω) = 1 − ω and Y (ω) = 0 if ω ∈ [0, 1/2) and Y (ω) = ω if ω ∈ [1/2, 1]

2. Find and compare the σ-algebra generated by the random Y  and E(X|Y) for the following problem: A die is rolled twice; Y is the first outcome and X is the sum of outcomes. How many elements do these σ-algebras have?

3. Suppose we have a risky asset with inital price S0  = $3 and at time 1, S = {$1, $5}. Consider a financial claim at time t = 1

F :=

Is F the payoff of a put or call option? What is the strike/exercise price?

4. Consider a non-dividend paying stock whose initial stock price is 62 and which has a log-volatility of σ = 0.20. The interest rate r = 10%, compounded monthly. Consider a 5-month option with a strike price of 60 in which after exactly 3 months the purchaser may declare this option to be a (European) call or put option.

Assume u = 1.05943 and d =  = 0.94390

(a) Compute the values of the binomial lattice for 5 1 month period.

(b)

(c)

Compute the appropriate risk-free rate.

Find the risk-neutral probability p˜ of going up?