5. (15 points total). Suppose that a stock's price at time ?' is expected to be uniformly distributed over integers in the range of 5 to 25, that is: q(x) - 20 5 5 x $ 25 otherwise Further, suppose that a put option with exercise price X = 16 trades on this stock with the following terminal payoff function: Pr - Max[X - 57. 0] a Calculate the probability that the put will be exercised. Calculate the expected value of the stock's price contingent on it exceeding the exercise price of the put. C- Calculate the conditional expected value of the put contingent on it being exercised. d. Calculate the expected time 7 value of the put. e. If time 7 = 2 and r = 0.01 what is the current value of the put? 6. (15 points total). Consider a one-time, two potential outcome framework where there exists Company Q stock current selling for $35 per share and a riskless $100 face value T-bill currently selling for $80. Suppose Company Q faces uncertainty, in that it will pay its owner either $30 or $75 in 1 year. Further assume that the physical probability that the stock will drop is 0.2. a List the risk-neutral probabilities for this payoff space. b . Compute values for the Radon-Nikodym derivative for this change of measure. C. Value call and put options on this stock, with exercise prices equal to X = 60. d. Does put-call parity hold for this example? 7 . (15 points total). Suppose that So - 0.7 to purchase a security, with potential payoffs given as follows: S1 = 3.S, = 1.5, and Sy = 0 such that under the physical probabilities, the expected value is Ep [S] = 0.85 and the variance equals Ep [S - E[S]]* = 0.80. a Find the physical probabilities in measure P. Find the risk-neutral probabilities in measure Q. Recall that under the risk-neutral probabilities, the expected value of the stock is equal to its current value (i.e., it is a martingale). Calculate the Radon-Nikodym derivatives for this change of measure. WE