We model the future value of a stock as a random variable X, where X has cumulative

distribution function

Question 3. (a) We model the future value of a stock as a random variable X, where X has cumulative distribution function Fx(x) = 1 - exp{-Ar}, for r > 0. (i) Calculate the probability density function for X. (ii) A European call option on the stock with strike price c is a contract that gives the buyer the right (but not the obligation) to buy one unit of the stock at price c, where c > 0. The buyer will only exercise this option if X > c, in which case the payoff is given by C = (X - c)+ = max{ X - c, 0}. Calculate E(C). (b) Crashes of the stock market are modelled as a Poisson process with rate 0.1 per year. If N is the number of crashes by time t, then M has a Poisson distribution with mean 0.It; so for r = 0, 1, 2, .. . .: P(N = ) = (0.It)" r! exp{-0.1t}. (i) Calculate the probability density function for the time until the first crash. (ii) What is the probability of precisely one crash in each of the next two years? (iii) Calculate P(No = 1/N10 2 1). (iv) For 0