Assume the stock price of a company follows the Geometric Brownian Motion (refer to the formula sheet for the equation of Geometric Brownian Motion), with expected stock return u=12% and volatility of stock return at o = 25%, The current stock price is $34. a) About such stock price S in general, show that Geometric Brownian Motion for S implies that S has lognormal distribution, i.e. In(S,) In(S.) ${(u-0.50?),oT). (5 marks) b) Specify the probability distribution for the log of the stock price, i.e. In(S), in six months ( i.e., at time T=0.5). (5 marks) c) What is the confidence interval at 95% level for the stock price in six months (We know that there is a 95% probability that a normally distributed variable has its value within 1.96 standard deviation around its mean.) (4 marks) d) What is the probability that a European call option on the stock with strike price $47 and a maturity date in six months will be exercised. (You may give an answer in terms of N(x), i.e. the probability of taking a value less than x by a variable with standard normal distribution. ) (4 marks) Assume the stock price of a company follows the Geometric Brownian Motion (refer to the formula sheet for the equation of Geometric Brownian Motion), with expected stock return u=12% and volatility of stock return at o = 25%, The current stock price is $34. a) About such stock price S in general, show that Geometric Brownian Motion for S implies that S has lognormal distribution, i.e. In(S,) In(S.) ${(u-0.50?),oT). (5 marks) b) Specify the probability distribution for the log of the stock price, i.e. In(S), in six months ( i.e., at time T=0.5). (5 marks) c) What is the confidence interval at 95% level for the stock price in six months (We know that there is a 95% probability that a normally distributed variable has its value within 1.96 standard deviation around its mean.) (4 marks) d) What is the probability that a European call option on the stock with strike price $47 and a maturity date in six months will be exercised. (You may give an answer in terms of N(x), i.e. the probability of taking a value less than x by a variable with standard normal distribution. ) (4 marks)