Solve the following question precisely

The price process of a non-dividend paying stock S, satisfies the following stochastic differential equation dS, = MS, di + 05, dW where I' is a Brownian motion under the real-world probability measure P. Let V(1) be the value at / of a self-financing portfolio, consisting of 1, stocks and ', cash bond. (i) Show that die y (1) = Q,d(e 's,) 137 (ii) Determine the conditions under which the discounted value e-") (() is a martingale. [3] [Total 6] (1) State the main assumptions underpinning the Black-Scholes model. [3] Consider a put option on a non-dividend paying stock when the stock price is 68, the exercise price is 9, the continuously compounded risk-free rate of interest is 2% per annum, the volatility is 209% per annum. and the time to maturity is three months. (ii) Calculate the price of the option using the Black-Scholes model. [4] (iii) Discuss how the price of the contract in part (ii) would change if the rate of interest increases. (There is no need to carry out further calculations.) [2] [Total 9]