4. An option (with strike of 8 euros) is written on an underlying stock with current price of 8 euros and without dividends. At each ensuing two semesters that stock price can move up by 40% with probability p=0.8 or down by 60% with probability 1p=0.2. Denote by (St)t=0,1,2 the stochastic process representing the underlying stock price, where t=1 stands for one semester and t=2 for two semesters. The risk-free interest rate available on the market is 4% per year with continuously compounded. (a) Is the stochastic process (St)t=0,1,2 describing the underlying stock dynamics a martingale with respect to the real world probability P ? (b) By letting S~t denote the underlying value at time t discounted at the present date, find a probability measure Q with respect to which (S~t)t=0,1,2 is a martingale. (c) What is the fair initial value of a European call option written on the underlying considered before with maturity T=1 year? And what is the fair value of the European put on the same underlying and with the same parameters? (d) Establish if it is optimal to exercise the American call on the same underlying and with the same parameters of the options of the previous item before maturity. What would then be its fair value