suppose over the period [0,T] a stock pays a dividend whose present value, at an interest rate r is D. Using no arbitrage arguments, prove the dividend-adjust put call relationship for European options is given by

Question 3 (a) (b) (60 Suppose over the period [0,T] a stock pays a dividend whose present value, at an interest rate r, is D. Using no arbitrage arguments, prove the dividend-adjust put-call relationship for European options is given by: 'rT clDlKe =plSo, where 80 is the stock price at t = 0, c and p are the respective prices of the call and put options at time t : O, K is the strike price and T is the maturity for both options. [4] Suppose the current stock price is $100 and assume that a 4% proportionate dividend will be paid to the stock holders in just under two months so the stock price will drop by 4% in two months. The volatility of the stock is or = 30% and the continuously compounded interest rate is 5%. Using a binomial tree in which each step is equivalent to 1 month and the other parameters are specied above, nd the price of (i) a European call option and (ii) a European put option. Assume both options have a strike price of $98 and a maturity of three months. [6] Derive the implied present value of the dividend D using (a) and (b). Does this implied value agree with a direct calculation of the present value of the expected dividend? If not, what does it indicates? [4] Now, suppose cl;D+Ke'l'T >p+So. Is there an arbitrage opportunity? If there is, what strategies do you employ to exploit the arbitrage prots? [6] [Tot al: 20 marks]