Problem 4: Options (16 marks] (a) In this course, we have learnt the so-called "put-call parity" that describes a relation amongst Cr(K), Per(K), K, r and S. Suppose r is no longer a constant yet a deterministic function r(5), where & [t, T), write down the corresponding form of the put-call parity and prove why it is of this form so that there is no arbitrage. righ, where (b) One typical option strategy is called "call-backspreads". Along position in a European-type call-backspread has the following payoff: profit DE Stock price loss Explain clearly how this strategy can be constructed with calls only. Write down also the payoff of this strategy at maturity for different ranges of St. (c) Given that the underlying stock price is $10 and the continuously compounded risk-free interest rate is 6%, you observe the following 6-month barrier option prices. Options Price (4) A down-and-in call with strike price 11 and barrier 9 28.2 A down-and-in put with strike price 11 and barrier 8 73.5 A down-and-out put with strike price 11 and barrier 8 83.1 A down-and-out call with strike price 11 and barrier 9| 43.9 Is there any arbitrage opportunity? If so, construct an arbitrage portfolio; if not, argue clearly why there is no such an opportunity. Problem 4: Options (16 marks] (a) In this course, we have learnt the so-called "put-call parity" that describes a relation amongst Cr(K), Per(K), K, r and S. Suppose r is no longer a constant yet a deterministic function r(5), where & [t, T), write down the corresponding form of the put-call parity and prove why it is of this form so that there is no arbitrage. righ, where (b) One typical option strategy is called "call-backspreads". Along position in a European-type call-backspread has the following payoff: profit DE Stock price loss Explain clearly how this strategy can be constructed with calls only. Write down also the payoff of this strategy at maturity for different ranges of St. (c) Given that the underlying stock price is $10 and the continuously compounded risk-free interest rate is 6%, you observe the following 6-month barrier option prices. Options Price (4) A down-and-in call with strike price 11 and barrier 9 28.2 A down-and-in put with strike price 11 and barrier 8 73.5 A down-and-out put with strike price 11 and barrier 8 83.1 A down-and-out call with strike price 11 and barrier 9| 43.9 Is there any arbitrage opportunity? If so, construct an arbitrage portfolio; if not, argue clearly why there is no such an opportunity