In this question, assume the Black-Scholes model. Consider a portfolio consisting of a long position in a European call option and a short position in a European put option. Both options have the same maturity T and strike price K, and are written on the same underlying asset. Let S(t) be the time-t price of the underlying.

a. (3 marks) Suppose S(t) has risen so far above K that it is almost guar- anteed that S(T) > K. Estimate the price of the portfolio at time t. Explain your mathematical reasoning.

b. (2 marks) Suppose S(t) has dipped so below K that it is almost guar- anteed that S(T) < K. Estimate the price of the portfolio at time t. Explain your mathematical reasoning.

c. (5 marks) Now suppose that the underlying price remains constant, i.e. S(t) = S(T). How does the value of the portfolio today differ from the final payoff (is it larger or smaller, or the same)? Explain, referring to your estimate in parts (a.) and (b.), the reason for the difference. Some marks are associated with the clarity of your explanation.