Suppose that the price S(t) of an asset follows a geometric Brownian motion with expected rate of return p and volatility o, so that for any t > 0 S(t) (5t S(0) where Z is a standard normal random variable. Moreover, suppose that the risk free interest rate is r (compounded continuously). (a) For this part of the question, suppose that S(0) = 50, y = 0.04, r = 0.05 and 0 = 0.5. Consider a put option with maturity T = 2 and strike K = 60. (i) Calculate the no-arbitrage price of this put option at time t = 0. (ii) Find the A of this put option at time t = 0. (iii) Consider a portfolio consisting of 100 of these put options. What position in the underlying asset do you have to add to your portfolio to make the portfolio A-neutral? Why might you want to do this? Also explain the connection to the replicating portfolio. For each part clearly state any results from lectures that you use. (b) Suppose K >0 and T > 0. Consider an exotic option with payoff (S(T) K)?. (i) Why might you be interested in investing in such an option? (ii) Find the no-arbitrage price for such an option at time t = 0. (iii) Find a formula for the A of this option at time t = 0. Suppose that the price S(t) of an asset follows a geometric Brownian motion with expected rate of return p and volatility o, so that for any t > 0 S(t) (5t S(0) where Z is a standard normal random variable. Moreover, suppose that the risk free interest rate is r (compounded continuously). (a) For this part of the question, suppose that S(0) = 50, y = 0.04, r = 0.05 and 0 = 0.5. Consider a put option with maturity T = 2 and strike K = 60. (i) Calculate the no-arbitrage price of this put option at time t = 0. (ii) Find the A of this put option at time t = 0. (iii) Consider a portfolio consisting of 100 of these put options. What position in the underlying asset do you have to add to your portfolio to make the portfolio A-neutral? Why might you want to do this? Also explain the connection to the replicating portfolio. For each part clearly state any results from lectures that you use. (b) Suppose K >0 and T > 0. Consider an exotic option with payoff (S(T) K)?. (i) Why might you be interested in investing in such an option? (ii) Find the no-arbitrage price for such an option at time t = 0. (iii) Find a formula for the A of this option at time t = 0