Consider a stock market with risk-free interest rate r where the price of stock X changes after every At time interval. It is known that at the end of each interval, the price of X will either go up by a factor u or go down by a factor d. 1.1. Suppose that the current price of X is So and the current price of an option on X with maturity At is f. Furthermore, suppose that the payoff from this option in the scenario of an upward price movement is fu and downward price movement is fd. Then, in a portfolio which is long A shares and short one option, find the value of A which makes it riskless. [107] 1.2. Next, using the no-arbitrage argument, find the relation between current option price f and parameters u,d, fu, far and At. Also provide the expression for risk-neutral probability (10%) p. 1.3. For So = 40, u = 1.2, d= 0.8, At = 3 months, r = 12% per annum, compute the price of a European put option with maturity 6 months and strike price 42. (10%] 1.4. Repeat the calculations to compute the price of an American put option with maturity 6 months and strike price 42. [10%] 1.5. Repeat the calculations in the case when stock X pays a continuous dividend yield of 2% per annum to find the corresponding European put option price. (10%] Consider a stock market with risk-free interest rate r where the price of stock X changes after every At time interval. It is known that at the end of each interval, the price of X will either go up by a factor u or go down by a factor d. 1.1. Suppose that the current price of X is So and the current price of an option on X with maturity At is f. Furthermore, suppose that the payoff from this option in the scenario of an upward price movement is fu and downward price movement is fd. Then, in a portfolio which is long A shares and short one option, find the value of A which makes it riskless. [107] 1.2. Next, using the no-arbitrage argument, find the relation between current option price f and parameters u,d, fu, far and At. Also provide the expression for risk-neutral probability (10%) p. 1.3. For So = 40, u = 1.2, d= 0.8, At = 3 months, r = 12% per annum, compute the price of a European put option with maturity 6 months and strike price 42. (10%] 1.4. Repeat the calculations to compute the price of an American put option with maturity 6 months and strike price 42. [10%] 1.5. Repeat the calculations in the case when stock X pays a continuous dividend yield of 2% per annum to find the corresponding European put option price. (10%]
