(a) (5 points) Consider a portfolio of options that consist of buying one call option on an underlying stock for 3 dollars and buying two put options on the same underlying stock for 4 dollars each. Assume that all options in this problem have the same maturity and the same strike price of 35 dollars. Draw a pavof diagram of this portfolio of options and describe when it is useful. The payoff diagram should represent net payoffs (i.e. the cost of options should be reflected in the diagram.) (b) (10 points) Find the price of an option on an underlying asset such that the payoff at termination is max{ST - K, 0] where STis the average price of a path of asset prices from time 0 to T (maturity). Assume that the current price of the underlying asset is $50 and K-850 (i.e. strike price is $50) and use a two period binomial model of the asset price where the stock price will go up next period by a factor of 1.5 or go down with factor .5, there are no dividends, and the lending/borrowing rate is 10%. (c) (10 points) Derive an approximate formula for the price of a European put option on an underlving asset for some maturity 1' vears by using a binomial model for the asset price with N periods where the current price of the asset is S, the up factor is u, the down factor is d, and the strike price is K and the risk free rate is r. Assume no dividends. Your formula should converge to the exact formula for the price as N-oo (a) (5 points) Consider a portfolio of options that consist of buying one call option on an underlying stock for 3 dollars and buying two put options on the same underlying stock for 4 dollars each. Assume that all options in this problem have the same maturity and the same strike price of 35 dollars. Draw a pavof diagram of this portfolio of options and describe when it is useful. The payoff diagram should represent net payoffs (i.e. the cost of options should be reflected in the diagram.) (b) (10 points) Find the price of an option on an underlying asset such that the payoff at termination is max{ST - K, 0] where STis the average price of a path of asset prices from time 0 to T (maturity). Assume that the current price of the underlying asset is $50 and K-850 (i.e. strike price is $50) and use a two period binomial model of the asset price where the stock price will go up next period by a factor of 1.5 or go down with factor .5, there are no dividends, and the lending/borrowing rate is 10%. (c) (10 points) Derive an approximate formula for the price of a European put option on an underlving asset for some maturity 1' vears by using a binomial model for the asset price with N periods where the current price of the asset is S, the up factor is u, the down factor is d, and the strike price is K and the risk free rate is r. Assume no dividends. Your formula should converge to the exact formula for the price as N-oo
