Assume the Black-Scholes model for a stock price {St, t >= 0} that does not pay dividends. consider an option whose payoff at time T is Ka-ST (Ka minus ST) if ST <= Kb and pays 0 otherwise, where Kb <= Ka.

a) use risk-neutral valuation to determine a formula for the price at time 0 of this option. [can use the lognormal result described in the slides without proving them]

b) Consider the derivative that pays ST-Ka (ST minus Ka) at time T if ST > Kb and pays 0 otherwise. 1) draw the payoff diagram of this derivative. 2) determine the payoff at time T of a portfolio consisting of a long position in this derivative and a short position in the option priced in part a).

c) use your answer in part (a) and (b-ii) to find a formula for the price of the derivative described in (b)