Problem 3.3 A stock price is currently $50. It is known that at the end of 6 months it will be either $55 or $45. The risk-free rate of interest with continuous compounding is 5% per annum. Calculate the value of a 6-month European call option on the stock with an exercise price K = $50.

Problem 3.2 Suppose that c1, c2 and c3 are the prices of European call options with strike prices K1, K2 and K3, respectively, where K3 > K2 > K1 and K3 K2 = K2 K1. All option have the same maturity. Show that c2 0.5(c1 + c3) This condition must be satisfied by all European option prices with the same maturity, and is known as the butterfly arbitrage condition. Hint: Consider a portfolio that is long one option with strike K1, long one option with strike K3, and short two options with strike price K2. Plot the payoff of this portfolio as a function of the stock price at maturity S(T ), and convince yourself that it is always positive.