(a) Write down an expression for the price of a European derivative in a Black-Scholes market in terms of an expectation under the risk-neutral measure, defining any additional notation that you use. (b) Consider an option on a non-dividend-paying stock when the stock price is $50, the exercise price is $49, the continuously compounded risk-free rate of interest is 5% per annum, the volatility is 25% per annum, and the time to maturity is six months. (i) Calculate the price of the option using the Black-Scholes formula, if the option is a European call. (ii) Determine the price of the option if it is an American call. (iii) Calculate the price of the option if it is a European put. (iv) Determine how the prices of the contracts in parts (i), (ii) and (iii) would change in the case of a dividend-paying underlying stock. [Note that you do not have to perform any further calculations.]