Problem 1 What is the most important contribution of the Black and Scholes option pricing formula? Why is it important? The Black and Scholes formula assumes that i) the interest rate of the riskless savings account is constant, ii) the volatility of the stock is constant, and iii) the noise term is the stock return dynamics is a normally distributed random variable. Please comment on how realistic these assumptions are. Why did Black and Scholes make these assumptions?

Problem 2 Let C(S)K,T ,r, be the Black and Scholes formula for the price of a European call option with current stock price S, strike price K, expiration date T, risk-free rate r, and return volatility . Write down the expressions for the derivatives C/S, 2C/S2 , C/r, C/T, C/. How do we call these derivatives? Are you able to derive sign restrictions on these derivatives? If yes, what are these restrictions?

Problem 3 Let P(S)K,T ,r, be the Black and Scholes formula for the price of a European put option with current stock price S, strike price K, expiration date T, risk-free rate r, and return volatility . Write down the put-call parity formula for European put and call options with the same strike price, expiration date, and underlying asset. Use the put-call parity and your answers in Problem 2 to derive the expressions for the derivatives P/S, 2P/S2 , P/r, P/T, P/. How do we call these derivatives? Are you able to derive sign restrictions on these derivatives? If yes, what are these restrictions?