1. Assume that the stock price St follows a log-normal distribution, meaning ln St is normal random variable. If the variance of ln St is σ 2 t , then use the no arbitrage condition on St to determine the mean of ln St.
2. For any α > 0, use the no arbitrage condition to determine the fair market value of the European style option whose payoff function is g(St) = S α t .
3. Starting from the Black-Scholes-Merton option price for the European call option, determine the asymptotic value as t → 0.
4. Starting from the Black-Scholes-Merton option price for the European call option, determine the asymptotic value as σ → 0 and σ → ∞.
5. Let Pcall(K, S, r, t, σ) denote the Black-Scholes price for a European call option. show that for any α > 0, one has the relation Pcall(αK, αS, r, t, σ) = αPcall(K, S, r, t, σ). What is the economic meaning of α?
6. Let Pcall(K, S, r, t, σ) denote the Black-Scholes price for a European call option. show that for any α > 0, one has the relation Pcall(K, S, αr, t/α, √ α · σ) = Pcall(K, S, r, t, σ). What is the economic meaning of α?
7. Let gcall and gput denote the payoff functions of a European call and European put option, respectively, with the input parameters. Show that gcall - gput = St - Ke-rt , and, from this relation, deduce the put-call parity without assuming that the stock price is log-normal, but rather only assuming the no arbitrage condition.
8. Use the integral representation of the Black-Scholes price for the European style call option (rather than the form involving the cumulative normal distribution function) to show that as K increases, the call option price decreases.
9. Consider a European capped call option whose payoff function is given by g(S, K, M) = min{max{S - K, 0}, 0} where K is the strike price of the European style call option and M is the capped price. Show that the present value of such an option is equal to Pcall(K, S, r, t, σ) - Pcall(K + M, S, r, t, σ).