Math 381– Financial Mathematics I – Fall Term 2014-15
Practice Problems: 13 December 2014
Directions. The following list of problems are similar to those which will be asked on the upcoming final exam.
1. Assume that the stock price St follows a log-normal distribution, meaning lnSt is normal random variable. If the variance of lnSt is σ
2 t , then use the no arbitrage
condition on St to determine the mean of lnSt.
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).
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