dstefan
03-22-2008, 01:18 PM
Chapter 1, Problem 13:
Which of the following two portfolios would you rather hold:
\bullet Portfolio 1: Long one call option with strike K = X-5 and long one call option with strike K = X+5;
\bullet Portfolio 2: Long two call options with strike K = X?
(All options are on the same asset and have the same maturity.)
Solution:
Note that being long Portfolio 1 and short Portfolio 2 is long a butterfly spread, and therefore will always have positive (or rather nonnegative) payoff at maturity. Therefore, if you are to ASSUME a position in either one of the portfolios (not to purchase the portfolios), you are better off owning Portfolio 1, since its payoff at maturity will always be at least as big as the payoff of Portfolio 2.
V(T) ~=~ V_1(T) - V_2(T) ~=~ \max(S(T)-(X-5),0) + \max(S(T)-(X+5),0) - 2 \max(S(T)-X,0) .
\begin{tabular}{|c|c|c|c|c|}\hline & S(T) < X-5 & X-5 < S(T) < X & X < S(T) < X+5 & X+5 < S(T) \\ \hline V(T) & 0 & S(T)-(X-5) & X+5-S(T) & 0 \\ \hline \end{tabular}
Which of the following two portfolios would you rather hold:
\bullet Portfolio 1: Long one call option with strike K = X-5 and long one call option with strike K = X+5;
\bullet Portfolio 2: Long two call options with strike K = X?
(All options are on the same asset and have the same maturity.)
Solution:
Note that being long Portfolio 1 and short Portfolio 2 is long a butterfly spread, and therefore will always have positive (or rather nonnegative) payoff at maturity. Therefore, if you are to ASSUME a position in either one of the portfolios (not to purchase the portfolios), you are better off owning Portfolio 1, since its payoff at maturity will always be at least as big as the payoff of Portfolio 2.
V(T) ~=~ V_1(T) - V_2(T) ~=~ \max(S(T)-(X-5),0) + \max(S(T)-(X+5),0) - 2 \max(S(T)-X,0) .
\begin{tabular}{|c|c|c|c|c|}\hline & S(T) < X-5 & X-5 < S(T) < X & X < S(T) < X+5 & X+5 < S(T) \\ \hline V(T) & 0 & S(T)-(X-5) & X+5-S(T) & 0 \\ \hline \end{tabular}