dstefan
03-22-2008, 01:14 PM
Chapter 1, Problem 11:
Consider a portfolio with the following positions:
\bullet long one call option with strike K_1 = 30 ;
\bullet short two call options with strike K_2 = 35 ;
\bullet long one call option with strike K_3 = 40 .
All options are on the same underlying asset and have maturity T. Draw the payoff diagram at maturity of the portfolio, i.e., plot the value of the portfolio V(T) at maturity as a function of S(T), the price of the underlying asset at time T.
Solution:
A butterfly spread is an options portfolio made of a long position in one call option with strike K_1, a long position in a call option with strike K_3, and a short position in two calls with strike K_2, where K_1 < K_2 < K_3 with K_2 = \frac{K_1+K_3}{2}; all optons have the same maturity and have the same underlying asset. The payoff at maturity of a butterfly spread is always nonnegative, and it is positive if the price of the underlying asset at maturity is between the strikes K_1 and K_3, i.e., if K_1 < S(T) < K_3.
For our particular example, the values of the three call options at maturity are, respectively,
C_1(T) ~=~ \max(S(T)-K_1,0) ~=~ \max(S(T)-30,0)
C_2(T) ~=~ \max(S(T)-K_2,0) ~=~ \max(S(T)-35,0)
C_3(T) ~=~ \max(S(T)-K_3,0) ~=~ \max(S(T)-40,0)
and the value of the portfolio at maturity is V(T) ~=~ C_1(T) - 2C_2(T) + C_3(T).
\begin{tabular}{|c|c|c|c|c|}\hline & S(T) < 30 & 30 < S(T) < 35 & 35 < S(T) < 40 & 40 < S(T) \\ \hline C_1(T) & 0 & S(T)-30 & S(T)-30 & S(T)-30 \\ \hline C_2(T) & 0 & 0 & S(T)-35 & S(T)-35 \\ \hline C_3(T) & 0 & 0 & 0 & S(T)-40 \\ \hline V(T) = C_1(T) - 2C_2(T) + C_3(T)& 0 & S(T)-30 & 40-S(T) & 0 \\ \hline \end{tabular}
Consider a portfolio with the following positions:
\bullet long one call option with strike K_1 = 30 ;
\bullet short two call options with strike K_2 = 35 ;
\bullet long one call option with strike K_3 = 40 .
All options are on the same underlying asset and have maturity T. Draw the payoff diagram at maturity of the portfolio, i.e., plot the value of the portfolio V(T) at maturity as a function of S(T), the price of the underlying asset at time T.
Solution:
A butterfly spread is an options portfolio made of a long position in one call option with strike K_1, a long position in a call option with strike K_3, and a short position in two calls with strike K_2, where K_1 < K_2 < K_3 with K_2 = \frac{K_1+K_3}{2}; all optons have the same maturity and have the same underlying asset. The payoff at maturity of a butterfly spread is always nonnegative, and it is positive if the price of the underlying asset at maturity is between the strikes K_1 and K_3, i.e., if K_1 < S(T) < K_3.
For our particular example, the values of the three call options at maturity are, respectively,
C_1(T) ~=~ \max(S(T)-K_1,0) ~=~ \max(S(T)-30,0)
C_2(T) ~=~ \max(S(T)-K_2,0) ~=~ \max(S(T)-35,0)
C_3(T) ~=~ \max(S(T)-K_3,0) ~=~ \max(S(T)-40,0)
and the value of the portfolio at maturity is V(T) ~=~ C_1(T) - 2C_2(T) + C_3(T).
\begin{tabular}{|c|c|c|c|c|}\hline & S(T) < 30 & 30 < S(T) < 35 & 35 < S(T) < 40 & 40 < S(T) \\ \hline C_1(T) & 0 & S(T)-30 & S(T)-30 & S(T)-30 \\ \hline C_2(T) & 0 & 0 & S(T)-35 & S(T)-35 \\ \hline C_3(T) & 0 & 0 & 0 & S(T)-40 \\ \hline V(T) = C_1(T) - 2C_2(T) + C_3(T)& 0 & S(T)-30 & 40-S(T) & 0 \\ \hline \end{tabular}