dstefan
06-14-2008, 04:30 PM
Chapter 3, Problem 16
A portfolio containing derivative securities on only one asset has Delta 5000 and Gamma -200. A call on the asset with \Delta(C) = 0.4 and \Gamma(C) = 0.05, and a put on the same asset, with \Delta(P) = -0.5 and \Gamma(P) = 0.07 are currently traded. How do you make the portfolio Delta--neutral and Gamma--neutral?
Solution:
Take positions of size x_1 and x_2, respectively, in the call and put options specified above. The value \Pi of the new portfolio is \Pi = V + x_1 C + x_2 P, where V is the value of the original portfolio. This portfolio will be Delta- and Gamma-neutral, if
\Delta(\Pi) = \Delta(V) + x_1 \Delta(C) + x_2 \Delta(P) = 5000 + 0.4 x_1 - 0.5 x_2 = 0
\Gamma(\Pi) = \Gamma(V) + x_1 \Gamma(C) + x_2 \Gamma(P) = -200 + 0.05 x_1 + 0.07 x_2~=~ 0
The solution of this linear system is x_1 = -4716.98 and x_2 = 6226.42
To make the initial portfolio as close to Delta- and Gamma-neutral as possible by only trading in the given call and put options, 4717 calls must be sold and 6226 put options must be bought. The Delta and Gamma of the new portfolio are
\Delta(\Pi) = \Delta(V) - 4717 \Delta(C) + 6226 \Delta(P) ~=~ 0.2;
\Gamma(\Pi) = \Gamma(V) + 4717 \Gamma(C) + 6226 \Gamma(P) ~=~ -0.03
To understand how well balanced the hedged portfolio \Pi is, recall that the initial portfolio had \Delta(V) = 5000 and \Gamma(V) = -200.
A portfolio containing derivative securities on only one asset has Delta 5000 and Gamma -200. A call on the asset with \Delta(C) = 0.4 and \Gamma(C) = 0.05, and a put on the same asset, with \Delta(P) = -0.5 and \Gamma(P) = 0.07 are currently traded. How do you make the portfolio Delta--neutral and Gamma--neutral?
Solution:
Take positions of size x_1 and x_2, respectively, in the call and put options specified above. The value \Pi of the new portfolio is \Pi = V + x_1 C + x_2 P, where V is the value of the original portfolio. This portfolio will be Delta- and Gamma-neutral, if
\Delta(\Pi) = \Delta(V) + x_1 \Delta(C) + x_2 \Delta(P) = 5000 + 0.4 x_1 - 0.5 x_2 = 0
\Gamma(\Pi) = \Gamma(V) + x_1 \Gamma(C) + x_2 \Gamma(P) = -200 + 0.05 x_1 + 0.07 x_2~=~ 0
The solution of this linear system is x_1 = -4716.98 and x_2 = 6226.42
To make the initial portfolio as close to Delta- and Gamma-neutral as possible by only trading in the given call and put options, 4717 calls must be sold and 6226 put options must be bought. The Delta and Gamma of the new portfolio are
\Delta(\Pi) = \Delta(V) - 4717 \Delta(C) + 6226 \Delta(P) ~=~ 0.2;
\Gamma(\Pi) = \Gamma(V) + 4717 \Gamma(C) + 6226 \Gamma(P) ~=~ -0.03
To understand how well balanced the hedged portfolio \Pi is, recall that the initial portfolio had \Delta(V) = 5000 and \Gamma(V) = -200.