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Gamma

The gamma of a portfolio is defined as the sensitivity of delta when the stock price changes,

$\displaystyle \Gamma _{\Pi }=\frac{\partial \Delta _{\Pi }}{\partial S}=\frac{\partial ^{2}\Pi }{\partial S^{2}}\text{.} $

We already mentioned that delta is a function of the underlying asset's price, therefore we expect the delta of a portfolio to change across time, as the price of the asset changes. Gamma will give us a quantitative insight on the magnitude of these changes. It shares a lot of similarities with the convexity adjustment that we examined when we discussed the duration of a bond.

We have already analyzed how a portfolio can be made delta neutral, by taking a position in the underlying asset. In order to achieve gamma neutrality, the underlying asset is not sufficient. This is due to the fact that

$\displaystyle \Gamma _{S}=\frac{d^{2}S}{dS^{2}}=0$.

In fact we need instruments that are nonlinear with respect to the underlying asset price in order to achieve gamma neutrality. Options are perfect candidates for this job. On the other hand, the fact that $ \Gamma _{S}=0$ implies that after we have made the portfolio gamma neutral we can turn into achieving delta neutrality by taking a position in the underlying asset. The value of gamma will not be affected by this position. We call this strategy delta-gamma hedging.

Say that we hold a portfolio with value $ \Pi _{1}$ which has a given delta, $ %% \Delta _{\Pi _{1}}$, and gamma, $ \Gamma _{\Pi _{1}}$. Our goal is to find the position we have to take in a derivative of the underlying asset, in order to achieve gamma neutrality. Say that we use an option with price $ C$ with known delta, $ \Delta _{C}$, and gamma, $ \Gamma _{C}$. We also use the underlying asset, which has price $ S$ to achieve delta neutrality [of course $ \Delta _{S}=1$ and $ \Gamma _{S}=0$]. The resulting portfolio will be delta-gamma neutral.

Gamma neutrality
We want to buy $ w_{C}$ units of the option. This makes the value of our composite position equal to

$\displaystyle \Pi _{2}=\Pi _{1}+w_{C}C$.

The linearity of the derivatives will therefore give that

$\displaystyle \Gamma _{\Pi _{2}}=\frac{\partial ^{2}\Pi _{2}}{\partial S^{2}}=\Gamma _{\Pi _{1}}+w_{C}\Gamma _{C}\text{.} $

To achieve gamma neutrality we need to sell $ \Gamma _{\Pi _{1}}/\Gamma _{C}$ units of the option,

$\displaystyle w_{C}=-\frac{\Gamma _{\Pi _{1}}}{\Gamma _{C}}$.

Delta neutrality
We want to buy $ w_{S}$ shares of the underlying asset. This will make the value of our composite position equal to
$\displaystyle \Pi _{3}$ $\displaystyle =$ $\displaystyle \Pi _{2}+w_{S}S=\Pi _{1}+w_{C}C+w_{S}S$  
  $\displaystyle =$ $\displaystyle \Pi _{1}-\frac{\Gamma _{\Pi _{1}}}{\Gamma _{C}}C+w_{S}S$.  

Apparently the delta of this composite position is

$\displaystyle \Delta _{\Pi _{3}}=\Delta _{\Pi _{1}}-\frac{\Gamma _{\Pi _{1}}}{\Gamma _{C}}%% \Delta _{C}+w_{S}$.

To make this position delta neutral we have to sell $ \Delta _{\Pi _{1}}-\Gamma _{\Pi _{1}}\Delta _{C}/\Gamma _{C}$ shares, or

$\displaystyle w_{S}=-\Delta _{\Pi _{1}}+\frac{\Gamma _{\Pi _{1}}}{\Gamma _{C}}\Delta _{C}%% $   .

The value of our composite portfolio $ \Pi =\Pi _{3}$ is

$\displaystyle \Pi =\Pi _{1}-\frac{\Gamma _{\Pi _{1}}}{\Gamma _{C}}C-\left( \Delta _{\Pi _{1}}-\frac{\Gamma _{\Pi _{1}}}{\Gamma _{C}}\Delta _{C}\right) S\text{.} $

We can verify that $ \Delta _{\Pi }=\Gamma _{\Pi }=0$, achieving a delta-gamma hedge.

For a European call option, the value of gamma is given by

$\displaystyle \Gamma _{C}=\frac{\mathcal{N}^{\prime }\left( d_{1}\right) }{S\sigma \sqrt{T}%% }\text{.} $

Graphically, figure 8.4 gives Gamma across different moneyness and maturity levels.

Figure 8.4: Behavior of a Call option Gamma. Part (a) gives the behavior of the gamma on a three options with specifications $ (K,\tau,r,\sigma)$ equal to $ (1,0.2,0,0.5)$ for the solid line, $ (1,0.2,0,1)$ for the thin line, and $ (1,0.5,0,0.5)$ for the dashed line. Part (b) gives the behavior of the gamma for a contract which is in-the-money [solid line], at-the-money [thin line], and out-of-the-money [dashed line].
\begin{figure}\centering %% {\epsfig{file=c:/miktex/mymiktex/derivatives/v2003/N... ...tex/mymiktex/derivatives/v2003/NET/HedgeGamT.eps,height=5cm}} %% %% \end{figure}

next up previous contents
Next: Vega Up: The Greeks Previous: Theta   Contents
Kyriakos 2003-03-17