Integrating Gamma

Quantitative Finance Level 1

Let $C_x$ be the call option with a strike price of $x$ .
Let $\gamma_{C_x}$ be the gamma of $C _ x$ .
What is

$\int_0^{ \infty } \gamma_{ C_ x } \, dx ?$

- \frac{1}{2}

-1 Cannot be determined 0

\frac{1}{2}

1 solution

Dionys Nipomici
May 30, 2020

Since $\gamma$ is the derivative of $\Delta$ , we have:

$\int_0^{\infty} \gamma dx = \Delta (\infty) - \Delta (0) = 1-0=1$

Remember that Gamma is the derivative of the delta with respect to a change in the underlying stock.

What you're thinking of is "For a fixed $x = P$ , $\int_0^\infty \gamma_{C_P} d S = [\delta_{C_P} ]_0^\infty$ .

However, in this question, note that we're integrating with respect to the Strike price.
There's a "change of variables" that's going on.

Calvin Lin Staff - 1 year ago