Why gamma and theta have opposite signs?

Question

I saw some textbooks use B-S equation to explain why gamma and theta have opposite signs in most of the cases. For example, John Hull's classic book.

The explanation is, first write B-S equation in terms of greeks:

$\frac{\partial V}{\partial t}+rS\frac{\partial V}{\partial S}+\frac{1}{2}\sigma^2S^2\frac{\partial^2 V}{\partial S^2}=rV$

$\Theta+rS\Delta+\frac{1}{2}\sigma^2S^2\Gamma=rV$

$\Theta+\frac{1}{2}\sigma^2S^2\Gamma=r(V-S\Delta)$

Do we need to assume r=0, in order to draw the conclusion that gamma and theta have opposite signs?

Answer 1

I think you are answering your own question.

Hull states: "When $\Theta$ is large and positive, $\Gamma$ tends to be large and negative and vice versa."

In practice, you can expect $r(V-S \Delta)$ to be quite small.

Answer 2

As time passes P/L line is getting closer and closer to expiration P/L line.

Volatility resists P/L line to get closer to expiration P/L line. You can think about Volatility as an elastic element btw P/L line and expiration P/L line. The higher is Volatility the bigger is the gap btw P/L line and the expiration P/L line.

Time and Volatility are opposite forces on P/L diagram.

Time pushes P/L line toward expiration P/L line.

Volatility pushes P/L line away from expiration P/L line.

Volatility affects daily P/L slices thickness. Higher volatility - thicker the daily slice - https://www.dropbox.com/s/s7pscwm2yp31sha/Screenshot%202020-08-05%2018.51.18.png?dl=0