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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?

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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.

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  • $\begingroup$ Theta and Gamma are also small in practice, so I am not sure if your last argument makes sense. $\endgroup$ – emcor Jun 12 '15 at 23:02
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Long options have a positive gamma because as price increases, call Delta approaches 1 from 0 put Delta approaches 0 from -1 (think of $S=0\to+\infty$).

Based on below numerical example, theta and gamma can have equal signs for both put and call.

If we set $r=0$, they have different signs.

enter image description here

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    $\begingroup$ You wrote: Theta is always negative. Are you sure? Even for a European put that is deep in the money? ;-) $\endgroup$ – noob2 Jun 12 '15 at 12:20
  • $\begingroup$ @noob2 Yes: image.slidesharecdn.com/thegreeks-090825155137-phpapp01/95/… The reason is that continuously compounded returns go from $(-\infty,+\infty)$, so even deep ITM put still has infinite "symmetric return potential". Also, as the vega is always positive for both put and call, an option with less maturity has less variance and hence a negative theta. $\endgroup$ – emcor Jun 12 '15 at 12:37
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    $\begingroup$ Try the following in a Black-Scholes calculator: European Put S=10, K=100, T=1 year, q=0, r=25%, sigma=10%. You will get a Theta of 0.053. That's a positive Theta. $\endgroup$ – noob2 Jun 12 '15 at 17:28
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    $\begingroup$ @noob2 I used this calculator: fintools.com/resources/online-calculators/options-calcs/… -- For $S=100$, call and put both have same signs for gamma and theta... if we set $r=0.0001$ then they have opposite signs... $\endgroup$ – emcor Jun 12 '15 at 21:39
  • $\begingroup$ @emcor: So do you agree with noob2 that Theta is not always negative for a European put deep in the money? It is not clear what you are trying to say. $\endgroup$ – Hans Jul 15 '17 at 21:06

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