# Showing that pnl from gamma and theta cancel

I've seen a few questions state without proof that $$0.5 \Gamma S^2 \sigma^2 = \Theta$$. That is, the gamma and theta pnls cancel out. For example:

Relationship between time decay and gamma

My question is straightforward: how do we show that from the greek formulas?

https://en.wikipedia.org/wiki/Greeks_(finance)#Formulas_for_European_option_Greeks

I must be missing something obvious here, or part of the story is missing.

edit: Now I see that the key assumption folks make is that the rate is zero, in which case the result is obvious. But if there's anything interesting to say about the gamma and theta relationship with nonzero rates, I'd be curious. For example, if we hedge with futures.

• The Black--Merton-Scholes PDE gives the relationship between gamma and theta. I am assuming you are working within a BMS world for now. Apr 9 at 14:00