Gamma of straddles

Quantitative Finance Level 1

True, False or It depends?

We know that the ATM straddle price can be approximated by the formula

$Y_{ATM} = \frac{ 1}{ 2000} S \sigma \sqrt{t}.$

Since gamma is the second (partial) derivative with respect to the underlying price $S$ , the gamma of the straddle is 0.