Theta of ATM Straddles

Using the straddle approximation formula , $Y_{ATM} \propto \sqrt{t}$ .

Theta is the derivative of the ATM straddle price wth rrspect to time: $\theta_Y = \dfrac{\partial Y}{\partial {t}}$ . On differentiating $Y_{ATM}$ , we get $\theta_Y \propto \frac{1}{\sqrt{t}}$ That is, $\theta_Y \times \sqrt{t} = \text{constant}$

When $t = 5$ , $~~\theta_Y = 128$

When $t = 20$ , $~~\theta_Y = \theta_Y$

$128 \times \sqrt{5} = \theta_Y \times \sqrt{20}$

Therefore, $\theta_Y = \boxed{64}$ $_\square$