Using Theta to find price

From the straddle approximation formula , $Y_ {ATM} \approx \frac{4}{5} S \sigma \sqrt{T}$ .

Hence, we can calculate that the theta value of the straddle is $\theta_{ATM} = \frac{4}{5} S \sigma \frac{1}{2 \sqrt{T} }$ . Thus $Y_{ATM} = 2T \theta_{ATM}$

Given that theta is 12 and there are 5 days to expiry, so $Y_{ATM} = 120$ .

How, for the straddle that is 20 days out to expiray, since $\sqrt{ \frac{T_1}{T_2} } = \sqrt{ \frac{20}{5} } = 2$ , hence the price would be $2 \times 120 = 240$ .