The ATM straddle formula

(a) Prove (9A.7) and the expansion (9A.8), including the error term $-\sigma^2\tau/24$. (b) For $F=24{,}500$, $\sigma=13\%$, $r=6\%$, 7 days, compute the exact ATM-forward straddle and the approximation, and their relative difference. Compare with $\sigma^2\tau/24$. (c) Compute the one-standard-deviation move and the expected absolute move in points, and the probability (normal model) that the index ends within $\pm$ the straddle.