Expiry-day gamma

(a) Derive (9.26) with an explicit bound on the $O(\tau)$ term when $r=0$. (b) With 30 minutes to a Tuesday expiry, Nifty at 25,000 and implied vol 12%, compute ATM gamma per unit and the width $\sigma\sqrt\tau S$ of the "gamma zone". (c) A dealer is short 1,000 lots of the 25,000 straddle. How many lots of Nifty futures must she buy to re-hedge after a 20-point rise? Compare the exact Black–Scholes answer with the linear gamma approximation and explain the difference.