Digital options

A cash-or-nothing call pays ₹1 if $S_T>K$; an asset-or-nothing call pays $S_T$ if $S_T>K$. (a) Price both under Black–Scholes with dividend yield $q$, by direct integration. (b) Show that a vanilla call is a portfolio of the two and recover (8.23). (c) Compute the price of a contract paying ₹1,00,000 if Nifty is above 25,000 on 29 Dec 2026, using $q$ implied from the future ($q=0.491\%$) and $\sigma=12.2\%$. What is the risk-neutral probability of the event? Compute the delta of this digital (rupees per index point). (d) Why are digitals hard to hedge near expiry? Relate to the gamma of the vanilla call.