A. Sell more shares of the index as the stock index rises.

CORRECT

The trader is long in the call of stock index. An increase in the underlying will increase the delta of his long position in index call.

to make the delta neutral, he then will have to short the shares of the underlying stock index. Shorting of the underlying stock gives a negative delta

the increasing delta on the long call will get neutralized by the negative delta on shorting the index stocks

B. Sell more shares of the index as the stock index falls.

INCORRECT

Since the trader is long on call option, it has a positive delta. This delta reduces when the index falls. Hence he need not short the index to make delta neutral since shorting will further reduce the delta

C. No need to rebalance the initial position in shares of the index.

INCORRECT

AS explained above, as the index rises the delta rises, to neutralize it a short position in the index stocks is required. Not rebalancing will lead to an increase in delta

D. Close their position in index shares and hold Treasury bills.

INCORRECT

The Challenge here is to make the position neutral to changes in delta and not to close the position completely