At what price can investors exercise an index option?

Investors can exercise an index option at the closing price of the index on the day of expiration. This is because index options are settled in cash, and the payout is determined based on the index's performance at the end of the trading day. The closing price reflects the final value of the index at which contracts are settled, making it the relevant price for the exercise of the option.

Regarding the other options: the lowest price of the day, the opening price of the index, and the average price do not represent the final position of the index on the day the option is exercised. Instead, they provide varying snapshots of the index throughout the trading day, which do not impact the cash settlement value that determines the execution price for index options. Therefore, the correct choice is the closing price of the index, as it is the standardized measure used to settle index options.