a. The overnight profit or loss of the market maker's delta-hedged position can be calculated as follows:

The market maker initially sold a put option with a delta of -0.3089 (or -0.3091, both values are close). This means for every $1 change in the stock price, the value of the put option will change by approximately -0.3089 (or -0.3091) dollars.

Since the stock price increased from $40 to $41, the delta loss would be: (-0.3089 * 100 shares) = -30.89 (or -30.91)

However, the market maker also holds a short position in the stock to hedge the put option. With the stock price increase, the short position would have gained: (1 share * ($41 - $40) * 100 shares) = 100

Therefore, the net overnight profit or loss is the sum of these two changes: Net Profit/Loss = Delta Loss + Stock Gain Net Profit/Loss = (-30.89) + 100 Net Profit/Loss = 69.11

So, the market maker has an overnight profit of $69.11.

b. To rebalance the delta-hedged position, the market maker must adjust the number of shares they hold in the stock. Since the stock price has increased, the delta has become less negative, meaning the market maker needs fewer shares to hedge the same put option position.

To maintain delta neutrality, the market maker should buy back some of the stock they had sold short. The number of shares to buy back can be calculated by taking the absolute value of the new delta and multiplying it by the number of shares hedged:

New Shares to Buy Back = |New Delta| * 100 shares New Shares to Buy Back = |-0.3089| * 100 shares New Shares to Buy Back = 30.89 shares (or 30.91 shares, depending on the chosen delta value)

Thus, the market maker should buy back approximately 30.89 shares (or 30.91 shares) to rebalance the delta-hedged position.