The market maker sold a 1-year European put option on 100 shares of a non-dividend-paying stock, which is currently trading at $40 per share. The volatility of the stock is 30%, and the continuously compounding interest rate is 8%. The market maker is using a delta-hedging strategy to mitigate the risk associated with the sold put option.

The next day, the stock price increases to $41 per share. We need to calculate the new delta hedge position after the stock price increase.

The delta of the put option at the new stock price ($41) is approximately -0.3091 (using the provided values from the put option delta calculator). This means for every 100 shares hedged, the market maker needs to sell about 30.91 shares of the stock to maintain a delta-neutral position.

With the stock price increase, the value of the put option has decreased due to lower intrinsic value. Using the Black-Scholes put option pricing model, we can see that the value of the put option has decreased from the initial sale. The new value of the put option can be found using the provided values:

BSPut(41, 40, 0.30, 0.08, 365/365, 0) = 2.8857

This indicates that the market maker's exposure to potential losses has reduced, as the put option is now less valuable.

To summarize, after the stock price rises from $40 to $41, the market maker should adjust their hedge by buying back approximately 30.91 shares of the stock to maintain delta neutrality. The value of the put option they sold has decreased from the original sale, now being worth approximately $2.8857.