Assume the Black-Scholes framework. Consider a 9-month at-the-money European put option on a futures contract.

The continuously compounded risk-free interest rate is 8%.

The futures price instantaneously decreases by 10. You are given:

(i) Using the delta approximation, you find that the option price increases by 4.148.

(ii) Using the delta-gamma approximation, you find that the option price increases by 4.231.

Calculate the exact price of the put option at the new futures price, i.e., after the initial futures price drops by 10.