6. You are given the following data: the three-month futures price for silver is $22, the risk-free rate is 6% per annum on a continuously compounded basis, and the volatility of the futures price of silver is 24%. Calculate the price of a three-month European put option on the spot price of silver struck at $24.

7. A futures price is currently $65 and its volatility is 10%. The risk-free interest rate is 8% per annum on a continuously compounded basis.

a. Use a three-step binomial tree to calculate the value of a nine-month European call option on the futures with a strike price of $60?

b. Use the same tree to calculate the value of the option if it is American, and show where early exercise is optimal if it is optimal to exercise the option early?

8. Consider a three-month put option on a futures with a strike price of $350 when the risk-free interest rate is 5% per annum on a continuously compounded basis. The current futures price is $320. What is a lower bound for the value of the futures option if it is (a) European and (b) American?