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Consider a 2 -month call futures option with a strike price of 40 when the risk-free interest rate is \(10 \%\) per annum. The current futures price is \(47 .\) What is a lower bound for the value of the futures option if it is (a) European and (b) American?

Consider a 4 -month put futures option with a strike price of 50 when the risk-free interest rate is \(10 \%\) per annum. The current futures price is \(47 .\) What is a lower bound for the value of the futures option if it is (a) European and (b) American?

A futures price is currently \(60 .\) It is known that over each of the next two 3 -month periods it will either rise by \(10 \%\) or fall by \(10 \%\). The risk- free interest rate is \(8 \%\) per annum. What is the value of a 6 -month European call option on the futures with a strike price of \(60 ?\) If the call were American, would it ever be worth exercising it early?

A futures price is currently 25, its volatility is \(30 \%\) per annum, and the risk-free interest rate is \(10 \%\) per annum. What is the value of a 9 -month European call on the futures with a strike price of 26 ?

A futures price is currently \(70,\) its volatility is \(20 \%\) per annum, and the risk-free interest rate is \(6 \%\) per annum. What is the value of a 5 -month European put on the futures with a strike price of \(65 ?\)

Suppose that a futures price is currently \(35 .\) A European call option and a European put option on the futures with a strike price of 34 are both priced at 2 in the market. The risk-free interest rate is \(10 \%\) per annum. Identify an arbitrage opportunity. Both options have 1 year to maturity.

Suppose that a futures price is currently \(30 .\) The risk-free interest rate is \(5 \%\) per annum. A 3 -month American call futures option with a strike price of 28 is worth \(4 .\) Calculate bounds for the price of a 3 -month American put futures option with a strike price of 28.

Can an option on the yen/euro exchange rate be created from two options, one on the dollar/euro exchange rate, and the other on the dollar-yen exchange rate? Explain your answer.

A corporation knows that in 3 months it will have $$\$ 5$$ million to invest for 90 days at LIBOR minus 50 basis points and wishes to ensure that the rate obtained will be at least \(6.5 \%\). What position in exchange-traded interest rate options should it take?

If the price of currency A expressed in terms of the price of currency B follows the process \\[ d S=\left(r_{\mathrm{B}}-r_{\mathrm{A}}\right) S d t+\sigma S d z \\] where \(r_{\mathrm{A}}\) is the risk-free interest rate in currency \(\mathrm{A}\) and \(r_{\mathrm{B}}\) is the risk-free interest rate in currency B. What is the process followed by the price of currency B expressed in terms of currency A?