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Derivatives
Exercise 10
Layout the intermediate steps systematically. Round the answers to 4 decimals.
1. A company has entered into a short range-forward contract on the S&P 500 that is two
months from maturity. The current value of the index is 870, the exercise price of the
call option is 900, the exercise price of the put option is 856.34429, the risk-free interest
rate is 8% per annum, and the volatility of the index is 20% per annum. Dividend
yield is 3% per annum with continuous compounding.
(a) Verify that the cost of the range-forward contract is 0.
(b) Suppose that the company also owns a portfolio of stocks that mirrors the index.
What is the total value of the portfolio and the range-forward contract in terms
of the index ST at maturity?
2. Suppose that the current exchange rate of British pound is 1.6000, the risk-free interest
rate in the United States is 8% per annum (continuous compounding), the risk-free
interest rate in Britain is 11% per annum (continuous compounding), and the volatility
is 14% per annum. Find the value of the following options.
(a) An insurance company agrees to buy 1 million GBP at an exchange rate of 1.5 if
the exchange rate has fallen below 1.6 at the end of one year.
(b) An insurance company agrees to pay 0.1 million USD if the exchange rate has
fallen below 1.6 at the end of one year.
(c) A regular European put option to buy 1 million GBP at an exchange rate of 1.6
at the end of one year.
3. Consider a stock currently standing at $50 per share. The dividend yield is 4% per
annum, the volatility is 30% per annum, and the risk-free rate is 6% per annum. What
is the value of a forward start European call option on the stock that will start in 3
months and mature in 6 months?
4. Calculate the price of a chooser option with strike price $30, time to choose 0.5 years,
and maturity 1 year. The current asset price is $28 and volatility is 20% per annum.
The dividend yield is 2% per annum. The risk-free interest rate is 5% per annum.