Consider a six month American put option on 100 shares of stock with a strike price of $110 per share. The spot price per share is $105 and the financing rate is 6% per annum on a continuously compounded basis. Finally, the annual volatility is 25%. Assume no dividend on the stock. In answering the questions below use a binomial tree with three steps.
1. Compute u, d, as well as p for the standard binomial model.
2. Value the option at time 0 using the binomial tree.
3. How would you hedge a short position in the put option at time 0 with a portfolio composed of shares of stock, and a cash borrowing or lending position?
4. An exchange rate is currently $0.8500 per unit of the foreign currency. The volatility of the exchange rate is 15%. Interest rate in $ is 5% while the interest rate in the foreign currency is 2%. Both rates are quoted per annum on a continuously compounded basis. Using the lognormal assumption, estimate the probability that the exchange rate in four months will be:
a. less than or equal to $0.800,
b. between $0.800 and $0.900, and
c. greater than $0.900.
5. Suppose that the result of a major lawsuit affecting a company is due to be announced in the next few seconds. The company’s stock price is currently $500 per share. If the ruling is favorable to the company, the stock price per share is expected to jump to $650. If it is unfavorable, the stock is expected to drop to $300 per share.
a. What is probability of a favorable ruling implied by the stock price and the two possible outcomes after the announcement?
b. Assuming that the volatility of the company’s stock after the ruling will be 20% for six months if the ruling is favorable and 40% if it is unfavorable, calculate the prices of six-month European call options with strike prices of $500, and $600. Finally calculate the implied volatilities corresponding to these two option prices. Assume that the company does not pay dividends and that the six-month risk-free rate is 5% on a continuously compounded basis.
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?