15.1) A portfolio is currently worth $10 million and has a beta of 1.0. An index is currently standing at 800. Explain how a put option with a strike price of 700 can be used to provide portfolio insurance. Index goes down to 700
10*(800/700)= 8.75 million
Buying put options= 10,000,000/800= 12,500
If you buy the options at 800, the value will be 12,500 times the index with a strike price of 700 therefore providing protection against a drop in the value of the portfolio below $8.75 million. Each contract is on 100 times the index, a total of 125 contracts would be required. 15.2) "Once we know how to value options on a stock paying a dividend yield, we know how to value options on stock indices and currencies." Explain this statement. A stock index is similar to a stock paying a dividend yield, only if the dividend yield is the dividend yield of the index. Currencies are similar to a stock paying a dividend yield, the dividend yield being the foreign risk-free interest rate. 15.3) A stock index is currently 300, the dividend yield on the index is 3% per annum, and the risk-free interest rate is 8% per annum. What is a lower bound for the price of a six month European call option on the index when the strike price is 290? (300e^-0.03*.5)- 290e^-.08*.5 = $16.90
15.4) A currency is currently worth $.80. Over each of the next two months it is expected to increase or decrease in value by 2%. The domestic and foreign risk-free interest rates are 6% and 8%, respectively. What is the value of a two-month European call option with a strike price of $.80? .8160
p= (e^.06-.08)*.08333 - .98 / 1.02-.98 = 0.4584
The purchase price of one unit of currency is $.0067
15.5) Explain how corporations can use range-forward contracts to hedge their foreign exchange risk when they are due to receive certain amount of a foreign currency in the future. Corporations can use range forward contracts to hedge their foreign exchange risk when they are due to receive a certain amount of foreign currency in the future by buying a put option with a strike price below the current exchange rate. Also they can sell a call option with a strike price above the current exchange rate. This ensures that the exchange rate obtained for the foreign currency is between the two strike prices reducing the risk. 15.6) Calculate the value of a three-month at-the-money European call option on a stock index when the index is at 250, the risk free interest rate is 10% per annum, the volatility of the index is 18% per annum, and the dividend yield on the index is 3% per annum. S_0 = 250
d_1= ln(250/250)+(.10-.03+.18^2/2).25 / .18√.25 = 0.2394
d_2 = d_1-.18√.25 = .1494
250N(0.2394)e^-.03*.25 - 250N(.1494)e^-.10*.25
=250 * .5946e^-.03*.25 - 250 * .5594e^-.10*.25
The call price is $11.15.
15.7) Calculate the value of an eight month European put option on a currency with a strike price of 0.50. The current exchange rate is .52, the volatility of the exchange rate is 12%, the domestic risk free interest rate is 4% per annum, and the foreign risk free interest rate is 8% per annum. S_0 = .52
d_1= ln(.52/.50)+(.04-.08+.12^2 / 2)* .6667 / .12√.6667 = .1771
d_2 = d_1 -.12√.6667 = .0791
.50N(-.0791)e^-.04*.6667 - .52N(-.1771)e^-.08*.6667
.50*.4685e^-.04*.6667 - .52*.4297e^-.08*.6667
The put option price is $.0162.
Time to maturity = 47/252 = 0.1865
Derivagem result = 10.23% of implied volatility
The price of the put = 2.25 + 126e^-.53*.1865= p+125.56e^-.03*.1865 p=2.1512
A European call has the same volatility as a European put when both have the same strike price. 15.24)
The value European Put option is 14.39.
American put option:
The value of an American put option is 14.97.