Part 1 - Multiple Choice Questions
Questions come from the Hull Test Bank
1. Which of the following is assumed by the Black-Scholes-Merton model?
2. A one-year call option on a stock with a strike price of $30 costs $3; a one-year put option on the stock with a strike price of $30 costs $4. Suppose that a trader buys two call options and one put option. The breakeven stock price below which the trader makes a profit is:
3. Which of the following statements is true?
4. An investor has a portfolio consisting of one European vanilla option only. The portfolio has a negative delta and a negative gamma. What is the option position in the portfolio?
5. What does N(x) denote?
6. You sell a European vanilla Put option with a strike price of $100, and you receive $10 in premium. No delta hedge was traded. Under what circumstances does the seller make a profit?
7. You buy a European vanilla Call option with a strike price of $50, and you pay $2 in premium. No delta hedge was traded. Under what circumstances do you make a profit?
8. Which of the following are always positively related to the price of a European call option on a stock? (Circle three)
9. Which of the following are always positively related to the price of an American put option on a stock? (Circle two)
10. Which statement is correct?
11. The ____ of an option of an underlying asset is the approximate rate of change of the option’s value with respect to the risk-free interest rate.
12. Which of the following European vanilla option strategies benefits from an increase in volatility of the underlying asset price?
13. Which of the following European vanilla option strategies benefits from lower market volatility in the underlying asset price?
14. Let K1 and K2 be two strike prices where K1 < K2 and T1 and T2 are two maturity dates where T1 < T2. Assuming the spot price, riskless interest rate and volatility are all positive and identical for all options, which of the following statements about vanilla option prices is always true?
15. The ____ of an option is the ratio of the change in the delta of the option to the change in the spot price of the underlying asset.
16. An investor buys a call option on a futures for gold. The contract size is 100 ounces. The strike price is 1,800. The investors exercises the option when the futures price is 1,840 and the most recent settlement price is 1,838. Which answer is true?
17. An investor buys a put option on a futures contract. Which answer is true?
18. An investor has a portfolio consisting of one European vanilla option only. The portfolio has a positive delta and a negative gamma. What is the option position in the portfolio?
19. The price of the underlying asset is expected to follow a one-step binominal process spanning six months. If u=1.07 and d = u-1, and the riskless interest rate for one year is 5% p.a., what is the current price of a European vanilla Call option expiring in one-year with an end of period value of Vu = $2.50 in the up-state and a value of Vd = $0.90 in the down state?
20. What is the lower bound for the price of a three-year European vanilla Call option on a non-dividend paying stock when the stock price is $50, the strike price is $45, and the risk-free interest rate is 3% p.a. continuously compounded?
21. What is the lower bound for the price of a three-year European vanilla Put option on a non-dividend paying stock when the stock price is $50, the strike price is $55, and the risk-free interest rate is 3% p.a. continuously compounded?
22. Which of the following describes what a company should do to create a range forward contract in order to hedge foreign currency that will be paid?
23. What is the expected growth rate of an index future price in the risk-neutral world?
24. What is the cash component of the payoff if a call option on futures on 50 units of the underlying asset is exercised?
Part 2 - Essay Questions
1. You observe a six-month European put option on the S&P500 Index.
2. You observe a six-month European call option on the Australian dollar-Malaysian ringgit.
3. A stock price is currently $30. At the end of six months, it will be either $35 or $25. The risk-free interest rate is 5% per annum with continuous compounding. Suppose ST is the stock price at maturity. What is the derivative’s value (call) that pays off ST2 at Time T using no arbitrage argument?
4. Suppose a trader buys 1,000 European put options. If your tree diagram is below, how many shares of the stock do you need to hedge the position for the first and second three-month period? For the second time period, consider both cases when the stock price moves up during the first period and the case when it moves down during the first period.
5. A stock index is currently at 1,200. The risk-free interest is 5% p.a. compounded quarterly. The stock index pays 10% dividends compounded semi-annually and has a volatility of 20%.
6. The goal of this question is to ensure students calculate the correct u, d, P, and discount.
7. The cost to IBM and KDB of accessing either fixed rate yen or the floating rate dollar market for a new issue is as follows:
Suppose IBM would like to borrow fixed rate yen, whereas KDB would like to borrow floating rate dollars.
Short Answer Essay
1. p = 38.8826 (Note: You can have large rounding error)
2. c = 0.1627
3. The derivative has a price of 946.6038 using the weird portfolio contruction and price of 946.5773 using the standard binomial tree valuation.
4. For the fist branch,, delta = -0.5936 and an investor needs to buy (long) 593.6 shares for delta hedging. For the second branch, the upper branch has a delta = 0. Thus, the investor needs to sell his 593.6 shares for delta hedging. For the lower branch, the delta equals -0.7680. Thus, the investor needs to buy an additional 174.4 shares.
5 (a) An European put has a price of 112.81.
(b) The investor could exercise the first node and lower second node early. However, the investor has a greater expected payoff to wait.
6 (a) u = 1.1052, d = 0.9048, p = 0.4157, and discount = 0.9877
(b) u = 1.1618, d = 0.8607, p = 0.4626, and discount = 0.9839
(c) u = 1.1303, d = 0.8847, p = 0.4967, and discount = 0.9900
7 (a) The total gain available for both parties equals 0.15%. If we split the gain evenly, then IBM and KDB get a 0.075% benefit.
(b) If IBM takes all the gain, then it can take the whole 0.15%. If KDB takes the whole gain, then IBM gets zero gain. You can draw a diagram and show the fix rate for the SWAP as x.
(c) If IBM takes the whole gain on the SWAP, then this translates to a savings of 19,687,500 yen annually. If KDb takes all the gain, then IBM saves nothing by entering the SWAP.