# Final Exam

## 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?

1. The return from the stock in a short period of time is lognormal.
2. The stock price at a future time is lognormal.
3. The stock price at a future time is normal.
4. None of the above.

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:

1. \$25
2. \$28
3. \$26
4. \$20

3. Which of the following statements is true?

1. The Black-Scholes-Merton model is used to price all vanilla options, because it is an exact closed form, analytical formula.
2. The Black-Scholes-Merton price of a European vanilla option is an expected value of the asset.
3. The Black-Scholes-Merton price of a European vanilla option is the expected profit from the option.
4. The Black-Scholes-Merton model allows an option to be priced assuming the investment is risky.
5. None of the above.

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?

1. Long Call option
2. Long Put option
3. Short Call option
4. Short Put option
5. It is impossible to have negative delta and negative gamma from a single European vanilla option.

5. What does N(x) denote?

1. The area under a normal distribution from zero to x.
2. The area under a normal distribution up to x.
3. The area under a normal distribution beyond x.
4. The area under the normal distribution between –x and x.

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?

1. Current spot price (St) is less than 90.
2. Terminal spot price (ST) is less than 90.
3. Current spot price (St) is greater than 90.
4. Terminal spot price (ST) is greater than 90.
5. Terminal spot price (ST) is greater than 110.

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?

1. Current spot price (St) is less than 48.
2. Terminal spot price (ST) is less than 48.
3. Current spot price (St) is greater than 52.
4. Terminal spot price (ST) is greater than 50.
5. Terminal spot price (ST) is greater than 52.

8. Which of the following are always positively related to the price of a European call option on a stock? (Circle three)

1. The stock price
2. The strike price
3. The time to expiration
4. The volatility
5. The risk-free rate
6. Dividends

9. Which of the following are always positively related to the price of an American put option on a stock? (Circle two)

1. The stock price
2. The strike price
3. The time to expiration
4. The volatility
5. The risk-free rate
6. Dividends

10. Which statement is correct?

1. The vega of a call option is negative and the vega of a put option is positive.
2. The vega of a call option is positive and the vega of a put option is negative.
3. The vega of a call option and a put option are both positive.
4. The vega of a call option and a put option are both negative.
5. None of the above

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.

1. vega
2. theta
3. gamma
4. rho
5. delta

12. Which of the following European vanilla option strategies benefits from an increase in volatility of the underlying asset price?

1. Short straddle
2. Short strangle
3. Short butterfly
4. All of the above
5. None of the above

13. Which of the following European vanilla option strategies benefits from lower market volatility in the underlying asset price?

1. Long straddle
2. Long box spread
3. Long butterfly
4. All of the above
5. None of the above

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?

1. American Call (K1, T1) > European Call (K1, T1)
2. European Call (K1, T1) > European Call (K2, T1)
3. European Call (K2, T1) > European Call (K1, T2)
4. American Call (K1, T1) > American Call (K1, T2)
5. None of the above

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.

1. Delta
2. Gamma
3. Vega
4. Theta
5. None of the above

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?

1. The investor receives a long position in the contract and \$4,000.
2. The investor receives a short position in the contract and pays \$3,800.
3. The investor receives a long position in the contract and \$3,800.
4. The investor receives a short position in the contract and receives \$3,800.
5. None of the above

17. An investor buys a put option on a futures contract. Which answer is true?

1. The investor takes a long position in the futures if he/she exercises the put option.
2. The investor takes a short position in the future if he/she exercises the put option.
3. The investor takes a short position in the futures if he/she does not exercise the put option.
4. The investor can choose whether to go long or short when he/she exercises the put option.
5. None of the above

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?

1. Long Call option
2. Long Put option
3. Short Call option
4. Short Put option
5. It is impossible to have positive delta and negative gamma from a single European vanilla option.

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?

1. 0.67
2. 0.98
3. 1.97
4. 1.92
5. 1.39

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?

1. \$0
2. \$0.70
3. \$8.87
4. \$5
5. None of the above

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?

1. \$0
2. \$0.27
3. \$9.30
4. \$5
5. None of the above

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?

1. Buy a put and sell a call on the currency with the strike price of the put higher than that of the call
2. Buy a put and sell a call on the currency with the strike price of the put lower than that of the call
3. Buy a call and sell a put on the currency with the strike price of the put higher than that of the call
4. Buy a call and sell a put on the currency with the strike price of the put lower than that of the call

23. What is the expected growth rate of an index future price in the risk-neutral world?

1. The excess of the risk-free rate over the dividend yield
2. The risk-free rate
3. The dividend yield on the index
4. Zero

24. What is the cash component of the payoff if a call option on futures on 50 units of the underlying asset is exercised?

1. (Current Futures Price – Strike Price) times 50
2. (Strike Price – Current Futures Price) times 50
3. (Most Recent Futures Settlement Price – Strike Price) times 50
4. (Strike Price – Most Recent Futures Settlement Price) times 50

## Part 2 - Essay Questions

1. You observe a six-month European put option on the S&P500 Index.

• Strike price equals 2,000
• S&P500 currently stands at 2,100
• Dividend yield on the S&P500 Index is 2% p.a. semi-annual compounding.
• Risk-free interest rate is 4% p.a. quarterly compounding.
• Volatility of the Index equals 15%

2. You observe a six-month European call option on the Australian dollar-Malaysian ringgit.

• Strike price equals 3 RM per AUD.
• The exchange rate currently stands at 3.10 RM per AUD.
• Malaysia has an interest rate of 5% p.a. semi-annual compounding.
• Australia has a 3% p.a. quarterly compounding.
• Volatility of the exchange rate equals 10%

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%.

1. Please draw a tree diagram to show the detailed calculations of a six-month European put option when the strike price is 1,250.
2. If this were an American option, which nodes would be exercised early?

6. The goal of this question is to ensure students calculate the correct u, d, P, and discount.

1. A stock index is currently at 1200. 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%. Please calculate the u, d, P, and discount for a six-month European call option with a strike price of 1250. Assume two-time steps.
2. A futures price is currently \$90. The risk-free interest rate is 6.5% p.a. compounded monthly. The volatility of the futures price is 30% p.a. continuously compounded. Please calculate the u, d, P, and discount for a nine-month call option on futures contract with a strike price of \$85. Assume three-time steps.
3. The exchange rate is 4.50 RM per USD. The United States has an interest rate of 2% while Malaysia has an interest rate of 6%. The exchange rate has a 30% volatility. Please calculate the u, d, P, ad discount for a four-month European put option with a strike price of 4.60 RM per USD. Assume two-time steps.

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:

 Company Fixed rate Yen Available Floating rate Dollar Available KDB 4.9% Libor + 0.8% IBM 4.5% Libor + 0.25%

Suppose IBM would like to borrow fixed rate yen, whereas KDB would like to borrow floating rate dollars.

1. Identify the overall spread (basis points) of the swap and at what rate should each party borrow to create the swap?
2. What is the fixed rate Yen at which IBM can borrow using interest rate/currency swap if KDB can borrow at floating rate of Libor+0.25%, alternatively what is the range of possible cost savings that IBM can realize through the swap with KDB?
3. Assuming a notional principle equivalent to \$125 million and a current exchange rate of Yen105/\$, what does these possible cost savings translate into in Yen terms?

## Answers

Multiple Choice

 1. b 2. d 3. a 4. c 5. b 6. d 7. e 8. a, d, e 9. a, e 10. c 11. d 12. c 13. c 14. b 15. b 16. c 17. b 18. d 19. d 20. c 21. b 22. d 23. d 24. c

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)