MBA Derivatives - New York University



MBA Derivatives

Six Problem Sets

Note: Unless otherwise indicated, ignore market imperfections such as bid-offer spreads, slippage, taxes, and the like.

Problem Set 1

1. What is the price of a European-style call written on CSCO (a non dividend-paying stock) assuming the following parameter values: S = $22, K = $20, r = 1.50%, T - t = 0.25, and ( = 30%? What are the delta, gamma, vega, rho, and theta of this call? What is the probability that the call will expire in-the-money? Using a Taylor-series expansion to the second order, what will be the new price of the call if the share price instantaneously increases from $22 to $23 per share?

2. What is the price of a European-style put option written on the British pound assuming the following parameter values: S = $1.3570, K = $1.3500, r = 1.50%, rf = 2.00%, T - t = 0.50, and ( = 19%? What is the probability that this put will expire in-the-money?

3. Briefly explain why the following weather option can not be priced using the no-arbitrage approach: A one-month European-style call option whose payoff is equal to $1 million x HDD, where HDD = MAX[0, 65 - A], where A is the arithmetic and equally-weighted average of S, where S is the sum of the daily highest and lowest temperature reading (measured in degrees Fahrenheit), divided by two, for each day during the one-month option life.

4. Suppose that a European-style cash-or-nothing call option with maturity T - t and strike K is worth A dollars. Suppose that a European-style asset-or-nothing call option with maturity T - t and strike K, written on the same underlying liquid asset, is worth B dollars. Then under the Black-Scholes economy, what is the value of a plain-vanilla European-style call option, also with maturity T - t, strike K, and same underlying asset?

Problem Set 2

5. What is the price of an American-style put option, written on a non dividend-paying stock, having the following parameter values. Use a 40-period lattice and the BOPM: S = $38, K = $40, r = 1.2%, T - t = 0.30, and ( = 35%? What are the option's delta, gamma and theta? Finally, price the same option using the trinomial option pricing model (TOPM). See for TOPM software.

6. Conduct the same exercises as in Problem 5 above, but permit the asset to exhibit a continuous leakage (() at the rate of 1.2% per annum.

7. Using Problem 2 (from Problem Set 1) above, illustrate the price of the corresponding British pound call option using put-call parity.

Problem Set 3

8. Assume that the relevant interest rate term structure is flat at 1.2% per annum with continuous compounding. Given the following option prices, produce an implied volatility matrix. In addition, provide the 3-month vol three months hence for the $50 strike. Note that the options are European-style calls written on a non dividend-paying stock whose current price is $49 per share.

Call Option Prices

Strike Prices 3-month 6-month 9-month 12-month

40 11.00 11.75 12.25 13.00

45 7.00 8.10 8.75 9.75

50 3.25 4.75 6.00 6.50

55 1.50 3.00 5.00 5.75

The next three problems (9 through 11) refer to the following information: The SP500 stock index is 990 and exhibits an implied volatility surface that is perfectly flat at 27% per annum. In addition, the relevant interest rate term structure is perfectly flat at 1.25% per annum with continuous compounding. The dividend yield on the index is 1.10%. A dealer of European-style SP500 options, which have a contract size of 250 index units, holds the following three-option book:

Short 15 one-month call contracts with strike 1000

Short 12 five-month put contracts with strike 975

Long 8 two-month call contracts with strike 1025

9. What are the book's delta, gamma and vega?

10. Suppose there are two traded SP500 options available in the market, with the first having delta = 0.58 (per index unit), gamma = 0.003 (per index unit), and vega = 1.40 (per %), and second having delta = 0.50 (per index unit), gamma = 0.004 (per index unit), and vega = 1.10 (per %). How many contracts (to the nearest whole number) of each option must the dealer trade in order to simultaneously gamma and vega hedge the book? (Assume that the entire volatility surface can only shift in a parallel fashion.)

11. Given your response in Problem 10, how many six-month SP500 futures contracts (again, to the nearest whole number) must the dealer now long/short in order to delta hedge the book? [The futures contract also has a multiplier of 250 index units.]

12. A dealer runs an option book containing scores of options with different maturity and strike prices. Describe how the dealer might go about vega hedging this book.

Problem Set 4

13. You enter a long position in a 6-month oil forward contract. The forward price is $30.25 per barrel. In three months, the forward price on newly minted 3-month oil forward contracts is $29.50 per barrel. If the relevant term structure is flat at 2% per annum with continuous compounding, then what is the value of your long position?

14. Suppose that the $LIBOR yield curve is flat at 3.87% per annum with continuous compounding. Consider a European-style swaption that gives the holder the right to pay 4% in a three-year swap starting in one month. The volatility measure for the forward swap rate is 14%. Swap payments would be made semiannually and the notional principal on the swap would be $10 million. What is the swaption price?

15. Consider a contract that caps the interest rate on a floating-rate $10 million loan at 5% per annum (with quarterly compounding) for three months starting in one year. This contract is a caplet and could be one element of a cap. Suppose that the interest rate for a three-month period starting in one year is 4.90% per annum with quarterly compounding); the current 15-month zero rate is 4.85% per annum with continuous compounding; and the volatility measure for the three-month rate underlying the caplet is 20% per annum. What is the price of the caplet?

Problem Set 5

16. You manage a large-cap stock portfolio with current market value of $1.2 billion. The beta of the portfolio is 0.97. You want to reduce your systematic risk quickly by 25%. If the SP500 stands at 1180, how many nearby SP500 futures contracts will you short?

17. The six-month s.a. $LIBOR is 2.2%, and the one-year s.a. $LIBOR is 2.4%. What is the swap rate on a one-year plain-vanilla interest rate swap? Also, demonstrate that your answer is the par rate.

18. A dealer wants to value an outstanding cross-currency interest rate swap. The dealer pays s.a. $LIBOR on a notional principal of $20 million, and receives 5% (with semiannual compounding) on a notional principal of 13 million British pounds. The swap has 1.75 years remaining to maturity. The s.a. $LIBOR curve was 3.5% (with semiannual compounding) three months ago. The current U.S. and U.K. LIBOR curves are flat at 3.6% and 4.9% (with continuous compounding), respectively. Finally, the spot exchange rate is $1.4580 per pound. What is the value of the swap to the dealer?

19. A BB-rated counter party has approached you about transacting a 5-year interest rate swap. Describe some ways in which you can credit enhance the county party.

Problem Set 6

20. An A-rated firm can issue five-year dollar-denominated fixed-rate debt at 4% on a bond equivalent basis or can issue a five-year floater at s.a. $LIBOR plus 30 basis points. A BBB-rated firm can issue five-year dollar-denominated fixed-rate debt at 5% on a bond equivalent basis or can issue a five-year floater at s.a. $LIBOR plus 90 basis points. All issues would have $25 million face value. Describe a swap transaction that would make both issuers better off. Be sure to detail the terms of the swap including the swap rates. Include a swap dealer who would command an overall spread of 4 basis points.

21. A corporate treasurer tells you that she has a $20 million, 3-year floating-rate liability that has a bullet principal and pays a semiannual coupon of s.a. $LIBOR plus 50 basis points. She would like to hedge the position. Present three trades that would accommodate her and briefly discuss the advantages and disadvantages of each trade.

22. Revisit Phase-1 of your project. Using your software, compute the gamma of the Asian option. Gamma is given by (Delta2 - Delta1)/(0.25). To get Delta1, calculate a second Asian option price using the stock's spot price + $0.25. Then compute the ratio of the difference between the second and first option prices over 0.25. To get Delta2, compute a third option price using the stock's spot price plus $0.50. Then compute the ratio of the difference between the third and second option prices over 0.25. Remember to use the same epsilon streams when computing the three option prices. The best way to accomplish this task is to run three price paths at a time, all using the same daily epsilons but with the first path starting at the original stock price (S), the second starting at S + $0.25, and the third starting at S + $0.50. Eventually you will obtain the three option prices. Call these C1, C2, and C3. So another way of expressing gamma is {[(C3 - C2)/(0.25)] - [(C2 - C1)/(0.25)]}/(0.25).

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