Sample Final Exam
Finance 353 Derivatives
Section I: Multiple Choice Questions. (5X10=50 points).
Please read the instructions below carefully before answering questions in section I.
Instructions: Please select one and only one answer for each of the following questions.
No credit will be given if multiple selections are made. The information provided in each
question may be more than enough for you to choose the right answer. Choose the correct
answer for each question by writing down the letter A, B, C, or D in the space provided
for each question. Each question is worth 5 points.
You do not need to write down the steps for any of the questions in section I. Full credit
will be given as long as the final answer is correct. However, a maximum of 3 points
will be given for partial credit even if the final answer is incorrect. You will NOT earn
full credit for the question if the steps you wrote are clearly wrong, even if the final
answer is correct.
Important: The following table contains the price and Delta of call options calculated
using the Black-Scholes model. All quantities are computed assuming a volatility of
30% per annum, continuously compounded risk-free interest rate 5% per annum,
maturity of 5 years, and dividend yield of 0. You may use the following information in
answering any of the questions in this exam.
Table 1: Option Price and Delta using Black-Scholes Model
Stock Price Strike Price BS Call Price BS Call Delta
200 50 161.1965 0.997237
200 100 125.0315 0.959191
200 150 95.07198 0.872218
200 200 71.91561 0.760555
100 200 12.38903 0.372517
100 100 35.95781 0.760555
150 100 78.2095 0.9053
Question 1-3 are based on the following information.
A financial institution has the following portfolio of OTC options on the S&P 500 index
with a total value of $70000. All options have a maturity of 6 months.
Table 2: Portfolio Positions and Portfolio Greeks used in Question 1-3
Type Position Delta of Option Vega of Option
Call-1100 -1000 0.50 1.8
Call-1500 -500 0.80 0.2
Put-1200 -2000 -0.40 0.7
Call-1450 -500 0.70 1.4
The S&P 500 index is currently 1400. The expected return on S&P 500 index is 8% per
annum, and the expected return on the above portfolio is 5% per annum. The risk-free
interest rate is 3% per annum. All rates are continuously compounded.
1. A traded option call-1400 is available with a Delta of 0.6 and a Vega of 1.0.
Which of the following positions in the S&P 500 index and the call-1400 will make the
portfolio both Delta-neutral and Vega-neutral?
A. A long position in 4000 call-1400 and a long position in 1950 shares of the stock.
B. A long position in 4000 call-1400 and a short position in 1950 shares of the stock.
C. A long position in 5000 call-1400 and a short position in 2550 shares of the stock.
D. A long position in 450 shares of the stock.
2. Suppose the financial institution took the position in the S&P 500 index as
described in answer D of question 1. Suppose also it will frequently rebalance its
position in the S&P 500 index to maintain Delta neutrality of the portfolio at all times
during the next 6 months. Assume frictionless market and no transaction cost. Which of
the following best describes the expected return of the Delta-neutral portfolio over the
next six months?
A. 8% per annum.
B. 3% per annum.
C. Lower than 8% per annum but higher than 3% per annum.
D. Higher than 8% per annum.
3. Suppose you currently have a short position in a stock with market value
$35,000,000. The stock has a beta of 1.2 with the S&P 500 index. Which of the following
position in the S&P500 index options listed in table 2 Delta-hedges the market risk of
your position in the stock?
A. A short position in 60000 call-1100. B. A short position in 62500 put-1200.
C. A long position in 62500 put-1200. D. A short position in 75000 put-1200.
4. Consider a non-dividend-paying stock with current price S0 100 . Suppose the
continuously compounded interest rate is r 5% per annum. Suppose the assumptions of
the Black-Scholdes model are all satisfied. All of the options described below have a
strike price of $100 and a maturity of 6 months. Which of the following scenario has the
highest implied volatility according to the Black-Scholes model?
A. An American call option on one share of the stock is traded at $11.385 when the stock
price is S0 $99 .
B. A European call option on one share of the stock is traded at $12.385 when the stock
price is S0 100 .
C. A European put option on one share of the stock is traded at $9.916 when the stock
price is S0 100 .
D. A European put option on one share of the stock is traded at $10.916 when the stock
price is S0 99 .
5. The price of a 5-year forward contract on a dividend paying stock is 128.4031.
What is the Black-Scholes price of a 5-year European call option on the stock with strike
price $100. Assume the continuously compounded interest rate is 5% per annum, and the
volatility of the stock return is 30% per annum.
(a) $35.9578 (b) $78.2095
(c) $77.8801 (d) Not enough information to
6. Firm B’s current total asset value is 4 million and it has 1 million shares
outstanding. Firm B is not expected to a pay dividend to equity holders in the next five
years. The historical volatility of firm B’s total asset is 0.30 per annum. Assume the
distribution of firm B’s total asset value satisfies the assumptions of the Black-Schole’s
model. Firm B has issued 20,000 convertible bonds, each with a maturity of T 5 years.
At maturity, each convertible bond can be redeemed for a face value of $100 or can be
exchanged for 50 shares of firm B’s stock at the bond holder’s choice. If firm B’s assets
are not enough to pay for all the convertible bonds, it will declare bankruptcy and the
convertible bond holders will have claim to its assets. Assume that the continuously
compounded interest rate is r 5% per annum. Which of the following is the best
estimate of the no-arbitrage price of the convertible bond?
A. $110.926 B. $189.074
C. $44.834 D. None of the above.
Question 7 and 8 are based on the following information:
The current price of ABC stock is S0 $200 , the volatility of the stock is 0.30 .
ABC stock is not expected to pay any dividend in the next five years. The continuously
compounded risk-free interest rate is 5% per annum. CDE bank has issued a $1 million
Equity-linked CD with a 5 year maturity. Each CD is structured to pay 100% principle
protection, and pay participation of the ABC stock’s appreciation (This means the CD
will pay a fraction of ABC stock’s return on top of the guaranteed principal payment if
ABC appreciates in value at the maturity date, and pays only the principal amount if the
ABC stock depreciates.).
7. What is the maximum amount of price participation that CDE bank could offer?
A. 76.06% B. 61.52%
C. 100% D. 50%
8. Which of the following is a perfect hedge of the risk associated with the issuance
of the equity linked CD if your answer in 7 is indeed the price participation offered by the
A. A long position of 2339 shares of the ABC stock.
B. A long position of 3075 shares of the ABC stock and a long position in 3075 at-the-
money put options of ABC stock.
C. A short position of 2339 shares of the ABC stock.
D. A long position of 3075 at-the-money put option of the ABC stock.
9. The manager of a blue chip growth stock mutual fund is trying to hedge the
market risk of $135 million portfolio position that has a beta of 1.37 and a correlation of
0.95 with the S&P 500 index. The current level of the S&P 500 Index is $1400, and the
current LIBOR is 4.2% per annum. How many S&P 500 index futures contracts will you
need to hedge the position (Hint: Each S&P 500 futures contract contains 250 S&P500
index)? (Assume you cannot take fractions of contracts in answering this question.)
A. 528 B. 366
C. 551 D. 382
10. Under which of the following circumstances is it NEVER optimal to exercise the
American option before maturity?
A. An exchange American call option whose strike asset is a dividend-paying stock
and the underlying asset is a non-dividend paying stock.
B. An American call option on a dividend paying stock.
C. A dollar denominated American put option on Euro (That is, it is An American
style currency put option on Euro, and the strike currency is US dollar.).
D. An American put option on a non-dividend paying stock.
Please read the instructions below carefully before answering questions in section II and
Instructions: Please write your answers to section II and section III only on the
separately provided Midterm Exam Book II. Any writing on this page will not be
graded. Please show your steps in answering questions in section II and section III.
Partial credits will be given even if the final answers are wrong.
Section II: Hedging Interest Rate Risks (30)
Firm ABC planed a project that would take nine months to complete, at a total cost of $100
million. Bank XYZ offered to provide a nine-month loan starting from Sep 20. The bank
insisted that the interest payment be made quarterly and the loan rate be 200 basis points
above the spot rate of 90-day LIBOR.
Interest payment for each of the three quarters was due at the end of each quarter. Principal
plus the interest payment for the third quarter was due at the end of the last quarter. The
Eurodollar price indexes are given in the table 3.
1) What is the annualized three-month spot rate of LIBOR on Sep 20 (for a three-month
loan initiated on Sep 20)? What is the annualized implied three-month forward rate of
LIBOR on Sep 20, for a loan initiated on Dec 20? What is the annualized implied
three-month forward rate of LIBOR on Sep20, for a loan initiated on Mar 20 of the
next year? (2+2+2=6 points in total)
Maturity of the Contract Eurodollar Price Index
Table 3: Eurodollar Price Indexes
2) Assume no arbitrage and a frictionless market. What is the annualized 6-month spot
rate of LIBOR on Sep 20 (for a three-month loan initiated on Sep 20)? What is the
annualized 9-month spot rate of LIBOR on Sep 20 (for a three-month loan initiated on
Sep 20)? (2+2=4 points in total)
3) Suppose firm ABC decided to use Eurodollar futures contracts to hedge the interest
rate risk. Which Eurodollar futures contracts should the firm use to hedge the interest
rate risk? How many contracts should the firm use? Should the firm take long
positions or short positions in the Eurodollar futures contracts? (2+2+1=5 points in
4) Suppose a swap dealer offers an interest rate swap contract to firm ABC, which
allows firm ABC to pay a fixed swap rate, and receive floating interest payment on
the 100-million principle. Terms of the swap contract is detailed in table 4. What is
the annualized no-arbitrage swap rate? (5 points)
Interest Rate Swap Term Sheet
Floating Rate Payment Fixed Rate Payment
Payer Swap Dealer Payer Firm ABC
Floating Rate Three-month LIBOR Fixed Rate ???
Notional $ 100 million Notional $ 100 million
Date Count 30/360 Date Count 30/360
Interval 3 month Interval 3 month
Maturity 9 months Maturity 9 months
Table 4: The Interest Rate Swap Term Sheet
An American Call Option in a Binomial Model (20 points).
Consider a stock with current price $80 per share, and a continuously compounded
dividend yield of 5% per annum. The continuously compounded interest rate is 3% per
annum. The stock return is lognormally distributed with volatility 0.40 per annum.
Consider an American call option on the stock with strike price $75 and maturity of 12
1) Calculate the u and d using McDonald’s recipe and set up a two-period binomial
model of the stock price movement. (You need to draw the binomial tree, and write down
the stock price on the binomial tree.)
2) Calculate the risk-neutral probability, and describe the optimal exercise rule of the
American call option by circling the nodes at which it is optimal to exercise the option on
the binomial tree you drew in part 1). Calculate the value of the American call at date 0.
3) Calculate the Delta of the American call option at date 0. Calculate the Delta of an
otherwise identical European call option. Which option has a higher Delta? Explain your