# Sample Midterm Solution to by benbenzhou

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```									                         Solution to Sample Midterm
Finance 353 Derivatives
Daytime MBA
Fall 2009

Section I: Multiple Choice Questions. (4X15=60 points).

1. Assuming frictionless market, which of the following cannot be the profit diagram of
any derivative contract?

Answer: C
Shorting the derivative contract creates an arbitrage opportunity.
2. Suppose you enter into a short futures contract to sell July Silver for \$5.20 per ounce
on the New York Commodity Exchange. The size of the contract is 5,000 ounces. The
initial margin is \$4,000, and the maintenance margin is \$3,000. Which of the following
prices will lead to a margin call?

A. \$5.45 per ounce.                                                     B. \$ 5.30 per ounce.
C. \$4.89 per ounce.                                                     D. \$5.10 per ounce.

Answer: A.
In order to receive a margin call, you need to lose \$1,000. Since one contract is 5,000
ounces, and you are shorting the contract, you will lose \$1,000 if silver price increase by
more than 1000/5000=\$0.20
Additional question: which of the above price would allow you to withdraw \$1,000 from
the margins account?         Answer: C

3. Assuming the risk-free interest is 9% per annum with continuous compounding and
that the dividend yield on a stock index varies throughout the year. In February, May,
August, and November, dividends are paid at a rate of 5% per annum. In other months,
dividends are paid at a rate of 2% per annum. Suppose the value of the index on July 31,
2006 is 300. What is the futures price (per share) for a contract deliverable on December
31, 2006?

A. \$265.41                                                              B. \$307.34.
C. \$305.04                                                              D. \$317.91

Hint: Use the average dividend yield.

Answer: B
Aug              Sep                Oct         Nov             Dec
5%               2%                 2%          5%              2%
Hence the average dividend yield is (5%+2%+2%+5%+2%)/5=3.2%. Price of the futures
5
( 0.090.032)*
contract is thus: F0,T  S 0 e ( r  )T  300 * e                 12
 307.3383
Question 4, 5, and 6 are based on the following information:

A stock is expected to pay a dividend of \$1 per share in 2 months and in 5 months. The
current stock price is \$50, and the continuous compounding risk free interest rate is 8%
per annum. An investor has just taken a short position in a 6-month forward contract on
the stock.

4. What is the arbitrage free price of the forward contract?
A. \$52.04                                             B. \$50.01
C. \$47.96                                             D. \$48.05

Answer: B.
The           arbitrage                free                    forward                    price   is
6                 4                 1
0.08*             0.08*             0.08*
F0,T  S0erT  FV ( D)  50* e        12
1* e        12
1* e        12
 50.007

5. Suppose your answer in question 4 is in fact the price specified in the investor’s
forward contract. What is the value of the forward contract?
A. \$0                                                B. \$50.01
C. \$47.96                                            D. \$2.00

Answer: A
The date 0 value of a forward contract is always 0.
6. Suppose your answer in question 4 is in fact the price specified in the investor’s
forward contract. Three months later, the price of the stock is \$48 and the risk free
rate of interest is still 8% per annum. What is the value of the short position of the
forward contract?

A. \$0                                                                  B. \$2.004
C. \$-2.00                                                              D. \$2.00

Answer: B
The           arbitrage                free                    forward                    price   is
3                 1
0.08*             0.08*
F0,T  S0erT  FV ( D)  48* e  1* e12
 47.963     12

However, the forward contract allows you to sell the same stock three months
later at \$50, thereby making a profit of 50.007-47.963 =\$2.044. Therefore the
3
 0.08
present value of the contract is 2.044  e                12
 \$2.004
7. Suppose the continuously compounded Euro-denominated Libor is 4.5% per annum,
the continuously compounded dollar denominated Libor is 3% per annum, and the
current exchange rate is 0.84 \$/€. Which of the following is mostly likely to be the 6-
month forward rate of dollar denominated in Euro traded in Frankfurt?

A. 0.8337 €/\$.                                  B. 1.1994 €/\$.
C. 1.2085 €/\$.                                  D. 1.1728 €/\$

Answer: B
Use      the         formula           for     forward        price   of     currencies:
0.030.045 6
F0,T  x0 e r r*T  0.84  e      0.8337 \$/€ Therefore the Euro-denominated
12

dollar price is 1/0.8337=1.1994 €/\$

8. Suppose the S&P 500 currently has a level of 900. The continuously compounded
return on a one-year T-bill is 5% per annum. You wish to hedge a \$135,000 thousand
portfolio that has a beta of 0.75 (relative to the S&P 500 index). Suppose you can trade
fractions of contracts. How many S&P 500 E-mini futures contracts should you short to
hedge your portfolio?

A. 3.0                                          B. 2.25
C. 2.14                                         D. 2.85

Hint: (One S&P 500 E-mini futures contract is of size \$50XS&P500 index.)

I     135000
Answer: B        Use the formula                   0.75  2.25
S0*
50  900
Question 9 and 10 are based on the following information:
The contract size of Gold futures traded on CBOT is 100 fine troy ounces (We will use
oz hereafter) per contract. The spot price and futures price on Feb 2, 2007 is as follows:

Price
Spot Price                                     710.00 \$/oz
April Futures                                  720.00 \$/oz
June Futures                                   735.00 \$/oz

A gold producer is expecting to sell 100,000 oz of gold in April and in June, respectively.
Assume interest rate is zero in answering question 9 and question 10.

9. Suppose the gold producer wants to use a stack hedge strategy to hedge the risk of gold
price fluctuations. Which of the following is a valid stack hedge strategy?

A. Purchase 1000 April futures on Feb 2, Accept delivery on the 1000 April futures on
April 2. Purchase 1000 June futures on April 2, and accept delivery on the 1000 June
futures on June 2.

B. Sell 1000 April futures on Feb 2, and deliver the 1000 April futures on April 2. Sell
1000 June futures on April 2, and deliver the 1000 June futures on June 2.

C. Sell 2000 April futures on Feb 2. Deliver 1000 April futures contracts on April 2 and
settle the rest of the 1000 April futures contracts by cash. Sell 1000 June futures on April
2, and deliver the 1000 June futures on June 2.

D. Sell 2000 June futures on Feb 2. Sell 100,000 oz gold on the spot market in April.
Deliver the 1000 June futures on June 2, and cash settle the rest of the 1000 June futures
on June 2.

Answer: C is stack hedge.
10. Under which of the following circumstances does the stack hedge strategy perform as
well as a strip hedge strategy?

A. On April 2, the spread between spot price and June futures price is -15 \$/oz (meaning
Spot price - June futures price =-\$15).

B. On April 2, the spread between spot price and June futures price is -10 \$/oz.

C. The spot price on June 2 is \$15 higher than the spot price on April 2.

D. The spot price on April 2 is \$10 higher than the spot price on Feb 2.

Answer: A.

The total gain on the stack hedge strategy is: F0,1   F0,1  P   F1,2 . The total gain on the
1

strip hedge strategy will be F0,1  F0,2 . So the difference between the two is
F  0,1    F0,2    P  F1,2  . Therefore if the spread between spot price on April 2 and the
1

June futures price is -15 \$/oz ( F0,1  F0,2  720  735  15 ), stack hedge perform as well
as strip hedge.

11. Suppose in March, you entered into 10 long positions of the September Eurodollar
futures contract when the price index is 92. Suppose also, on the last trading day of the
Sep Eurodollar futures contract1, the continuously compounded 3-month Libor is 6% per
annum. Which of the following is the gain on your position? (Please ignore marking to
market in answering this question.)

A. -\$48869.35                                      B. \$49607.92
C. \$48869.35                                       D. \$488.6935
1
0.06
Answer: The 3-month effective interest in Sep is e 4  1  0.015113 . Therefore the
price index of the Eurodollar futures contract must be: 100 1.5113 4  93.9548 . The
gain on the Eurodollar futures in therefore: 93.9548  92  2500  4886.935 . Since you
have 10 contracts, you should choose C.

1
This is usually the second London bank business day immediately preceding the third Wednesday of the
contract month (September).
12. A stock that pays 4% continuous dividend yield currently sells for \$86.00. A 10-
month European call option with a strike of \$95.00 has a premium of \$5.7405, and a
European put option with the same strike price and maturity date is traded at \$14.6371.
What is the continuously compounded risk free rate that excludes arbitrage?

A. 3.5% per annum.                                            B. 3.13% per annum.
C. 3.75% per annum.                                           D. 2.95% per annum.

Answer: C
Using the             put-call        parity:   c  p  S 0 e T  Ke  rT ,   we   have   5.7405  14.6371
10              10                      10                                        10
0.04             r                     r                                    0.04
 86  e         12
 95e      12
. That is, 95 e       5.7405  14 .6371  86  e
12                                        12
 92 .0772 .
ln 92.0772 / 95
Therefore we have: r                      0.0375  3.75%
10 / 12

13. Which of the following actions can never be optimal?

A. Exercising before maturity an American put option on a stock that does not pay any
dividend.
B. Exercising before maturity an American call option on a stock that pays continuous
dividend.
C. Exercising an American call option on a stock at maturity just after a large dividend is
paid.
D. Exercising an American put option on a stock at maturity just after a large dividend is
paid.

Answer: C
You should exercise just before the dividend is paid. Since the option is about to expire,
the time value and option value of the option are both zero. Therefore it is optimal to
exercise the call and get the dividend.
14. A stock price is currently \$40. It is known that at the end of one month it will be
either \$42 or \$38, each with 50% probability. The stock does not pay any dividend during
the next month. The risk-free interest rate is 8% per annum with continuous
compounding. What is the value of a 1-month European call option with a strike price of
\$39?

A. \$1.689                                                  B. \$1.701
C. \$ 1.133                                                 D. \$1.126

Answer: A
We can use the risk-neutral probability method here. R  e 0.08 / 12  1.00669 ,
38
1.00669 
*                 40  0.5669 . Therefore the value of the call is c  1 0.5669  3
0
42 38                                                        R

40 40
0.08 / 12
e             0.5669  3  1.6893

15. The risk free interest rate is 10% per annum, the current price of the stock index ABC
is \$50, and volatility of the stock is 30% per annum. The Black-Sholes price of a 3-month
at-the-money European put option on the stock is _____.

A. \$3.61                                           B. \$2.37
C. \$0.047                                          D. \$0.072

Answer:      r  0.1, S 0  50 ,   0.30 , T  3 / 12  0.25, K  50 .      Using BS formula,
 S0 
ln  rT 
d1  
Ke  1
p  Ke  rT N  d 2   S 0 N  d1  ,   where                     T  0.2417                and
 T     2
 S0 
ln  rT 
d2  
Ke  1
  T  0.0917 . We have p  2.37
 T       2
( N  0.0917  and N  0.2417  can be obtained from the normal distribution table. For
example,                                      N 0.0917   N 0.09   0.17  N 0.10   N 0.09 
 0.5359  0.17  0.5398  0.5359   0.5366 ,                                            similarly,
N 0.2417   N 0.24   0.17  N 0.25   N 0.24   0.5948  0.17 * 0.5987  0.5948 
 0.5955 .                   Therefore                     N  0.0917   1  N 0.0917   0.4634 ,
N  0.2417   1  N 0.2417   0.4045 )
Section II. Constructing Arbitrage Portfolios (10X2=20 points)

The following scenario creates arbitrage opportunities for investor. Construct an arbitrage
portfolio and complete the Cash Flow Table of the portfolio constructed to verify it is
indeed an arbitrage opportunity.

1. The current exchange rate between Euro and Yen is 0.02 €/¥. The continuously
compounded interest rate of Euro is 4% per annum, and the continuously compounded
interest rate of Yen is 1% per annum. The price of one futures contract with 6-month
maturity is 254,000 (One Yen futures contract contains ¥ 12,500,000).

Portfolio       Cash Flow at Date 0                                     Cash Flow at Maturity
Strategy
In Yen             In euro                              In Yen                     In Euro
Borrow €                                         0.01
6
0.04
6
12 .5M  e             12
 0.02                              0.24875  e          12

 0.24875 M                                                      0.253778 M

Buy ¥           0.24875 / 0.02      0.24875M
 12.4375M

Lend ¥           12.4375M
6
0.01
12.43766 M  e        12

 12.5M

Short       1                                                            12.5M                    0.254M
futures
Total                                                                                              €221.7

The no arbitrage price of a futures contract of Yen is                                       F0,T  x0 erEUROrYEN T
6
0.03
 12 ,500 ,000  0.02  e     253778 .3 . Therefore the futures contract is over-priced.
12

We sell the futures, and buy Yen today.
2. The current price of the stock is \$100 per share. The continuously compounded interest
rate is 5% per annum. The stock is expected to pay quarterly dividend in the amount of
\$2. The price of an at-the-money call option with 6 months to maturity is \$3, and the
price of a put with the same strike price and maturity is \$2.25. (Assume both the call and
the put expires immediately after the second dividend payment.)

Portfolio Strategy                  Cash Flow at Date 0               Cash Flow at Maturity
ST >=K               ST <K
Long 1 put                          -2.25                             0                    100- ST
Long 1 share of stock               -100                              ST                   ST
3.926 (PV(D))
Short 1 call                        3                                 100-ST                  0
Short Ke  rT bond                  97.53                             -100                    -100
Total                               2.206                             0                       0

The put-call parity implies: c  Ke  rT  p  S 0  PV ( D ) . Using the prices given above:
6
 0.05
 rT
c  Ke           3  100 * e          12
 100 .531 , and the right hand side of the parity is:
3                  6
0.05             0.05
p  S 0  PV D  2.25  100  2  e                   12
 2e         12
 98.32 . Therefore the put is
 rT              T
under-priced: c  Ke  S 0 e  p , we’d like to buy low and sell high, that is long put,
long stock, short call, and short bond.
Section III: Binomial Option Pricing Models. (20 points)
Show your steps. Partial credits will be given even if the final answer is wrong.

Consider a non-dividend paying stock XYZ. Suppose the historical standard deviation of
the log annual return of the stock is   0.30 . Suppose the continuous compounded
interest rate per annum is 8%, and the current price of the stock is S 0  \$100 .

1) Construct the following two-step binomial tree of the stock price movement over a
period of 6 months.

S uu  uuS0
( Node uu)
S u  uS0
( Node u)
S du  duS0
S 0  100
( Node du)
S d  dS 0
( Node d )
S dd  ddS 0
( Node dd )

Compute the appropriate parameters of the model (that is, u , d , and the effective one-
period interest rate R ) from the historical moments of the stock return data.
(Hint: Use the formula: u  e r  h h , d  e r  h h , R  erh )

u                   d                  R
1.1853              0.8781             1.020

2) Compute the delta of an at-the-money call option at node u, and d. Which is greater?

Node                   0                       u                      d
Δ                      0.6074                  1                      0.1513
3) Compute the price of the call option.

* 
Rd
ud
 0.4626 , c  2  * Cuu  2 * 1   *Cud  10.275
R
1
 2


4) Compute the price of a put option with the same strike price and maturity. Using put-
6
0.08
call       parity,     c  p  S 0  Ke rT  100  100  e         12
 3.921 .   Therefore
p  c  3.921  6.354

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