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Derivative Securities, Fall 2023
FNCE4040, Sample Midterm Exam 1 Solutions
FNCE 4040 Derivative Securities
Sample Midterm Exam 1 Solutions
Name:
ID#:
Section#:
Instructions:
•
You have 80 minutes to answer the questions.
•
The exam is closed-book: you are not allowed to consult any books, notes, readings, material
distributed in class, or other resources. You are allowed to consult the formula sheet that has
been distributed for this exam.
•
For your calculations, you are allowed to use a blank Excel spreadsheet (no pre-programmed
macros or pre-populated cells), or a scientific calculator.
•
The exam is an individual effort. Students must not compare questions, answers, or solution
methods with anyone during the exam.
•
The honor code applies.
©
Buffa-Garc´
ıa, Leeds School of Business
Page 1 of 6
Derivative Securities, Fall 2023
FNCE4040, Sample Midterm Exam 1 Solutions
Short Questions –
30 points
1. The seller of a put option has the right to not exercise the option at the expiration date.
(a) True
(b)
False
✓
2. An American option can be exercised at the expiration date, as well as at any point in time
before it.
(a)
True
✓
(b) False
3. If an option is out-of-the money, its time value can be negative.
(a) True
(b)
False
✓
4. The payoff of a butterfly spread is equal to:
(a) max(
S
T
−
K
1
,
0)
−
max(
S
T
−
K
2
,
0) + 2
×
max(
S
T
−
(
K
1
+
K
2
)
,
0)
(b) max(
S
T
−
K
1
,
0
) + max(
S
T
−
K
2
,
0
)
−
2
×
max(
S
T
−
(
K
1
+
K
2
)
/
2
,
0
)
✓
(c) max(
S
T
−
K
1
,
0)
−
max(
S
T
−
K
2
,
0) + max((
K
1
+
K
2
)
/
2
−
S
T
,
0)
(d) max(
S
T
−
K
1
,
0) + max(
S
T
−
K
2
,
0)
−
max(2
×
S
T
−
(
K
1
+
K
2
)
/
2
,
0)
5. Consider buying a put option with a strike of
$
50, which costs
$
7. What would be your profits
if the underlying asset trades for
$
57 at the expiration date? What would be your profits if
the underlying asset trades for
$
41 at the expiration date?
Solution.
The profit for a general value of the underlying asset at the expiration date is
max(50
−
S
T
,
0)
−
7. So, if
S
T
= 57, the profit is
−
$7. If
S
T
= 41, the profit is
$
2.
6. A call option on MSFT with a strike of
$
40 and a one year maturity is trading for
$
7.19.
MSFT is trading for
$
40, and the annual risk-free rate is 1%. What should be the price of a
put option with the same strike and maturity, in the absence of arbitrage?
Solution.
From the Put-Call Parity
P
0
+
S
0
=
C
0
+
PV
0
(
K
), we can the no-arbitrage price
of the put option as
P
0
=
C
0
+
PV
0
(
K
)
−
S
0
= 7
.
19 +
40
(1 + 1%)
−
40 = 6
.
79
.
©
Buffa-Garc´
ıa, Leeds School of Business
Page 2 of 6
Derivative Securities, Fall 2023
FNCE4040, Sample Midterm Exam 1 Solutions
Term Structure of Interest Rates –
35 points
1. The following three bonds are available for trading:
Maturity
Coupon Rate
Face Value
Price
(
years
)
(
annual
)
(
$
)
(
$
)
Bond A
3
10%
1000
986.89
Bond B
3
20%
1000
1242.59
Bond C
2
30%
1000
1421.18
Assume all bonds pay coupons annually.
(a) What is the term structure in this economy? Determine
r
1
,
r
2
, and
r
3
.
Solution.
The set of spot rates
r
1
,
r
2
and
r
3
that define the term structure must satisfy:
986
.
89 =
100
1 +
r
1
+
100
(1 +
r
2
)
2
+
1100
(1 +
r
3
)
3
1242
.
59 =
200
1 +
r
1
+
200
(1 +
r
2
)
2
+
1200
(1 +
r
3
)
3
1421
.
18 =
300
1 +
r
1
+
1300
(1 +
r
2
)
2
Solving the above system of equations yields
r
1
= 5%,
r
2
= 7%, and
r
3
= 11%.
(b) What are the forward rates in this economy? In particular, spell out what rate you can
get for an investment at date 1 that matures at date 2, for an investment at date 2 that
matures at date 3, and for an investment at date 1 that matures at date 3.
Solution.
The forward rates,
f
1
,
2
and
f
2
,
3
, are given by
(1 +
r
2
)
2
= (1 +
r
1
)(1 +
f
1
,
2
)
⇒
f
1
,
2
=
(1 +
r
2
)
2
(1 +
r
1
)
−
1 =
(1
.
07)
2
1
.
05
−
1 = 9
.
04%
(1 +
r
3
)
3
= (1 +
r
2
)
2
(1 +
f
2
,
3
)
⇒
f
2
,
3
=
(1 +
r
3
)
3
(1 +
r
2
)
2
−
1 =
(1
.
11)
3
(1
.
07)
2
−
1 = 19
.
45%
(1 +
r
3
)
3
= (1 +
r
1
)(1 +
f
1
,
3
)
2
⇒
f
1
,
3
=
(1 +
r
3
)
3
(1 +
r
1
)
1
2
−
1 =
(1
.
11)
3
1
.
05
1
2
−
1 = 14
.
13%
(c) Suppose that we have a fourth bond, bond D, that pays annual coupons of 5%, face value
of
$
1000, and a maturity of 3 years. What is the no-arbitrage value of bond D?
Solution.
With the given term structure, we value the bond using the present value
formula:
V
D
=
50
1 +
r
1
+
50
(1 +
r
2
)
2
+
1050
(1 +
r
3
)
3
= 859
.
04
©
Buffa-Garc´
ıa, Leeds School of Business
Page 3 of 6
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Derivative Securities, Fall 2023
FNCE4040, Sample Midterm Exam 1 Solutions
(d) How would you replicate the future cash flows of bond D by trading bonds A, B, and
C given above? Give an explicit description of the units of each bond in the replicating
portfolio.
Solution.
Let (
b
A
, b
B
, b
C
) denote the units of Bond A, Bond B, and Bond C, respectively.
The future cash flows of the replicating portfolio need to match those of bond D in each
period:
t
= 1 :
100
b
A
+ 200
b
B
+ 300
b
C
= 50
t
= 2 :
100
b
A
+ 200
b
B
+ 1300
b
C
= 50
t
= 3 :
1100
b
A
+ 1200
b
B
+ 0
b
C
= 1050
Solving for (
b
A
, b
B
, b
C
) jointly, one obtains that the replicating portfolio is
b
A
= 1
.
5,
b
B
=
−
0
.
5 and
b
C
= 0. So, we need to:
•
buy 1.5 units of Bond A;
•
sell 0.5 units of Bond B;
•
not trade Bond C.
Note that the cost of this portfolio is:
1
.
5
×
986
.
89
−
0
.
5
×
1242
.
59 = 859
.
04
(the fair value of bond D)
(e) You discover that bond D is currently traded for
$
855-
$
860 (bid-ask prices). Can you
create an arbitrage strategy that involves trading bond D, as well as the other three
bonds given above? If so, give an explicit description of the units of each bond in your
arbitrage strategy, as well as the arbitrage profit.
Solution.
Since the cost of the replicating portfolio is in between the bid and the ask
prices of that bond, no arbitrage opportunity exists.
©
Buffa-Garc´
ıa, Leeds School of Business
Page 4 of 6
Derivative Securities, Fall 2023
FNCE4040, Sample Midterm Exam 1 Solutions
Option Trading –
35 points
1. Consider the following options on the same underlying asset with expiration in two months:
•
Call option with strike price
$
80 is currently priced at
$
14.6;
•
Put option with strike price
$
80 is currently priced at
$
7.2;
•
Put option with strike price
$
95 is currently priced at
$
13.8.
The annual risk-free rate is 3% and the term structure is flat (consider annual compounding).
The underlying asset is not expected to pay any dividends in the next two months.
(a) Based on the options above, what should be the no-arbitrage price of the underlying
asset?
Solution.
Using the Put-Call Parity
P
0
+
S
0
=
C
0
+
PV
0
(
K
) for the options with strike
price
K
1
= $80, we can solve for the no-arbitrage price of the underlying asset:
S
0
=
C
0
(
K
1
) +
PV
0
(
K
1
)
−
P
0
(
K
1
) = 14
.
6 +
80
(1 + 3%)
2
/
12
−
7
.
2 = 87
.
01
(b) Note that at the moment there are no call options with strike price
$
95 (on the same un-
derlying asset and with expiration in two months) that are traded in the market. Based
on the options above, what should be the no-arbitrage price of that call option with strike
price
$
95?
Solution.
Combining the Put-Call Parities for the options with strike prices
K
1
= $80
and
K
2
= $95, we obtain that the no-arbitrage price of a call option with strike price
$
95 is
C
0
(
K
2
) =
C
0
(
K
1
) +
P
0
(
K
2
)
−
P
0
(
K
1
)
−
(
K
2
−
K
1
)
(1 + 3%)
2
/
12
= 14
.
6 + 13
.
8
−
7
.
2
−
15
(1
.
03)
1
/
6
= 6
.
27
(c) You realize that someone has just written some call options with strike price
$
95, which
are now traded for
$
7.4. Is there an arbitrage strategy that you can implement without
trading the underlying asset? Give an explicit description of the units of each asset in
your arbitrage strategy, as well as the arbitrage profit.
Solution.
From part (b) we can infer the replicating portfolio for the the call option
with strike
K
2
= $95:
•
buy one call option with strike
K
1
= $80;
•
buy one put option with strike
K
2
= $95;
•
sell one put option with strike
K
1
= $80;
•
sell a risk-free zero-coupon bond with face value of
$
15 and maturity 2 months.
Since the cost of this replication portfolio is
$
6.27, the call option with strike
K
2
= $95
is overvalued. Therefore, the arbitrage strategy consists in shorting the call option with
strike
K
2
= $95 and buying its replicating portfolio. The following cash flow table reports
the cash flows of the arbitrage strategy and its profit.
©
Buffa-Garc´
ıa, Leeds School of Business
Page 5 of 6
Derivative Securities, Fall 2023
FNCE4040, Sample Midterm Exam 1 Solutions
In two months
Strategy
Today
S
T
<
80
80
⩽
S
T
<
95
S
T
⩾
95
Short Call (
K
2
= $95)
7
.
4
0
0
−
(
S
T
−
95)
Long Call (
K
1
= $80)
−
14.6
0
S
T
−
80
S
T
−
80
Long Put (
K
2
= $95)
−
13.8
95
−
S
T
95
−
S
T
0
Short Put (
K
1
= $80)
7.2
−
(80
−
S
T
)
0
0
Short ZC-Bond (
F
= $15)
14.93
−
15
−
15
−
15
Net Cash Flows
1.13
0
0
0
(d) (***,
10 extra points
) If the current market price of the underlying asset is actually 5%
higher than the no-arbitrage price you obtained in part (a), is there a more profitable
arbitrage strategy than the one you identified in part (c)? Give an explicit description of
the
most
profitable arbitrage strategy that you can create by trading the options above,
bonds, and the underlying asset.
Solution.
If the price of the underlying asset
S
0
= 87
.
01
×
1
.
05 = 91
.
36, we can compute
deviations from the two Put-Call Parities. From the Put-Call Parity with
K
1
= $80, the
arbitrage profit is exactly the difference between the price of the underlying asset and
the no-arbitrage one obtained in part (a): $91
.
36
−
$87
.
01 = $4
.
35. From the Put-Call
Parity with
K
2
= $95, the no-arbitrage price of the underlying asset is
S
0
=
C
0
(
K
2
) +
PV
0
(
K
2
)
−
P
0
(
K
2
) = 7
.
4 +
95
(1 + 3%)
2
/
12
−
13
.
8 = 88
.
13
So, the arbitrage profit obtained by using Put-Call Parity with
K
2
= $95 to construct a
replicating portfolio for the underlying asset is: $91
.
36
−
$88
.
13 = $3
.
23.
Therefore, the most profitable arbitrage strategy is:
•
sell the underlying asset
•
buy one call option with strike
K
1
= $80;
•
sell one put option with strike
K
1
= $80;
•
buy a risk-free zero-coupon bond with face value of
$
80 and maturity 2 months.
The following cash flow table reports the cash flows of the arbitrage strategy and its
profit.
In one-year
Strategy
Today
S
T
<
80
S
T
⩾
80
Short Underlying Asset
91.36
−
S
T
−
S
T
Long Call (
K
1
= $80)
−
14
.
6
0
S
T
−
80
Short Put (
K
1
= $80)
7.2
−
(80
−
S
T
)
0
Long ZC-Bond (
F
= $80)
−
79
.
61
80
80
Net Cash Flows
4
.
35
0
0
©
Buffa-Garc´
ıa, Leeds School of Business
Page 6 of 6
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Issuing Stock
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ETHICS Lou Hoskins and Shirley Crothers are
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authorization for 400,000,000 shares of common
stock to be sold to the general public. Lou and Shirley
have decided to establish par of $0.03 per share in
order to appeal to a wide variety of potential
investors. Lou and Shirley believe…
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Your grandfather has agreed to deposit a certain amount of money each year into an account paying 7.90 percent annually to help you
go to graduate school. Starting next year, and for the following four years, he plans to deposit $2,350, $8,600, $7,200, $6,600, and
$12,150 into the account. How much will you have at the end of the five years? (Round answer to 2 decimal places, e.g. 15.25.)
Future value at end of five years
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* CengageNOww2 | Online teachin x
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Providing for Doubtful Accounts
At the end of the current year, the accounts receivable account has a debit balance of $969,000 and sales for the year total $10,990,000.
a. The allowance account before adjustment has a credit balance of $13,100. Bad debt expense is estimated at 1/4 of 1% of sales.
b. The allowance account before adjustment has a credit balance of $13,100. An aging of the accounts in the customer ledger indicates estimated
doubtful accounts of $41,900.
C. The allowance account before adjustment has a debit balance of $4,600. Bad debt expense is estimated at 3/4 of 1% of sales.
d. The allowance account before adjustment has a debit balance of $4,600. An aging of the accounts in the customer ledger indicates estimated
doubtful accounts of $38,200.
Determine the amount of the adjusting…
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ccounting | Spring 2023
uiz
Course Home - kar X
K
Chapter 9 Quiz
OA. $18,200
OB. $18,800
OC. $17,500
O D. $20,300
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Question 5 of 10
X G At year-end, Snow
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Buddy Company uses the allowance method to account for uncollectible receivables. At the beginning of the year,
Allowance for Bad Debts had a credit balance of $800. During the year Buddy wrote off uncollectible receivables of
$2,100. Buddy recorded Bad Debts Expense of $2,800. Buddy's year-end balance in Allowance for Bad Debts is
$1,500. Buddy's ending balance of Accounts Receivable is $20,300. Compute the net realizable value of Accounts
Receivable at year-end.
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2022 Spring
PROBLEM 1
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Use the Excel Template File to answer Problem 1 and Problem 2
Exam II Excel Template File. Problem1 and Problem2.xlsx
nnouncements
The accounts listed below for the year ended 2021, in random order, are for Polanyi Knowledge, Inc.
Construct a classified balance sheet for the business in good form.
yllabus
Modules
Cash
32,900
Discount on bonds payable
103.000
rades
Accumulated depreciation-Buildings 630,600
Wages payable
92,000
eople
Accounts Receivable
90,200
Note payable due 2023
300,000
uJa
Taxes payable
27,000
Retained earnings
980,000
Inventory
180,000
Bonds payable
500,000
Additional paid in capital common
Common stock
264,400
150,000
stock
Long-term investments
77,000
Preferred stock
110,000
Buildings
1,500,000
Accounts payable
167.000
Additional paid in capital preferred
Accumulated depreciation:
13,400
stock
210,000
equipment
Equipment
760,000
Land
310,600
Prepaid…
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Sheffield Corp. uses the percentage-of-receivables basis to record bad debt expense and concludes that 3% of accounts receivable will become
uncollectible. Accounts receivable are $430,800 at the end of the year, and the allowance for doubtful accounts has a credit balance of $2,864.
(a) Prepare the adjusting journal entry to record bad debt expense for the year.
(b) If the allowance for doubtful accounts had a debit balance of $887 instead of a credit balance of $2,864, prepare the adjusting journal entry for
bad debt expense.
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