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Derivative Securities, Fall 2023
FNCE 4040, Sample Midterm Exam 2 Solutions
FNCE 4040 Derivative Securities
Sample Midterm Exam 2 Solutions
Problem 1 — Oratio Dominica
(40 points)
1. Short-answer questions.
(a) The risk-neutral probability
p
that the stock goes up over the next period is always higher
(i.e., more optimistic) than the true probability
q
.
i. True
ii.
False
✓
(b) It is never optimal to exercise an American call option early (i.e., before the expiration
date) if the underlying asset is not expected to pay any dividends before the expiration
date.
i.
True
✓
ii. False
(c) The ∆ of a call option
i. is always larger than 1
ii.
is always smaller than 1
✓
iii. can be larger or smaller than 1
(d) Consider a stock price with annual volatility sigma of 65% and an annual risk-free rate
of 6%. What is the state price of the down state
π
d
in a binomial model where the stock
price changes proportionally to its previous value and each period corresponds to three
weeks?
i. 0.593
ii. 0.472
iii.
0.526
✓
Solution.
(1+
r
f
) = (1+0
.
06)
3
/
52
= 1
.
003367,
u
=
e
0
.
65
√
3
/
52
= 1
.
169,
d
= 1
/u
= 0
.
855.
p
=
(1 +
r
f
)
−
d
u
−
d
= 47
.
18%
⇒
π
d
=
1
−
p
1 +
r
f
= 0
.
526
(e) The current price of AAPL is
177 and its annual volatility is 28%. The annual risk-free
rate is 7%.
If the AAPL price can only go up or down over the next month, what is
the current price of an American put option with strike price
190 that expires in one
month?
Solution.
u
=
e
0
.
28
√
1
/
12
= 1
.
084 and
d
= 1
/u
= 0
.
922.
Hence,
S
u
= 191
.
90 and
S
d
= 163
.
26. The payoff of the put option, max(
K
−
S
T
,
0), is therefore given by:
P
u
= 0
and
P
d
= 26
.
74.
Given that (1 +
r
f
) = (1 + 0
.
07)
1
/
12
= 1
.
005654, the risk-neutral
probability
p
is equal to
p
=
(1 +
r
f
)
−
d
u
−
d
= 51
.
47%
.
The price today of the American put option is given by
P
0
= max
p
×
0 + (1
−
p
)
×
26
.
74
1 +
r
f
,
190
−
177
= max(12
.
90
,
13) = 13
.
Buffa-Garc´
ıa, Leeds School of Business
Page 1 of 4
Derivative Securities, Fall 2023
FNCE 4040, Sample Midterm Exam 2 Solutions
2.
Gold is trading for
100. Assume that it is expected to appreciate by 20% over the next year.
You also know that the volatility of gold’s returns is 25%, and that the risk-free rate is 10%.
(a) The price of a put option with a strike of
K
= 100 on gold with a one year maturity,
according to the Black-Scholes model, is
5.63.
(b) One should short 0.3063 units of gold, and lend
36.26.
(c) The ∆ of the firm’s portfolio is
−
0
.
3063, so taking a long position in gold of 0.3063 units
would make the portfolio ∆-neutral.
Buffa-Garc´
ıa, Leeds School of Business
Page 2 of 4
Derivative Securities, Fall 2023
FNCE 4040, Sample Midterm Exam 2 Solutions
Problem 2 — Fifty-cents
(30 points)
1. The most direct way to approach this problem is via risk-neutral probabilities (i.e. normalized
state prices). In our problem we had
ˆ
p
u
=
r
−
d
u
−
d
=
1
.
01
−
0
.
6
1
.
4
−
0
.
6
= 0
.
5125;
ˆ
p
d
= 0
.
4875
.
Starting at expiration we note that by basic principles the values of the put will be given by
the following table.
8.53
3.45
0.00
0.00
14.05
7.14
0.00
21.60
14.80
29.20
2. This problem was intended to go through a dynamic replication argument in full At each date
t
, we look for a portfolio that buys ∆
t
units of the stock, and invest
B
t
dollars in the bond,
so that
∆
t
S
tu
+
B
t
1
.
01 =
P
tu
;
∆
t
S
td
+
B
t
1
.
01 =
P
td
;
where
S
tu
and
P
tu
denote the prices of the underlying asset and the put in the up state (from
node
t
), and
S
td
and
P
td
denote the prices of the underlying asset and the put in the down
state (from node
t
).
Solving one gets the following dynamic trading strategy:
Deltas
Cash positions
-0.2651
-0.1276
0
21.79
12.38
0.00
-0.6025
-0.4405
32.13
25.64
-1.0000
39.60
3. The value of the put with a strike of
45 is
10.87, so the extra protection will cost Fifty-cents
a bit more than
2. The value trees for the put with
K
= 45 is included below.
Put tree
10.87
4.61
0.00
0.00
17.67
9.56
0.00
26.55
19.80
34.20
Solving as before for the replicating portfolio, one gets the following dynamic trading strategy:
Deltas
Cash positions
-0.3263
-0.1707
0
27.18
16.56
0.00
-0.7082
-0.5893
38.91
34.31
-1.0000
44.55
Buffa-Garc´
ıa, Leeds School of Business
Page 3 of 4
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Derivative Securities, Fall 2023
FNCE 4040, Sample Midterm Exam 2 Solutions
Problem 3 — Trader Smurf
(30 points)
Note that the volatility of monthly returns from investing in mushrooms of 8.66% translates into
an annual volatility of 30%.
The following Greeks are relevant for this problem:
Position in options
100
−
80
50
Type
Calls
Puts
Puts
Strike
100.00
90.00
95.00
Call price
7.4054
9.7790
9.7217
Put price
8.9955
2.6879
6.4322
Delta call
0.5114
0.7213
0.6066
Delta put
−
0
.
4886
−
0
.
2787
−
0
.
3934
Gamma
0.0196
0.0233
0.0189
Vega
27.0700
16.1226
26.1087
1. The value of the portfolio of Trader Smurf is, in 1000s:
V
= 100(7
.
41)
−
80(2
.
69) + 50(6
.
43) = 847
With the risk-free asset, his portfolio amounts to a little over
1.3 mill.
2. The delta of the portfolio Trader Smurf owns is
∆ = 100(0
.
5114)
−
80(
−
0
.
2787) + 50(
−
0
.
3934) = 53
.
77
Thus, all Trader Smurf needs to do is to short 53.77K mushrooms.
3. First note that the Γ of the portfolio of options is:
Γ = 100(0
.
0196)
−
80(0
.
0233) + 50(0
.
0189) = 1
.
0365
Let
x
denote the number of units of the underlying asset Trader Smurf buys, and
y
the number
of units of the
K
= 95 options that he needs to trade.
Trader Smurf should hedge by creating a portfolio that is both delta and gamma neutral:
∆ = 53
.
77 +
x
+
y
(
−
0
.
3934) = 0;
Γ = 1
.
0365 +
y
(0
.
0189) = 0;
so
x
=
−
75
.
4 and
y
=
−
55
.
1, i.e., Trader Smurf needs to sell all his holdings of the calls with
K
= 95 (and a few more, his portfolio has a long position of +50, he should end up at
−
5
.
1),
as well as short
−
75
.
4 mushrooms.
4. Trader Smurf should hedge by creating a portfolio that is both delta and vega neutral.
Let
x
denote the number of units of the underlying asset Trader Smurf buys, and
y
the number
of units of the
K
= 95 options that he needs to trade. Trader Smurf should pick
x
and
y
such
that
∆ = 53
.
77 +
x
+
y
(
−
0
.
3934) = 0;
vega = 2722 +
y
(26
.
1) = 0;
so
x
=
−
94
.
8 and
y
=
−
104
.
2, i.e., Trader Smurf needs to sell all his holdings of the calls with
K
= 95 and short 54.2 more, as well as short
−
94
.
8 mushrooms.
Buffa-Garc´
ıa, Leeds School of Business
Page 4 of 4
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Return on Security 2
Return on Security 3
1
.14
.199
.199
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.36
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.099
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.36
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.049
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state of Econ.
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Stock II
0.20
0.06
0.55
0.47
0.25
0.23
The market risk premium is 8%, and the risk free rate is 6%.
Economy
Recession
Normal
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Exhibit 5.5
USE THE INFORMATION BELOW FOR THE FOLLOWING PROBLEM(S)
Stock
Rit
Rmt
ai
Beta
A
10.6
15
0
0.8
Z
9.8
8.0
0
1.1
Rit = return for stock i during period t
Rmt = return for the aggregate market during period t
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a.
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