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Unformatted text preview: of the above. 8 Solution: (c)
In our usual notation,
1
1
S (0)
+ r − δ + σ2 T
d1 = √
ln
K
2
σT
1
1
1
50
=
+ 0.05 − 0.02 + × 0.342 ×
= 1.44,
ln
40
2
4
0.34 1/4
√
d2 = d1 − σ T = 1.27.
The standard normal tables give us
N (d1 ) = 0.9253, N (d2 ) = 0.8983. Finally,
VC (0) = S (0)e−δT N (d1 ) − Ke−rT N (d2 ) = 10.55.
5. In this problem, use the BlackScholes pricing model.
Consider a bear spread consisting of a 20−strike put and a 25−strike put. Suppose that
σ = 0.30, r = 0.04, δ = 0, T = 1 and S (0) = 15.
What is the price of this bear spread?
(a) About $4.36
(b) About $4.80
(c) About $9.16
(d) About $13.96
(e) None of the above
Solution: (a)
Here, you sell a 20−strike put and buy a 25−strike put.
For the 20−strike put, we have
1
15
1
d1 =
[ln( ) + (0.04 + · 0.32 ) · 1] = −0.6756;
0.3 · 1
20
2
d2 = −0.9756.
The price of the 20−strike put is
VP (0, 20) = Ke−rT N (−d2 ) − S (0)e−δT N (−d1 ) = 4.79698.
For the 25−strike put, we have
1
15
1
d1 =
[ln( ) + (0.04 + · 0.32 ) · 1] = −1.4194;
0.3 · 1
25
2
d2 = −1.7194.
The price of the 25−strike put is
VP (0, 25) ≈ 25e−0.04 N (−d2 ) − 15e−0.02 N (−d1 ) = 9.16076
So, the price of the bear spread is
VP (0, 25) − VP (0, 20) = 9.16076 − 4.79698 = 4.36378. 9 6. Assume the BlackScholes framework. Let the current price of a share of nondividendpaying stock be equal to S (0) = 100; let its volatility be σ = 0.3.
Consider a gap call option with expiration date T = 1, with the trigger price 100 and
the strike price 90.
You are given that the continuously compounded riskfree interest rate equals r = 0.04
per annum.
Let VGC (0) denote the price of the above gap option. Then,
(a) VGC (0) < $13.20
(b) $13.20 ≤ VGC (0) < $15.69
(c) $15.69 ≤ VGC (0) < $17.04
(d) 17.04 ≤ VGC (0) < $21.25
(e) None of the above.
Solution: (d)
In our usual notation, the BlackScholes formula for the price of a gap call option reads as
VGC (0) = S (0)−δT N (d1 ) − K1 e−rT N (d2 )
where
1
1
d1 = √ [ln(S (0)/K2 ) + (r − δ + σ 2 )T ],
2
σT
√
d2 = d1 − σ T .
In the present problem,
1
0.09
[ln(100/100) + (0.04 +
)] ≈ 0.28;
0.3
2
d2 = −0.02.
d1 = So, the price equals
VGC (0) = 100N (0.28) − 90e−0.04 N (−0.02) = 18.49.
7. Assume the BlackScholes setting. Alice wagers to pay one share of stock to Bob if the price
of nondividendpaying stock in 1 year is above $100.00. Assume S (0) = 100.00, σ = 0.3,
and r = 0.04. What is the time−0 value of this bet?
(a) About $61.15
(b) About $81.15
(c) About $91.15
(d) About $100
(e) None of the above.
Solution: (a) or (e)
The BlackScholes price of this asset call is
V 3 [S (0), 0] = S (0)N (d1 ) 10 with
d1 = 1
0.3 ln( 100
1
) + 0.04 + 0.09
100
2 = 0.2833. So,
V 3 [S (0), 0] = 100N (0.28) = 61.03.
Note: If you use the standard normal “calculator”, you get 61.15....
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This note was uploaded on 08/04/2013 for the course M 339W taught by Professor Cudina during the Fall '12 term at University of Texas.
 Fall '12
 CUDINA
 Math

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