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Unformatted text preview: aiting to exercise the option,
because exercising gives us a share of AOL, which does not pay interest in form of a dividend.
However, by early exercising we will forego the interest we could earn on Apple. Therefore, it is
again never optimal to exercise the American put option early.
Question 9.16. a) Short call perspective: Transaction
Sell Call
Buy put
Buy share
Borrow @ rB
TOTAL t=0
+CB
− PA
−SA
+ Ke − rBT
C B − P A − S A + Ke − rBT time = T
ST ≤ K
0
K − ST
ST
K
0 time=T
ST > K
K − ST
0
ST
−K
0 In order to preclude arbitrage, we must have: C B − P A − S A + Ke − rBT ≤ 0 62 Chapter 9 Parity and Other Option Relationships b) Long call perspective: Transaction
Buy call
Sell put
Sell share
Lend @ rL
TOTAL time = T
ST ≤ K
0 t=0 −CA
ST − K
+ PB
B
− ST
+S
− rLT
+K
− Ke
A
B
B
− rLT
0
− C + P + S − Ke time=T
ST > K
ST − K
0
− ST
+K
0 In order to preclude arbitrage, we must have: − C A + P B + S B − Ke − rLT ≤ 0
Question 9.18. a)
Since the strike prices are symmetric, lambda is equal to 0.5. Therefore, to buy a long
butterfly spread, we buy the 95strike call, sell two 100 strike calls and buy one 105 strike call.
C (K1 ) − C (K 2 ) 13.2 − 8.7
= 0 .9
=
100 − 95
K 2 − K1 and C (K 2 ) − C (K 3 ) 8.7 − 5.4
= 0.66
=
105 − 100
K3 − K2 and C (K 2 ) − C (K 3 ) 9 − 5.20
= 0.76
=
105 − 100
K3 − K2 Convexity holds.
b)
C (K1 ) − C (K 2 ) 12.90 − 9
= 0.78
=
100 − 95
K 2 − K1 Convexity holds.
c)
A market maker can buy a butterfly spread at the prices we sell it for. Therefore, the
above convexity conditions are the ones relevant for market makers. Convexity is not violated
from a market maker’s perspective.
d)
Let us check the conditions for the retail seller of a butterfly spread (she is the one who
buys the 100 strike option). Please note that this is the same position as that of a market maker
who buys the butterfly spread. We have:
C (K1 ) − C (K 2 ) 12.90 − 9.20
= 0.74 and
=
100 − 95
K 2 − K1 Therefore, convexity is indeed violated. 63 C (K 2 ) − C (K 3 ) 9.2 − 5.20
= 0 .8
=
105 − 100
K3 − K2 Chapter 10
Binomial Option Pricing: I
Question 10.2.
a) Using the formulas of the textbook, we obtain the following results:
∆ = 0 .7
B = −53.8042
price = 16.1958 b)
If we observe a price of $17, then the option price is too high relative to its theoretical
value. We sell the option and synthetically create a call option for $19.196. In order to do so, we
buy 0.7 units of the share and borrow $53.804. These transactions yield no risk and a profit of
$0.804.
c)
If we observe a price of $15.50, then the option price is too low relative to its theoretical
value. We buy the option and synthetically create a short position in an option. In order to do so,
we sell 0.7 units of the share and lend $53.8042. These transactions yield no risk and a profit of
$0.696.
Question 10.4.
The stock prices evolve according to the following picture:
169 callpayoff: 74 104 callpayoff: 9 64 callpayoff: 0 130
100
80 Since we have two binomial steps, and a time to expiration of one year, h is equal to 0.5.
Therefore, we can calculate with the usual formulas for the respective nodes:
t=0, S=100 t=1, S=80 t=1, S=130 64 Chapter 10 Binomial Option Pricing: 1 ∆ = 0.691 ∆ = 0.225 ∆ =1 B = −49.127
price = 19.994 B = −13.835
price = 4.165 B = −91.275
price = 38.725 Question 10.6.
The stock prices evolve according to the following picture:
169 putpayoff: 0 104 putpayoff: 0 64 putpayoff: 31 130
100
80 Since we have two binomial steps, and a time to expiration of one year, h is equal to 0.5.
Therefore, we can calculate with the usual formulas for the respective nodes:
t=0, S=100
∆ = −0.3088
B = 38.569
price = 7.6897 t=1, S=80
∆ = −0.775
B = 77.4396
price = 15.4396 t=1, S=130
∆=0
B=0
price = 0 Question 10.8.
We must compare the results of the equivalent European put that we calculated in exercise 10.6.
with the value of immediate exercise. In 10.6., we calculated: t=1, S=80
∆ = −0.775
B = 77.4396
price = 15.4396
immediate exercise =
max(95  80, 0 ) = 15 t=1, S=130
∆=0
B=0
price = 0
immediate exercise
= max(95 − 130,0) = 0 65 Part 3 Options Since the value of immediate exercise is smaller than or equal to the continuation value (of the
European options) at both nodes of the tree, there is no benefit to exercising the options before
expiration. Therefore, we use the European option values when calculating the t = 0 option price:
t=0, S=100
∆ = −0.3088
B = 38.569
price = 7.6897
immediate exercise = max (95  100, 0 ) = 0 Since the option price is again higher than the value of immediate exercise (which is zero), there
is no benefit to exercising the option at t = 0. Since it is never optimal to exercise earlier, the
early exercise option has no value. The value of the American put option is identical to the value
of the European put option.
Question 10.10.
We can calculate for the different nodes of the tree: delta
B
Call premium
value of early exercise node uu node ud = du
1
0.8966
92.5001
79.532
56.6441
15.0403
54.1442
10.478 node dd
0...
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 Spring '14
 NguyenThiMaiLan
 Derivatives

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