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0
0 Using these values at the previous node and at the initial node yields: delta
B
Call premium
value of early exercise t=0
0.7400
55.7190
18.2826
5 node d
0.4870
35.3748
6.6897
0 node u
0.9528
83.2073
33.1493
27.1250 Please note that in all instances the value of immediate exercise is smaller than the continuation
value, the (European) call premium. Therefore, the value of the European call and the American
call are identical. 66 Chapter 10 Binomial Option Pricing: 1 b)
We can calculate similarly the binomial prices at each node in the tree. We can calculate
for the different nodes of the tree: delta
B
Put premium
value of early exercise node uu node ud = du
0
0.1034
0
12.968
0
2.0624
0
0 node dd
1
92.5001
17.904
20.404 Using these values at the previous node and at the initial node yields: delta
B
Put premium
value of early exercise t=0
0.26
31.977
5.979
0 node d
0.513
54.691
10.387
8.6307 node u
0.047
6.859
1.091
0 c)
From the previous tables, we can see that at the node dd, it is optimal to early exercise the
American put option, because the value of early exercise exceeds the continuation value.
Therefore, we must use the value of 20.404 in all relevant previous nodes when we back out the
prices of the American put option. We have for nodes d and 0 (the other nodes remain
unchanged): delta
B
Put premium
value of early exercise t=0
0.297
36.374
6.678
0 node d
0.594
63.005
11.709
8.6307 The price of the American put option is indeed 6.678.
Question 10.12.
a) We can calculate u and d as follows:
u = e (r −δ )h +σ
d = e (r −δ )h −σ h
h = e (0.08 )×0.25+0.3×
= e (0.08 )×0.25−0.3× 0.25
0.25 = 1.1853
= 0.8781 67 Part 3 Options b)
We need to calculate the values at the relevant nodes in order to price the European call
option: delta
B
Call premium t=0
0.6074
20.187
4.110 node d
0.1513
4.5736
0.7402 node u
1
39.208
8.204 c)
We can calculate at the relevant nodes (or, equivalently, you can use putcallparity for
the European put option):
European put
delta
B
Put premium t=0
0.3926
18.245
2.5414 node d
0.8487
34.634
4.8243 node u
0
0
0 For the American put option, we have to compare the premia at each node with the value of early
exercise. We see from the following table that at the node d, it is advantageous to exercise the
option early; consequently, we use the value of early exercise when we calculate the value of the
put option.
American put
delta
B
Put premium
value of early exercise t=0
0.3968
18.441
2.5687
0 node d
0.8487
34.634
4.8243
4.8762 node u
0
0
0
0 Question 10.14.
a)
We can calculate the price of the call currency option in a very similar way to our
previous calculations. We simply replace the dividend yield with the foreign interest rate in our
formulas. Thus, we have: delta
B
Call premium node uu node ud = du
0.9925
0.9925
0.8415
0.8415
0.4734
0.1446 node dd
0.1964
0.1314
0.0150 Using these call premia at all previous nodes yields: delta
B
Call premium t=0
0.7038
0.5232
0.1243 node d
0.5181
0.3703
0.0587
68 node u
0.9851
0.8332
0.2544 Chapter 10 Binomial Option Pricing: 1 The price of the European call option is $0.1243.
b) For the American call option, the binomial approach yields: delta
B
Call premium
value of early exercise node uu node ud = du
0.9925
0.9925
0.8415
0.8415
0.4734
0.1446
0.4748
0.1436 node dd
0.1964
0.1314
0.0150
0 Using the maximum of the call premium and the value of early exercise at the previous nodes
and at the initial node yields:
t=0
node d
node u
delta
0.7056
0.5181
0.9894
B
0.5247
0.3703
0.8374
Call premium
0.1245
0.0587
0.2549
value of early exercise
0.07
0
0.2540
The price of the American call option is: $0.1245.
Question 10.16.
aa)
We now have to inverse the interest rates: We have a Yendenominated option, therefore,
the dollar interest rate becomes the foreign interest rate. With these changes, and equipped with
an exchange rate of Y120/$ and a strike of Y120, we can proceed with our standard binomial
procedure. delta
B
Call premium node uu node ud = du
0.9835
0.1585
119.6007 17.4839
9.3756
1.0391 node dd
0
0
0 Using these call premia at all previous nodes yields: delta
B
Call premium t=0
0.3283
36.6885
2.7116 node d
0.0802
8.4614
0.5029 node u
0.5733
66.8456
5.0702 The price of the European Yendenominated call option is $2.7116. 69 Part 3 Options ab) For the American call option, the binomial approach yields:
node uu node ud = du
delta
0.9835
0.1585
B
119.6007 17.4839
Call premium
9.3756
1.0391
Value of early exercise 11.1439
0 node dd
0
0
0
0 Using the maximum of the call premium and the value of early exercise at the previous nodes
and at the initial node yields: delta
B
Call premium
value of early exercise b) t=0
0.3899
43.6568
3.1257
0 node d
0.0802
8.4614
0.5029
0 node u
0.6949
81.2441
5.9259
5.4483 For the Yendenominated put option, we have: delta
B
Put premium
value of early exercise node uu node ud = du node dd
0
0.8249
0.9835
0
102.1168
119.6007
0
5.7287
17.2210
0
3.1577
15.8997 We can clearly see that early exercise is never optimal at those stages. We can therefore c...
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 Spring '14
 NguyenThiMaiLan
 Derivatives

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