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Unformatted text preview: 0.000 1.819 0.909
3.169 0.000
0.000 0.000
1.804 79 0.000 0.000
0.000 Part 3 Options 7.713 6.711
11.194 5.414
10.263 15.742 3.582
9.006 15.146
21.342 0.000
7.110 14.420
21.373 27.745 14.114
21.762 28.506
34.444 29.575
35.580 40.857 42.080
46.757
52.196 American Put 0.000
0.000
0.000
0.000
0.000
0.458
4.353 7.979 0.909
3.201
5.477
10.548 16.468 0.000
0.000
0.000
3.582 9.131
15.651 22.509 0.000 0.000
1.804 6.870
11.637 0.000
0.000 1.835 0.000
0.000 0.000
7.110 14.668
22.254 29.575 14.114
22.254 29.575
36.159 29.575
36.159 42.080 42.080
47.406
52.196 Question 11.18. We chose the stock of IBM from June 1st 1997 to May 31st 2002 (http://finance.yahoo.com).
Please note that we must calculate the continuously compounded returns before we can calculate
the weekly standard deviation. (c.f. table 11.1 of the textbook). We obtain the annual standard
deviation by multiplying our weekly estimate by the square root of 52. 80 Chapter 11 Binomial Option Pricing: II In particular, this yields:
period
97/6 – 02/05 weekly annual
0.050 0.360 97/6 – 98/5
98/6 – 99/5
99/6 – 00/5
00/6 – 01/5
01/6 – 02/5 0.040
0.049
0.052
0.063
0.040 0.287
0.355
0.376
0.457
0.287 97/6 – 97/11
97/12 – 98/5
98/6 – 98/11
98/12 – 99/5
99/6 – 99/11
99/12 – 00/5
00/6 – 00/11
00/12 – 01/5
01/6 – 01/11
01/12 – 02/5 0.039
0.041
0.043
0.056
0.055
0.050
0.060
0.067
0.037
0.041 0.279
0.298
0.311
0.401
0.395
0.364
0.432
0.486
0.268
0.297 We can observe a time trend in the volatility estimate: Volatility is rising throughout the nineties,
up to a record level of over 45 percent in 2000/2001. Only the last year of data shows a reversal
in the volatility.
There does not seem to be an additional pattern to be detected when we conduct the semiannual
volatility estimates. 81 Part 3 Options Question 11.20. We will use the methodology introduced by Hull, which is described in the main textbook.
We can calculate:
u = 1.2005
d = 0.8670
p = 0.4594
n = 4.0000 S = 50.0000
F = 46.0792
t = 1.0000
h = 0.2500 American Call 95.7188
50.7188 79.7304
35.6215 69.1230 66.4127
23.1772 24.1230 59.3195 57.5771 14.3195 13.4682 50.0000 47.9597 8.4551 7.2380 49.9170
4.9170 43.3480 41.5791 3.7876 2.2141 34.6340 36.0474 0.9970 0.0000 30.0263
0.0000 26.0315
0.0000 82 K=
dividend =
r=
sigma =
time to div 45.00
4.0000
0.0800
0.3255
0.2500 Chapter 11 Binomial Option Pricing: II European
Call 95.7188
50.7188 79.7304
35.6215 66.4127 69.1230 23.1772 24.1230 59.3195 57.5771 14.2721 13.4682 50.0000 47.9597 49.9170 8.4338 7.2380 4.9170 43.3480 41.5791 3.7876 2.2141 34.6340 36.0474 0.9970 0.0000 30.0263
0.0000 26.0315
0.0000 83 Chapter 12
The BlackScholes Formula
Question 12.2
N
8
9
10
11
12 Call
3.464
3.361
3.454
3.348
3.446 Put
1.718
1.642
1.711
1.629
1.705 The observed values are slowly converging towards the BlackScholes values of the example.
Please note that the binomial solution oscillates as it approaches the BlackScholes value.
Question 12.4
a)
T
1
2
5
10
50
100
500 Call Price
18.6705
18.1410
15.1037
10.1571
0.2938
0.0034
0.0000 The benefit to holding the call option is that we do not have to pay the strike price and that we
continue to earn interest on the strike. On the other hand, the owner of the call option foregoes
the dividend payments he could receive if he owned the stock. As the interest rate is zero and the
dividend yield is positive, the cost of holding the call outweighs the benefits.
b)
T
1
2
5
10
50
100
500 Call Price
18.7281
18.2284
15.2313
10.2878
0.3045
0.0036
0.0000 84 Chapter 12 The BlackScholes Formula Although the call option is worth marginally more when we introduce the interest rate of 0.001,
it is still not enough to outweigh the cost of not receiving the huge dividend yield.
Question 12.6
a) Using the BlackScholes formula, we find a callprice of $ 16.33. b) We determine the one year forward price to be:
F0,T(S) = S * exp(r*T) = $ 100 * exp(0.06*1) = $ 106.1837 c)
As the textbook suggests, we need to set the dividend yield equal to the riskfree rate
when using the BlackScholes formula. Thus:
C(106.1837, 105, 0.4, 0.06, 1, 0.06) = $ 16.33
This exercise shows the general result that a European futures option has the same value as the
European stock option provided the futures contract has the same expiration as the stock option.
Question 12.8
a)
We have to be careful here: Now we have to take into account the dividend yield when
calculating the 9month forward price:
F0,T(S) = S * exp((rdelta)*T) = $ 100 * exp((0.080.03)*0.75) = $ 103.8212.
b)
Having found the correct forward price, we can use equation (12.7) to price the call
option on the futures contract: C(103.8212, 95, 0.3, 0.08, 0.75, 0.08) = $ 14.3863
c)
The price we found in part b) and the prices of the previous question are identical. 12.7a,
12.7b and 12.8b are all based on the same BlackScholes formula, only the way in which we
input the variables differs.
Question 12.10
Time decay is measured by the greek letter theta. We will show in the following that the
statement of the ex...
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This document was uploaded on 03/11/2014 for the course FIN 402 at FPT University.
 Spring '14
 NguyenThiMaiLan
 Derivatives

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