About Statistics Solutions 2022

Statistics
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Derivatives on Options
1) Given :
S = $ 20
R = 2 % = 0.02
T = 9 months = 9 / 12 years
U = 1.4 , d = 0.6
A) (i) h* = Cu – Cd / ( Su – Sd )
20 x 1.4 = $ 28
20 x 0.6 = $ 12
Cu = Max(28-20,0) = $8
Cd = Max(12-20,0) = $0
H* = 8-0/20*(1.4-0.6) = 8/16 = 0.5
(ii) Amount = $8
B = 8 – h*Su = -6
(iii) C0 = h*S + B e-rT
= 20*0.5 – 6 x e-0.02x9/12 = $ 4.09
(iv) ∏ = erT – d / u – d = e0.02x9/12 – 0.6 / 1.4 – 0.6 = 0.519
C0 = ( ∏ Cu + ( 1 - ∏ ) Cd) e-rT
= ( 0.519 x 8 + ( 1 – 0.519 ) x 0 ) e-0.02x9/12 = $ 4.09
It can be observed that the values obtained are similar in both the cases.
(B) Put Option
Pu = Max ( 20 – 28 , 0 ) = $ 0
Pd = Max ( 20 – 12 , 0 ) = $ 8

(i) H * = 0 – 8 / 28 – 12 = - 0.5
(ii) B = - 0.5 x ( -1) x 12 = 6
(iii) P0 = 20 x 0.5 + 6 x e -0.015
= 10 + 6 x 0.985 = 15.91 $
(iv) ∏ = 0.519
(v) P0 = ( ∏ Pu + ( 1 - ∏ ) Pd ) e –rT = ( 1 – 0.519 ) x 8 x e -0.015
= 3.79 $
(C) Here, P0 + C0 = 15.91 + 4.09 = 20 = S
Hence, this relation is true.
(2) Given :
S = $ 75
T = 1.5 years
R = 2 % = 0.02
A) Price = 17.17 $ if dividend is 0 %
B) Price = 71.7 $ if dividend is 3 %
C) Price = 0.37 x 75 = 27.75 $
D) Price = 15.15 $
E) Price = 0.97 x 75 = 72.75 $
F) Yes. If the income is paid at discrete points, then it will lead to a rise in prices.
3)
a) Traditional Methodology
u = 1.22 , d = 0.82 , ∏ = 0.58
0 0.25 0.5 0.75 1
623
394 165
249.64 105 166
158 44
100 100 100 100
42 44
17.64 27.87 11.7
7.4 11.7
3.16

b) Alternative methodology
u = 1.23 , d = 0.826 , ∏ = 0.5
0 0.25 0.5 0.75 1
186 229
151.29 154
123 125 153.75
100 100 103
82.6 83 102
68.2 68
56 68.8
46
c) The values of upper part are higher and lower part are lower using traditional method.
4)
A)
0
50
100
150
200
250
300
350
Series1
B) and C)
i)
Mean
47.251120
57
Standard Error
2.3226109
13
Median
42.537889
9
Mode #N/A
Standard Deviation 23.226109