According to a simplified model, a certain stock goes either up $1 (U) or down $
ID: 3209512 • Letter: A
Question
According to a simplified model, a certain stock goes either up $1 (U) or down $1 (D) each day, and is equally likely to go up or down each day. Use Pascal's triangle to solve this problem; if you happen to have learned other notice or other methods from another course do not use them. What is the probability that after 9 days, the stock price will have had no net change? What is the probability that 9 days, the net change in the stock price will be exactly a $3 increase? What is the probability that after 9 days, the net change in the stock price will be at most $1? What is the probability that after 9 days, the net change in the stock price will be at least $2? Are the events in parts c and d of this problem mutually exclusive, and how can you tell? Are they complementary, and how can you sell?Explanation / Answer
Day
Possibility
1
1
1
1
1u, 1d
1
2
1
2
2u,1u1d,2d
1
3
3
1
3
3u,2u1d,1u2d,3d
1
4
6
4
1
4
1
5
10
10
5
1
5
1
6
15
20
15
6
1
6
1
7
21
35
35
21
7
1
7
1
8
28
56
70
56
28
8
1
8
1
9
36
84
126
126
84
36
9
1
9
9u,8u1d,7u2d,6u3d,5u4d,
4u5d,3u6d,2u7d,1u8d,9d
End of day1, possible it can either go up or down
End of day2, 2Up or 1Up1Down or 2Down
End of day3, 3U, 2U 1D, 1D 2U, 3D. and the net value for each of these are 3,1,-1,3
Similarly for day 9 9u,8u1d,7u2d,6u3d,5u4d,4u5d,3u6d,2u7d,1u8d,9d and the net value - 9,7,5,2,1,-1,-1,-3,-5,-7,-9
As per the pascal triangle 9 ups possible via 1 combination; 8up and 1 down via 9 combinations, 7u and 2 down via 36 combinations and so on till 0 ups and 9 downs via 1 combination. It leads to the below probability mass functn
Net value
x
9
7
5
3
1
-1
-3
-5
-7
-9
ways
1
9
36
84
126
126
84
36
9
1
512
They are not complementary events as, complementary is possible only for 2 outcomes. Here there are many events possible like P(x >3), P(x <0) and so on.
Day
Possibility
1
1
1
1
1u, 1d
1
2
1
2
2u,1u1d,2d
1
3
3
1
3
3u,2u1d,1u2d,3d
1
4
6
4
1
4
1
5
10
10
5
1
5
1
6
15
20
15
6
1
6
1
7
21
35
35
21
7
1
7
1
8
28
56
70
56
28
8
1
8
1
9
36
84
126
126
84
36
9
1
9
9u,8u1d,7u2d,6u3d,5u4d,
4u5d,3u6d,2u7d,1u8d,9d
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