Question 1158491: Hi Professor:
I have a question related to the probability, a stock price is currently at $10, what will be the probability the stock price will be above $10 after 5 days? Assume the stock price will be randomly moved according to the following exact percentages: -3%, -2%, -1%, 1%, 2% and 3% (so there are total 6 different possible percentages movement and the movement are purely random among each day).
My approach to this problem is the following: there are total 6 different movements, and the total trials are 5 days, so the total permutations will be 6^5=7776, but I just can't continue the rest due to my limited math knowledge, could you help me out on this?
Thank you for reading my email and I am looking forward to hearing from you soon.
Alex
California
Hi Tutors:
Thank you for answering my questions (#1158324) above (at Answer 781251).
I think the answer (781251) just solved one of the permutation, what if the percentage change were repeatable negative percentages, these percentage changes are all independent, so it could have like: -3% every single day, or -3% for 3 days then another 2 days -2%, etc. I wanted to know what are the probability under all the permutations, what are the chances that the stock price at the end of 5th day will be above $10?
again, thanks
Alex
Answer by greenestamps(13200)   (Show Source):
You can put this solution on YOUR website!
original response deleted... I'm looking at this further....
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In my earlier attempt at solving this, I tried to simplify the problem by treating all the percent increases and decreases as being relative to the original price. That way, for instance, increases of 3% and 3% would exactly balance decreases of 1%, 2%, and 3%.
However, that is probably not how the problem was intended.
If the percentage increases or decreases are treated as multipliers, there is NO permutation of 5 of the 6 possible percentage changes that results in an ending price exactly equal to the starting price.
So I looked at the problem again....
Certainly the other tutor missed the point of the problem, finding the percent increase or decrease if each of the changes is applied once.
It appears to me that a purely analytic solution would be extremely tedious, making it necessary to examine each permutation of 5 of the 6 percentage changes.
So I built an excel spreadsheet with all 6^5=7776 permutations of 5 of the 6 and identified the ones that produced a product greater than 1.
ANSWER: 3588 of the 7776 permutations of 5 of the 6 percentage changes produce a product greater than 1. Therefore, the probability that the stock price will be above the original $10 after 5 days is 3588/7776, or about 46.142%.
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In response to the reader's question, here is how to make the spreadsheet.
The 6 numbers you are using repeatedly are 1.03, 1.02, 1.01, 0.99, 0.98, and 0.97.
There will be 6^5 = 7776 rows in the table.
In column A there will be 6^4=1296 entries of 1.03, followed by 1296 entries of 1.02, followed by... etc.
In the first 6^4=1296 entries in column B, there will be 6^3=216 entries of 1.03, followed by 216 entries of 1.02, followed by ... etc.
Then you can copy those first 1296 entries in column B 5 more times to complete column B.
So you can see building the spreadsheet is not really as big a task as it seems, because you can do a huge amount of copying and pasting.
Similarly, in column C, you have repeated blocks of 36 entries of 1.03, followed by 36 entries of 1.02, ... etc.
And in column D the repeated pattern is blocks of 6 of each of the 6 numbers.
And finally in column E the 6 numbers repeat in a cycle through the whole 7776 entries.
Now the spreadsheet shows each possible permutation of 5 of the 6 possible numbers exactly once.
Now in column F, make each entry the product of the entries in columns A through E of that row. Those numbers, being the product of 5 numbers between 0.97 and 1.03, will all be close to 1. We are of course interested in how many of them are greater than 1.
To count how many of the 7776 entries in this column are greater than 1, there are 2 steps.
(1) in row 1, column G, enter
=IF(F1>1,1,0)
Then copy that formula down through all 7776 rows of column G.
That will put a 1 in column G wherever the value in column F is greater than 1 and a value of 0 wherever that value is less than 1.
Then to count the number of permutations for which the product is greater than 1, simply sum the entries in column G.
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