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Pavel314

Joined: 29 Apr 2005
Posts: 78

Posted: Sun Apr 30, 2006 2:01 pm    Post subject: Cumulative Changes Probability

This problem came up during a discussion of trading on the stock market but
the solution probably has wider application. As a disclaimer, I realize that
the past performance of a stock is no indication of future performance
because of many factors in the economy as a whole, and that I am not looking
for a system to beat the market, just a formula to compute probability.

GIVENS: For a particular stock, we have the closing values at the end of
each trading day for a year. From this, we compute the daily changes and
find they have a normal distribution with a mean of m and a standard
deviation of s.

PROBLEM 1: What is the probability that the stock will have increased in
value at least 10% at the end of the next three weeks, i.e., 15 trading
days? The probability should be given as a function of P, m, s, and 15,
the number of days in the observation period. Where P is the current stock
price and c_i is the change in value for trading day i, I would state the
problem as:

P + c_1 + c_2 + c_3 + ... + c_15 >= 1.1 * P

I've thought about expressing the c_i in terms of confidence intervals but I
get into problems with the summation.

PROBLEM 2: What is the probability that the stock will have increased in
value at least 10% at any time before the end of the next three weeks, i.e.,

P + c_1 >= 1.1 * P OR
P + c_1 + c_2 >= 1.1 * P OR
P + c_1 + c_2 + c_3 >= 1.1 * P OR
...
P + c_1 + c_2 + c_3 + ... + c_15 >= 1.1 * P

The more general statement of the problem would use n days in the
observation period as opposed to the 15 I've used above. Thanks for any
light you can shed on this.

Paul
Pavel314

Joined: 29 Apr 2005
Posts: 78

Posted: Tue May 02, 2006 10:47 am    Post subject: Re: Cumulative Changes Simulation

"Pavel314" <Pavel314@NOSPAM.comcast.net> wrote in message
news:ZLednXWtfoYuXsnZnZ2dnUVZ_sKdnZ2d@comcast.com...
 Quote: This problem came up during a discussion of trading on the stock market but the solution probably has wider application. As a disclaimer, I realize that the past performance of a stock is no indication of future performance because of many factors in the economy as a whole, and that I am not looking for a system to beat the market, just a formula to compute probability. GIVENS: For a particular stock, we have the closing values at the end of each trading day for a year. From this, we compute the daily changes and find they have a normal distribution with a mean of m and a standard deviation of s. PROBLEM 1: What is the probability that the stock will have increased in value at least 10% at the end of the next three weeks, i.e., 15 trading days? The probability should be given as a function of P, m, s, and 15, the number of days in the observation period. Where P is the current stock price and c_i is the change in value for trading day i, I would state the problem as: P + c_1 + c_2 + c_3 + ... + c_15 >= 1.1 * P I've thought about expressing the c_i in terms of confidence intervals but I get into problems with the summation. PROBLEM 2: What is the probability that the stock will have increased in value at least 10% at any time before the end of the next three weeks, i.e., 15 trading days? P + c_1 >= 1.1 * P OR P + c_1 + c_2 >= 1.1 * P OR P + c_1 + c_2 + c_3 >= 1.1 * P OR ... P + c_1 + c_2 + c_3 + ... + c_15 >= 1.1 * P The more general statement of the problem would use n days in the observation period as opposed to the 15 I've used above. Thanks for any light you can shed on this.

Since I couldn't solve this problem theoretically, I wrote a Q-Basic program
to simulate the situation under various scenarios. The first parameter is
the value of one standard deviation of the historical daily change as a
percent of the initial stock price. The second works into the two problem
statements above; the first checks to see if the stock has appreciated 10%
at the end of 15 days while the second checks to see if it appreciated 10%
on any day within the 15-day trading period. I ran each of the six scenarios
1,000,000 times; the results are shown below:

StdDev/Price +10% at End of Period +10% Within Period
1% 1,238
1,576
2% 66,772 97,892
4% 238,162 379,117

The moral seems to be that you should take your profit while you can because
there's a significant chance that the stock will go back down.

Paul
Michael Zedeler
science forum beginner

Joined: 29 Nov 2005
Posts: 17

Posted: Sat May 06, 2006 7:23 pm    Post subject: Re: Cumulative Changes Probability

Pavel314 wrote:
 Quote: This problem came up during a discussion of trading on the stock market but the solution probably has wider application. As a disclaimer, I realize that the past performance of a stock is no indication of future performance because of many factors in the economy as a whole, and that I am not looking for a system to beat the market, just a formula to compute probability. GIVENS: For a particular stock, we have the closing values at the end of each trading day for a year. From this, we compute the daily changes and find they have a normal distribution with a mean of m and a standard deviation of s.[...] The more general statement of the problem would use n days in the observation period as opposed to the 15 I've used above. Thanks for any light you can shed on this.

Here is a starting point:

http://en.wikipedia.org/wiki/Black-scholes

Regards,

Michael.
--
Which is more dangerous? TV guided missiles or TV guided families?
Get my vcard at http://michael.zedeler.dk/vcard.vcf
Pavel314

Joined: 29 Apr 2005
Posts: 78

Posted: Sun May 07, 2006 1:17 pm    Post subject: Re: Cumulative Changes Probability

"Michael Zedeler" <michael@zedeler.dk> wrote in message
news:oJ67g.162\$uS7.137@news.get2net.dk...
 Quote: Pavel314 wrote: This problem came up during a discussion of trading on the stock market but the solution probably has wider application. As a disclaimer, I realize that the past performance of a stock is no indication of future performance because of many factors in the economy as a whole, and that I am not looking for a system to beat the market, just a formula to compute probability. GIVENS: For a particular stock, we have the closing values at the end of each trading day for a year. From this, we compute the daily changes and find they have a normal distribution with a mean of m and a standard deviation of s.[...] The more general statement of the problem would use n days in the observation period as opposed to the 15 I've used above. Thanks for any light you can shed on this. Here is a starting point: http://en.wikipedia.org/wiki/Black-scholes Regards, Michael.

Thank you, that's exactly what I was looking for.

Paul

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