# Calculating probability of landing on an area of a distribution from any other area

I wish to calculate the probability of landing anywhere under a distribution curve, after already moving to a different "area" under the curve.

For example, suppose I have a large dataset of how much a stock, X, moves from its open price each day, in dollars, and come up with this:

In dollar amounts, how much price closes upwards/downwards relative to its daily open price:

1+               50
0.75 to <1       100
0.5 to <0.75     200
0.25 to <0.5     500
0 to <0.25       800
-0.25 to <0      800
-0.5 to <-0.25:  500
-0.75 to <-0.5:  200
-1 to <-0.75:    100
-1-:             50



So this distribution is of course generated from the "open" price, and its likelihood to end up in any of the areas, such as 0.25 to <0.5 is 500 of the total # of days analyzed.

However, how can I be able to analyze the movement after it already "deviates" from the open? Say, let's say price right now is at the 0.25 to <0.5 area, and the day has not closed yet.

How can I calculate the probabilities of a move to 0.5 to <0.75, or a move downwards instead, while I am currently at the 0.25 to <0.5 level, for example?

• I think that what you're asking about is the conditional probability of closing at some bracket given that some other bracket has been reached at some point in the day; do you have this data in your data set, or only open/close prices? Oct 4 '20 at 11:32
• @ItamarMushkin That sounds quite like what I'm trying to figure out how to do. I have the dataset of how much the close price deviates from the open price, which would give me a distribution seen close to what I've tried to emulate above. My struggle is, the % of time to close within each bracket is of course calculated from the open price. The probability is of course not the same, when price already moves, to say, the 0.5 to <0.75 bracket mid-day. Therefore I'd love to be able to calculate the "updated" probability of its likelihood to close within any higher/lower bracket given this move. Oct 4 '20 at 19:30

Let $$t$$ be the fraction of today's expected trades that haven't happened yet, calculated from the charts of historical trade volumes at different times of day.
Then $$\sqrt{t}$$ times the standard deviation for one day's price move gives the standard deviation for the remainder of today, according to the standard assumptions of Brownian motion or geometric Brownian motion.
So one simple reasonable way to convert the above table for 1 day into a table for $$t$$ days is to keep the right column fixed, and multiply the ranges in the left column of the table by $$\sqrt{t}$$.