Assuming a Stock's price changes in a random manner. If you buy this Stock, you are required to set a stop-return and stop-loss price. I am looking for the equation that shows the probability for each day since you buy the Stock (Buy Date), of crossing the stop-return and stop-loss price. In other words; I want the formula that allows me to chart a line showing the probability the Stock price crossing any stop-loss (on way down) or stop-return on way up.
The stop-loss price can be any price below the Buy price. The stop-return price can be any price higher than the Buy price. Known Information: 1. The Buy Price. 2. The Standard Deviation (or Historical Volatility) of the Price on the date you bought the Stock.
For example, on the x-axis of the chart it displays each day starting with the Buy Date. There are 2 lines. One line shows the probability of the random price crosses the stop-loss price going down for each day. The other line shows the probability the random price crosses the stop-return price going up on the same day.
My suspicion is that the solution is some log-normal density function. All help is appreciated.
Hint: You are probably expected to assume an independent daily normal distribution of price changes (in percent or in absolute dollars?). The variance on one day is the square of the standard deviation and variances add. As a function of the day number $n$, the number of standard deviations of movement decreases as $\frac 1{\sqrt n}$. For each span of days, figure out how many s.d. you have to move and look it up in your normal probability table.