Estimated probabilities of stock price movements

The tables below use the historical volatility of a stock price to calculate its probabilities of reaching or not reaching target prices by specified dates, and for other probability calculations described below. 

The calculators are grouped together for convenience. Their explanations are farther down the page.

Instructions

1. Enter the Current Price and Annualized Volatility in the first two boxes below. 
2. Click Calculate to calculate all the fields in all the calculators. 
3. The initial calculation assigns default values to some fields. You can manually change any value in an input box and recalculate. All of the Calculate buttons recalculate the entire page. 

• Numeric fields accept the characters .0123456789.
• Date fields accept dates formatted as MM/DD/YYYY using the characters /0123456789.  
• In the tables below, P_Event means the Probability of the event occurring.

Probabilities of being Above and Below a target price in the future

Price envelope a stock could be expected to stay within, 68% probability

Number of days to reach a target price with at least 16% probability

References

• How To Make Money In Stock Options, by Norman Saint-Peter, Prentice-Hall, 1984.

Explanations and commentary

1) Above/Below calculator

The possible prices that a stock will have on a date in the future take the form of a probability distribution commonly known as a normal or bell-shaped curve. Its mean (center, average, and most likely value, the tallest point on the curve) is today's Current Price. Its standard deviation is the Annualized Volatility adjusted to the time period under consideration.

The Above/Below calculator computes areas under the normal curve to the left and to the right of the Target Price. Future prices close to today's Current Price are likely. Future prices far away from today's price are unlikely, but the probability of them occurring does increase as you go farther into the future because there is more time for the stock to fluctuate to those extreme values.

The probabilities that the stock will be Above/Below the Current Price are always 50/50 indefinitely into the future. For a Target Price that is above the Current Price, the probability of the stock rising above that price can approach but can never reach 50%. Likewise for the stock falling below a Target Price that is below the current price.

2) Price Envelope calculator

In a normal distribution (bell-shaped curve), about 68% of values are in the area between 1 standard deviation below the mean (average) and 1 standard deviation above the mean.

The Annualized Volatility is by definition the standard deviation of the normal distribution of the possible stock prices at the end of 1 year. Thus, about 68% of the possible values fall between -1SD and +1SD. This can be restated two different ways (both assuming that the stock behaves consistently, which it might not): 1) at the end of any 1 year, there is a 68% probability that the ending price will be within 1SD of the starting price, and 2) in about 68% of years, the ending price will be within 1SD of the starting price.

The Annualized Volatility, however, can be adjusted so it applies to any length of time other than a year. For example, at the top of this page you'll see the volatility for 1 day. The probability calculations based on the normal curve still apply, but for the new time interval.

The "price envelope" table calculates the -1SD and +1SD stock prices for time intervals ending on the specified dates. It calls them the Lower and Upper Boundary points, but they don't actually confine the stock price! They're just the upper and lower points within which the stock price has a 68% chance of being on the various dates.

Because each day that a stock price fluctuates is one more day that it can wander farther from the starting price, the envelope tends to widen over time, which is why the upper and lower "boundary" points gradually diverge farther and farther away from each other and away from the current price. 

The "magic" numbers in the table of 16, 32, 68, and 84% all arise from the fact that the total probability must be 100%. If 68% of the numbers fall between +/-1SD (a statistical property of the normal curve), then 32% must fall outside that range. Half of them, 16%, are above the +1SD point, and the other 16% are below -1SD. And finally, if 16% of the values are above the +1SD point, then the total amount below +1SD must be 84%.

If you do these calculations using a starting price of $100 and an Annualized Volatility of 40%, you'll find something strange. At the end of 1 year, the Lower Boundary is about $67 (which is $100 - $33), and the Upper Boundary is about $149 (which is $100 + $49). Why are the two values not an equal distance from the starting price? 

This is due to the lognormal calculation of returns, which is different from the way most people think of percentage price changes.

The usual way of thinking about a price change is that 40% of $100 is $40, so the boundaries should be calculated like this:

• $100 + $40 = $140
• $100 - $40 = $60

However, when the logarithmic Volatility is 40% or 0.40, it is not an arithmetic 40%. Instead, it is used as a positive and negative exponent in the following equations:

UpperBoundaryPrice_Ending = Price_Starting * e^+0.40 . Which leads to $100 * e^+0.40 = $149.16

LowerBoundaryPrice_Ending = Price_Starting * e^-0.40 . Which leads to $100 * e^-0.40 = $67.04

3) Number of Days to Reach calculator

This calculation uses the same price envelope as the LowerBoundary and UpperBoundary calculations, but in a different way. It calculates at what point in time the 1 standard deviation point (as it gradually expands away from the current price) first equals the Target Price.

As mentioned in the notes for the "boundary" calculations, the probability that the price will actually have reached or exceeded a particular +/-1 standard deviation price is only 16%. Thus, calling this calculation "number of days to reach" is questionable at best, not just because the probability is low but because the choice to use 1 standard deviation as the definition of "reach" is arbitrary.

The lognormal price distribution affects this calculation the same as it does the others. For example, given a starting price of $100, the number of days to reach $110 is not the same as the number of days to reach $90.

As the price rises above $100, a 1% move becomes worth more than $1. As the price drops below $100, a 1% move becomes worth less than $1. So sequential 1% upward moves cover more ground (arithmetically) than sequential 1% downward moves. Thus, the stock is likely to reach $110 faster and more easily than it is likely to reach $90.

You'll discover that the number of days for the stock to double and reach $200 (a 100-point move) is the same as for it to be cut in half to reach $50 (a 50-point move). That's the nature of logarithmic returns. The Black-Scholes option valuation equation and most other financial equations use logarithmic return calculations rather than the arithmetic ones most of us are familiar with.

Other general notes

1) All of these calculations have validity ONLY if the volatility of the stock price during the future time period under consideration is the same as its historical volatility in the past (or the same as the estimated volatility that you use, if you enter an alternative volatility figure). If the volatility estimate does not accurately predict the future volatility (which it often does NOT), then these calculations, and the Black-Scholes option valuation model itself, fall apart and are completely useless. 

2) The program might not catch or warn about all types of possible errors in the input data. Make sure your inputs match the formats of the examples shown. 

3) Most options traders lose money. If you use these calculations in the course of making option trading decisions, you will probably lose money, not just because the calculations have questionable predictive power (see Note 1), but because options trading itself is typically a losing proposition. There is a high probability that you will lose money trading options no matter what calculations you do or what methods you try, and that is probably the most accurate prediction of the future that you'll find anywhere on this page.

4) Questions, suggestions, comments are welcome in the discussion forum.