**A StockOpter White Paper**

This document discusses what stock price volatilities are and how they are calculated. The volatility of the stock price is an input into StockOpter.com’s calculations of option values (using the Black-Scholes option-pricing model), value-at-risk (VaR), and probabilities associated with various stock prices. There are additional white papers on the Black-Scholes option-pricing model and VaR calculations at blog.stockopter.com. A discussion of probability calculations can also be found there at the end of the white paper on the lognormal random walk model for stock prices.

**Volatility Defined**

The volatility of a stock price is a measure of its variability. It is defined as the annualized standard deviation of the change in “ln” of the stock price (ln is the natural, or base *e*, logarithm of a number). For why change in ln is an appropriate measure, see the discussion of the lognormal random walk model for stock prices (Part I) here.

When the change in ln is small, the change in ln of the stock price is approximately the same as the percent change in the stock price. It is probably due to this that volatilities are normally expressed as percents. For example, if the annualized standard deviation of the change in ln of the price of a stock is 0.3, by convention we say that the stock’s volatility is 30%.

All of the ways in which StockOpter.com uses volatility are forward-looking. However, the future volatility of a stock is likely to resemble its past volatility, especially if one compares the near future to the recent past. Historical volatilities are calculated from historical prices. They are normally calculated from day-to-day price changes, though they are expressed on an annualized basis.

There are tradeoffs involved in how far back to look. The further back one looks in time, the more data one has to work with, and the less one has to worry about uncertainty in the estimate of the volatility due to small sample size. On the other hand, the world changes, and the volatility of the price of a stock several years ago may not be a very good guide to the volatility of the price of that stock today. Generally, six months of data seems to be a good compromise.

One might want to use somewhat less historical data if one wanted to avoid using data from some past period that there is reason to believe is not representative of what to expect in the future. One would want to distinguish here between the near future and the more distant future. One might not expect a past period of high variability to be repeated in the next month, for example, yet consider it highly likely that some such periods would be seen over a long time horizon.

The volatility of the stock price is an input in calculating the value of an option using the Black-Scholes model. An implied volatility is calculated by turning this on its head; one starts with the value of the option as an input, and solves the Black-Scholes model for the volatility. It turns out that one cannot actually solve the equation; an iterative trial-and-error procedure is used to determine the volatility that produces the observed price.

An option trader must have a forward-looking view of volatility, based only in part on the volatility seen in the recent past. Option traders use their estimates of future volatilities to price options. One can, in turn, calculate the implied volatility of an option to see what the trader pricing the option thinks the volatility of the stock price will be over the tenor of the option. Traders do this too; they want to know what other traders are thinking.

All of the purposes for which StockOpter.com uses volatility would seem to be best served by implied rather than historical volatility, due to the forward-looking nature of implied volatility. However, there are several potential problems with using implied volatilities. These include:

- Liquidity can be a problem. An estimate of future volatility based on only a few traded options can hardly be called a market consensus. Options will typically have greater liquidity for shorter tenors than for longer tenors.
- Employee stock options typically have much longer tenors than the tenors of any regularly traded options.
- The prices of options that are very far in the money or out of the money are not very sensitive to the volatility input, so implied volatilities calculated from the prices of such options are not very reliable.
- One must be careful to exclude profit from the price for an option used to calculate an implied volatility. One can approximate this by averaging bid and ask prices.

**Differing Time Intervals**

One problem in using historical data is what time interval to use. Although we ultimately want the standard deviation of year-to-year changes, we don’t want to use annual prices as inputs, due to the very small number of observed prices that we would be using. Historical volatilities are normally calculated from daily prices, collected at a consistent time of day (generally at closing). One calculates the ln of each price, then calculates the difference in ln of each price, starting with the second price in the sample, from that of the day before, and then calculates the standard deviation of these differences.

However, the result is the standard deviation of daily changes in price, and what we want is the standard deviation of annual changes in price. We can convert by making use of the fact that the standard deviation is proportional to the square root of the time interval. For an explanation of why this is the case, see the discussion of the lognormal random walk model for stock prices (Part II) here.

Thus, we can calculate the volatility by dividing the standard deviation of daily changes in ln by the square root of a one-day interval expressed in years. However, we need to be careful about the “expressed in years” part. The vast majority of change in price takes place during trading hours. As a result, we should use the fraction that one trading day is of one trading year. A common convention is that there are 252 trading days in a year. Thus, we calculate the volatility by dividing the standard deviation of daily changes in ln by the square root of 1/252, or, what is mathematically equivalent but easier, multiplying it by the square root of 252.