# What is the standard deviation for a stock?

One suggestion for when to take profits is when a stock rises 1 standard deviation. But what is the standard deviation for a stock looking forward? Is the mean the mean of the SPY, the mean of this stock, something else? And over what period of time?

If you assume that relative returns are normal (which many financial models do) then you'd take the standard deviation of the historical relative returns (not prices) over some time period.

Since you're looking at returns, and the "expected" (average) return of a stock would be zero, that would be your average.

The longer the time period, the more "accurate" the standard deviation, so it's up to you how far back you want to go. If you use daily prices, then you'd multiply that standard deviation by `SQRT(252)` to convert to an annual standard deviation.

As an example, say you get a daily standard deviation of returns of 0.0126, which would mean an annual standard deviation of 0.2 (`0.0126 * SQRT(252)`) or 20%. So you would sell when the price goes up 20% from the current price.

Since the interpretation of 1 SD is "the probability that the price moves by more than this percentage is about 16%, it's not a terrible suggestion to sell since the probability of that happening is fairly low (meaning you "got lucky") but it's a bit naive.

Another option is to look at the implied volatility of options on the stock. Implied volatility is essentially the market's prediction of future standard deviation of returns, so the interpretation would be the same as the calculation of historical volatility, meaning that the market thinks there's about a 16% change that the price moves more than that in one year.

Unfortunately, implied volatility is not constant across all strikes and maturities, but you could pick something that's relatively liquid (e.g. an option that expires in 3 months) and with a strike close to the current stock price ("at-the-money").