# Transforming log return volatility into standard return volatility

If I have a forecasted volatility of the log returns of say, 0.03, this is obviously transformed relative to the log I took of the returns. It strikes me that I should raise e to the power of the volatility I forecasted to in order to get back something "normal" looking. When I do this I get ~1.03, which seems really, really high. This would tell me I'm doing something wrong.

What is the right way to transform log return volatility back into non-logged volatility?

Thank you!

It isn't strictly speaking possible to convert a log vol to a normal vol, although it may be possible to get a rough idea. I am assuming you only have the vol of log returns but not the actual time series here. If you had the original time series, then you would just calculate the standard deviation of the prices to get the normal vol. I assume this is coming from the interest rate arena where we look at normal vols for interest rate options, but used to use log normal models.

If you have one time series with a vol of log returns of 0.03, there is more than one possible normal vol with the same vol of log returns. For example, if you double all the prices/rates that form the original time series, the vol of log returns is unchanged, but the normal vol will double. There is clearly no unique conversion formula that always works.

If you know that the the log volatility is much less than 100% (as in your case) and that the time series has more or less stayed around the same level of the time period of interest, then a ballpark approximation would be $$normal vol \approx P_{average} \times logvol$$

Where $P_{average}$ is the average price/rate over the time series of interest. This can be shown by writing down the expression for standard deviation of price, dividing the prices within the sum by the average price, and then multiplying outside the sum by the average price. If the average price is a reasonable approximation to any given price in the time series, then the approximation is ballpark OK. Where the low volatility is needed is where you say that a "percent return" is about the same as a "log return" - this is valid when returns are much less than 100%.

• I got the forecast from a GARCH model of log returns for a stock. I think it's the same as what you described. So I suppose the only thing this forecast would be good for is to forecast direction and not actual numbers right? – user20664 Jun 26 '16 at 20:02
• The log ratio is pretty close to the percent change for moves much less than 100%, so it should be intuitively thought of as a forecast for the percent change over a day, assuming the returns were daily. – RiskyScientist Jun 27 '16 at 8:38

If you have the vol of the log returns, that means you have the normal vol. You can then use that vol in the the lognormal distribution formula to get the volatility of the time series of prices.

$$\left[\exp(\sigma^2) - 1 \right]\exp(2\mu+\sigma^2)$$