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# Explaining Volatility Measurement

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Stocks move up and down, all day, every day. How far and how fast those moves are defines volatility: the variability of returns. We call the volatility "number" , which represents the degree of volatility, the standard deviation of returns (which is the square root of variability), and the calculation for it is linear (so it is somewhat flawed, but a good "guide") and complicated. Let me instead of using numbers describe this calculation in words in an attempt to make subjective sense of it.

First you calculate the mean of a stock price over a certain period: so we take the closing prices (you can make adjustments for volume, but let's keep it simple so we can get the concept) over a certain period of time. Now to make it easy, view this "mean" as just the current price. Then we take the next price and subtract it from the mean, giving us the move in price between the two. From here we simply do things mathematically to arrive at "an average move" and then normalize it to get a "relative average move".

To normalize the deviation (difference between the mean and the next price) first we square it (this makes the deviations always positive), then we add them all up, then we divide this sum by the number of observations (this gives us an average variance), then we take the square root (because we squared the deviation in the first place) to get the normalized or "standard" deviation. Of course to make it in all percentage terms we really take the log normal returns instead of stock prices, but this shouldn't change your mental picture of the above described.

When the standard deviation or volatility is annualized we get a number like 25%. This says that we can expect (based on linear analysis of past returns) a stock to move 25% up or down from its current price in one year's time. This incorporates 66% of all observations; so in a way, you can say that you would expect the stock to move up or down 25% with a 66% probability.

If you wanted to be 95% sure that the stock would be in a certain range, you would simply double the standard deviation (called two standard deviations) to 50%.

In using the log normal returns of closing prices, we only get a one-dimensional picture of volatility. For example, I mentioned that volatility was picking up a few days ago. To the untrained eye this may have been difficult to see if you only looked at the closing prices. What I was talking about was intra-day volatility: stocks may have wound up the same at the end, but during the day they were more volatile than usual.

One assumption that these calculations make is that all stocks have an even or "normal" distribution. This means that if you graph the number of observations over price, the graph forms a nice curve that is bell shaped. This is not reality, but option pricing assumes it is. This is where experience comes in.

So that is how we measure volatility. When the VIX goes from 17 to 18 it is really saying "option prices are implying that the index will move 18% in one year with a 66% probability".

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No positions in stocks mentioned.

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