Feb 21

Price levels versus Price returns

by David Harper, CFA, FRM, CIPM

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Yesterday a reader asked a question about a chart in an article I wrote a long time ago about the limits of volatility. The chart (shown below) plots two stock price paths: a red path produces a volatility of 10% and a green path that produces a volatility of 0%.

The idea was to illustrate a weakness of the traditional volatility metric: it doesn't care about the direction of volatility. Price upticks count the same as downticks. But the reader's question was:

"The volatility of the green line is said to be 0. But the standard deviation of this line is not 0. What is the definition of volatility you are using to make this statement?"

It's a good question because it highlights the difference between stock price levels and stock price returns. The reader is correct: in the green line above, the standard deviation of stock prices is nonzero. But the standard deviation of returns is still zero. In fact, all of the price paths below produce the same volatility, zero:

In terms of price levels, only the blue flat line (constant price at $100) has a zero standard deviation. But volatility is not dispersion of prices, volatility is the dispersion of the price returns. Specifically, volatility is the (instantaneous) standard deviation of logarithmic returns. Given today's stock price S(i) and yesterday's stock price S(i-1), the periodic return is the natural log (ln) of the ratio of prices:

This periodic return is the continuously compounded return. Volatility is the standard deviation of a series of these returns. So if the ratio of returns is constant, the volatility is zero. In the green series above, the stock price levels are the following: 100.00, 110.52, 122.14, 134.99, 149.18, 164.87, 182.21, 201.38, 222.55, 245.96. Each price is 20% (continuously compounded) greater than the previous price. The returns are constant at 20%, so they exhibit no dispersion. Same with the other series.

If you would like to take a closer look at volatility, I prepared an 18 minute screencast here. We walk step-by-step through the calculation of historical volatility (i.e., annualized standard deviation of logarithmic returns). Plus, you can download the spreadsheet.

In reality, we would never observe such series. Arbitrage would immediately interject volatility into the above- and below-normal return series. For example, buyers would start purchasing the stock plotted in green (it's a guaranteed +20%), so the price would jump up. Conversely, the stock that plots a certain negative (-) 20% series would immediately drop due to selling.

Comments

  1. Jose Mendes 17 Oct 2007

    Hi,

    Aren’t the stock prices exhibited showing a 10% (continuosly compounded) instead?

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