Volatility—One Measure of a Market

There are lots of ways to view and compute the volatility of a market. One frequently used is called either historical or statistical volatility, typically abbreviated as HV or SV. I prefer the term statistical volatility and that’s what we’ll use throughout the text. This is a simple if mildly tedious calculation.When we want to find the 30-day statistical volatility of an asset, we start by taking the daily change in the price of the asset, expressed as the closing price divided by the previous day’s closing price, for each of the past 30 days.

To satisfy the lognormal nature of the model, we then take the natural logarithm,loge, of each of these 30 price changes, and next compute the standard deviation of the set of these logarithms.The result of this calculation is the immediate 30-day statistical volatility.Volatilities are almost always stated as annual figures, though, so we’ve one final step yet to go. To annualize our calculation, we must multiply our result by the square root of the number of trading days in the year.

Some authors, Larry McMillan among them, use 260 for this figure in every year. Some use 256, whose square root is 16 (very conveniently), some count up the actual number of trading days in the current year. All these work well for our purposes, so just choose your favorite.I know some very masochistic people, but I’ve yet to meet any trader who actually sits down and manually performs these calculations each day (or each week, or ever for that matter) for all the markets that might be of interest at any given time.

SVs are available from commercial quotation services, from many brokerages (be sure to specify the term of the volatility to your broker, 20-day, 60-day, 6-month, or whichever you prefer), and from a considerable and growing body of Internet sites. Some of these sites provide volatility figures at no charge, some have nominal fees, but a few will try to take a hefty bite out of your wallet and are to be avoided.

Traders whom you know personally and who use volatility calculations in their trading will surely have them available.If you’re comfortable with spreadsheet programs, you can easily and cheaply acquire your own database of historical prices, a means of acquiring each day’s closing prices (even at no cost, on sundry Internet sites), and whip up the appropriate spreadsheet. You’ll have all the SVs you want every day, with just a few clicks of the mouse.

There are also several worthwhile commercial software packages, ranging in price from modest to ridiculous, that include historical price databases and volatility calculators, among a large number of other features.Some Fruits of the Volatility Tree:One of the many profitable implications of volatility is the concept that we can calculate the theoretical fair value of an asset’s various options.

Granted,the values we’ll obtain from this calculation are, again, only approximations, and almost assuredly will not match the actual prices of the options that we see in the marketplace. We can easily trade and prosper with these approximations,though, because “the market” is the sum of the actions of people over time, and, in broad, it bears no resemblance whatever to a precision instrument.

In addition, we must remember in the first place that the lognormal curve is only a model of the future distribution of prices, an approximation to be respected and applied as we find useful . . . but not worshipped.Let’s be candid with each other, shall we? The real world is not lognormal.It would be terrifically convenient if it were, but it isn’t. Just because the theoretical prices of a market’s options don’t match the actual prices we observe in the market, does this imply a failure of the model?

It does not,for the actual price of any option at any time represents the various market participants’ collective view of what the price of that option “should” be at that specific point in time. For clarification of this point, we can look to the heating oil market.Everyone who trades heating oil is well aware that heating oil distributors begin building their inventories in the late summer and that consumers, particularly homeowners, tend to buy the bulk of their heating oil in the autumn.

Most years, due to this concentration of demand, heating oil prices tend to rise during one or another portion of the time from July through early October. In anticipation of this, option traders in heating oil will on occasion be bidding rather eagerly for these months’ call options even earlier in the year, and the actual market price of these calls will acquire a premium, sometimes a very large one, over their computed theoretical fair values.In order to use the disparity between an option’s theoretical fair value and its actual market value to our advantage, we first need to measure this disparity.