*This article was originally published in The Option Strategist Newsletter Volume 8, No. 7 on April 8, 1999. *

Statistics are used to estimate stock price movement (and futures and indices as well) in many areas of financial analysis. For example, we have written extensively about the use of probabilities to aid us in choosing viable option strategies. Stock mutual fund managers often use volatility estimates to help them determine how risky their portfolio is. The uses are myriad. Unfortunately, almost all of these applications are wrong! Okay, maybe wrong is too strong of a word, but almost all estimates of stock price movement are overly conservative. This can be very dangerous if one is using such estimates for the purposes of, say, writing naked options or engaging in some other such strategy in which stock price movement is undesirable.

The problem lies in using the normal or lognormal distribution to predict stock price movements. Such an assumption does not allow for the occasional, wild days that many stocks, some indices, and the relatively rare futures contract undergoes. The normal distribution pretty much says that a stock can’t rise or fall by more than 3 standard deviations (don’t worry about what a “standard deviation” is, if you don’t know). In fact, according to math, the probability of something that behaves according to the normal distribution (the “classic” bell curve is a normal distribution) moving three standard deviations is 0.0013 (or just a little more than one tenth of one percent). So, if there are 2500 optionable stocks, say, then we would expect maybe 3 of them to move three standard deviations on any given day.

In reality, it turns out that not only do a much greater number than that move that far, but some of them move 7, 8 or more standard deviations. The normal distribution’s probability of something being able to move 8 standard deviations is : 0.000000000000000629

This is so small that one would expect to see only a handful of such occurrences in a century, if prices were truly distributed via the normal distribution. If the normal distribution were the correct format for stock prices, such a small probability would indicate that one would not see an 8-standard deviation move until millions of trading days had passed. However, I can point out several on nearly any trading day, and I don’t have to use some low-priced, oddball stock that dropped from 1 to 3/4 or some such nonsense as an example. These moves occurred this past Monday, April 5th:

Stock Last Sale Change Std Devs. Aspect Devt (ASDV) 8 –14-3/8 –31.2 Axent (ANT) 8 –12 –11.2 Ameritrade (AMTD) 91-5/8 +29-1/8 +8.6 CheckPoint (CHKP) 28-3/4 –10-3/4 –8.4 Sabre Gp. (TSG) 55 +8-1/2 +8.0

And these are just a few. In the above list, all three that fell in price were on earnings warnings, and the two that rose were just swept up in the latest version of the internet mania. All in all, 58 stocks had moves of greater than four standard deviations on Monday! Monday was not any sort of special day, although it did fall during earnings warnings season and in the midst of the internet mania.

So, just to make sure we didn’t use a bad sample, we looked at the implied of the “market” and found that – since $VIX has been trading – the lowest market volatility readings were in January and July, 1993. The lowest single day was July 25th, 1993. On that day, twelve stocks had moves of more than four standard deviations. They included some “big” names like Adaptec (ADPT), Bethlehem Steel (BS), U.S. Steel (X), Chiquita Brands (CQB), and Novell (NOVL).

Earlier, I said not to worry about what a “standard deviation” is if you don’t already know, but you do need to know one additional fact – the only way we can tell how many standard deviations a stock has moved is to use its historical volatility – say, the 20-day historic volatility, for example – in the measurement. Thus, a 4-point move for a non-volatile stock like Bethlehem Steel in 1993 was a large move in terms of standard deviations, but it pales in comparison to Ameritrade’s gains last Monday.

Just for fun, we went back to last October 8th – the day the market bottomed (and it was a very volatile day) to see how many stocks had moves of four standard deviations or more. There were 33 – not as many as last Monday – but that seemingly low number reflects the fact that many stocks’ 20-day historical volatilities were already well inflated by October 8th, 1998. On that day, the Utility Index ($UTY) fell over 14 points, which was about 5.5 standard deviations. American Power Conversion (APCC) was up over six points that day, to 36-7/8 – a gain of over five standard deviations.

So, we can say with a great deal of certainty that stocks do not conform to the normal distribution. Actually, the normal distribution is a decent approximation of stock price movement most of the time, but it’s these “outlying” results that can hurt anyone using it as a basis for a nonvolatility strategy.

Scientists working on chaos theory have been trying to get a better “handle” on this. If you’re interested, there was a good article (not too technical) in Scientific American magazine (“A Fractal Walk Down Wall Street”, February, 1999 issue). The article met some criticism from followers of Elliot Wave theory, in that they claim the article’s author is purporting to have “invented” things that R. N. Elliott discovered years ago. I don’t know about that, but I do know that the article addresses these same points in more detail.

So what does this mean for our option strategies? On the surface, it means that if we use the normal (or lognormal) distribution for estimating the probability of a strategy’s success, we may get a big move in the stock that we didn’t originally view as possible. If we’re long straddles, that’s great. (In fact, last Wednesday, we had a double “winner” in this regard in that ROSI was down 4.4 standard deviations while ESRX was up a similar amount – we were long straddles in both).

However, if we’re short naked options, then there could be a nasty surprise in store. That’s one reason why we rarely sell naked options on stocks – they can make moves of this sort too often. At least with indices, such moves are far less frequent (although the Dow drop of over 550 points in October, 1997, was a move of seven standard deviations, and the crash of ‘87 was about a 16 standard deviation move).

There are two approaches that one can take. One would be to invent another method for estimating stock price distributions. Suffice it to say that that is not an easy task, or someone would have made it well-known already. There have been many attempts – including some in which a large history of stock price movements is observed and then a distribution is fitted to them. The problem with accounting for these occasional large price moves is that it is perhaps an even more grievous error to over-estimate the probabilities of such moves.

The second approach is to continue to use the normal distribution, because it’s fast and accessible in a lot of places. Then, either rely on option buying strategies (straddles, e.g.) – knowing that you have a chance at better results than the “statistics” might indicate -- or adjust your calculations mentally for these large movements if you are using option selling strategies.

These extreme movements should be figured into the pricing of an option, but they really are not – at least not by most models. The Black-Scholes model, for example, uses a lognormal distribution. Personally, I feel that the Black- Scholes model is an excellent tool for analyzing options and option strategies, but one must understand that it may not be affording enough probability to large moves by the underlying.

Does this mean that most options are underpriced, since traders and market makers are using the Black- Scholes model (or similar model) to price them? Without getting too technical, the answer is that “yes, some options – particularly out-of-the-money options – are probably underpriced”. However, one must understand that it is still a relatively rare occurrence to experience one of these big moves – it’s just not as rare as the lognormal distribution would indicate. So, an out-of-the-money option might be slightly underpriced, but probably not enough to make any real difference in the long run.

In fact, futures options in grains, gold, oil, and other markets which have experienced large, sudden rallies in the past display a distinct volatility skew. That is, outof- the-money call options trade at significantly higher implied volatilities than do at-the-money options. Ironically, there is far less chance of one of these hyperstandard- deviation moves occurring in commodities than there is in stocks! – at least if history is a guide. So, the fact that some out-of-the-money futures options are expensive is probably an incorrect, over-adjustment for the possibility of large moves.

In summary, this discussion should show you that probability analysis is an inexact science because markets behave in ways that are very difficult to describe mathematically. However, probability analysis is also necessary for the option strategist – otherwise, he would be “in the dark” as to the likelihood of profitable outcomes for his strategy. As we showed in the last issue of The Option Strategist, our probability estimates for straddle buying have been pretty much in line with what actually happened in the marketplace (87% of our straddle recommendations have hit their breakeven points prior to expiration). Those results, of course, include some situations in which stocks moved much farther than we could have hoped for, and some in which the stock was more stagnant than we thought1, but overall – in a diversified set of positions, the option strategist can use the normal or lognormal distribution with the proviso that he understands it is not “gospel”.

*This article was originally published in The Option Strategist Newsletter Volume 8, No. 7 on April 8, 1999. *

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