Jan

14

There are many uses of the lognormal distribution of price changes in markets, and most of them are untested and non-predictive.

There is the use by experts to try to show that options are not fairly priced. There is the use by doomsdayists to show that catastrophe is much more common than might be predicted. There is the use by ignoramuses who don't know the meaning of the term at all, but use it as a catch word to back up their ideas. They think that it is de rigeur to lose 90% every few years because this is counterbalanced by the fantastic upsides from trend following and capturing the big moves that the large people are always ready to engulf.

There is the use by academics who would somehow adjust for the possibility of negative changes in a normal distribution that is not possible with a lognormal distribution.

There is a use by other academics to create a smoke screen of complexity about such things as in the enclosed article to back up statements like, "this means that an observed path of trailing historical volatility fluctuates around its mean with % difference to the mean of order 1/ square root of n. For example, a graph of trailing 30 day historical volatility on a perfectly lognormal stock price with actual volatility 0.30 will fluctuate around a mean value within a few $ if it's 0.30. Also, more than half the time, it will be within a band approximately given by 0.25 and 0.35." Apparently the variance of the variance is calculated, a central limit theorem is applied and cutoff points related to this approximation upon approximation is pulled out of the hat.

As always in such self aggrandizing presentations of quantitative knowledge ("the professor is a good trader," they say among the academics and "the trader is a Professor," they say on the trading floor after the rout), the hidden agenda is that one will not complete that sentence.

But one posits that the lognormal distribution of prices is not more predictive of the distribution of price changes of the indexes between days than a model based on a normal distribution of % price changes. Furthermore, a normal distribution of yearly % price changes gives as good a fit as a lognormal % distribution.

The test for this would have to be a predictive one, using rolling predictions based on a back window of say the last year for the daily % price changes or the last 20 years for the yearly predictions. But as a first step, random drawings should be taken of % price changes. They should be compared with the actual predictions of the distributions that would be created by a retrospective normal and lognormal model. Measures of the departure from the predictions should be evaluated.

One wonders in the absence of such tests what it will take for the lognormal boys to ever falsify their predictions. How many years will be gone without big % changes before they conclude that fat tails are not anymore likely than would have been predicted by a normal model? How long will they rely on the 1987 move in stock prices to prove that things are highly kurtotic and thus justify their pretensions and livings?


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