If you plot a histogram of price variations, you see it is quite well fit by a log-normal distribution for about 99% of the daily price variations, and it is something like 1% of the daily variations that are much larger than the prediction says they should be. Since log-normal fits quite well for 99% of the variations, this pretty much means that anything other than a log-normal will fit way less of the data than does a log-normal. That’s why they don’t use a different distribution.
The 1% of price variations that are too large are essentially what are called “black swans.” The point of Taleb’s talking about black swans is to point out that this is where all the action is, this is where the information and the uncertainty are. On 99 out of 100 days you can treat a stock price as if it is log-normally distributed, and be totally safe. You can come up with strategies for harvesting small gains from this knowledge and walk along picking small coins up from trading imperfections and do well. (The small coin usually cited is the American $0.05 coin called a nickel.)
But Taleb pointed out that the math makes picking up these nickels look like a good idea because it neglects the presence of these high variation outliers. You can walk along for 100 days picking up nickels and have maybe $5.00 made, and then on the 101st day the price varies way up or way down and you lose more than $5.00 in a single day! Taleb describes that as walking along in front of steam rollers picking up nickels. Not nearly as good a business as picking up nickels in a safe environment.
Sorry can’t give you a reference. I wrote code a few years ago to look at this effect. I found that code and here is one figure I plotted. This is based on real stock price data for QCOM stock price 1999 through 2005. In this figure, I am looking at stock prices about 36 days apart.
Stock price volatility from data. The random variable is log(P2/P1), the log of the later price to the earlier price. In this plot, P2 occurs 0.1 years after P1. A histogram is plotted with a logarithmic axis for the histogram count. A gaussian (bell curve) is fitted to the histogram, with a logarithmic y-axis, a gaussian is just a parabola. You can see the fit is great, except for some outliers on the positive side. Some of these outliers are quite a lot higher than the fitted gaussian, these are events that occur MANY TIMES more often that a log-normal distribution would suggest.
If you plot a histogram of price variations, you see it is quite well fit by a log-normal distribution for about 99% of the daily price variations, and it is something like 1% of the daily variations that are much larger than the prediction says they should be. Since log-normal fits quite well for 99% of the variations, this pretty much means that anything other than a log-normal will fit way less of the data than does a log-normal. That’s why they don’t use a different distribution.
The 1% of price variations that are too large are essentially what are called “black swans.” The point of Taleb’s talking about black swans is to point out that this is where all the action is, this is where the information and the uncertainty are. On 99 out of 100 days you can treat a stock price as if it is log-normally distributed, and be totally safe. You can come up with strategies for harvesting small gains from this knowledge and walk along picking small coins up from trading imperfections and do well. (The small coin usually cited is the American $0.05 coin called a nickel.)
But Taleb pointed out that the math makes picking up these nickels look like a good idea because it neglects the presence of these high variation outliers. You can walk along for 100 days picking up nickels and have maybe $5.00 made, and then on the 101st day the price varies way up or way down and you lose more than $5.00 in a single day! Taleb describes that as walking along in front of steam rollers picking up nickels. Not nearly as good a business as picking up nickels in a safe environment.
Interesting, and somewhat in line with my impressions—but do you have a short reference for this?
Sorry can’t give you a reference. I wrote code a few years ago to look at this effect. I found that code and here is one figure I plotted. This is based on real stock price data for QCOM stock price 1999 through 2005. In this figure, I am looking at stock prices about 36 days apart.
Thanks, that’s very useful!
Stuart, since you asked I spent a little bit of time to write up what I had found and include a bunch more figures. If you are interested, they can be found here: http://kazart.blogspot.com/2014/12/stock-price-volatility-log-normal-or.html
Cheers!