### Fake data, part 2: The black swan distribution

*May 2017: This is from my other blog that's no longer online. The original comments are no longer available, but you are welcome to add more.*In the previous part when I wrote about the logistic distribution, I observed that it fit the S&P 500 daily returns pretty well out to 5 standard deviations (much better than the normal distribution) but broke down in the tails beyond.That has been part of a weeks-long quest to find a distribution that fits, with the criterion that it must be closed-form; that is, we shouldn't need numerical methods to solve for the cumulative distribution or its inverse. Why do this? Because I want to generate an endless stream of artificial market data for analysis. Not only do I want my artificial market to exhibit behavior that is statistically similar a real-world market, but I also want closed-form expressions for ease of programming into Excel or whatever I'll use for analysis. The normal distribution fails on both counts.A normal distribut…