levy five or more standard deviations ("5-sigma events")

The Black–Scholes model of option pricing is based on a normal distribution. If the distribution is actually a fat-tailed one, then the model will under-price options that are far out of the money, since a 5- or 7-sigma event is much more likely than the normal distribution would predict.^[5]

Fat-tailed distribution - Wikipedia, the free encyclopedia

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Levy flight from a Cauchy Distribution compared to Brownian Motion (below). ... by five or more standard deviations ("5-sigma events") have lower probability, ...

Lévy distribution - Wikipedia, the free encyclopedia

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For the more general family of Lévy alpha-stable distributions, of which this ... Like all stable distributions, the Levy distribution has a standard form f(x;0,1) which ...

Volatility and the Square Root of Time

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Friday, April 17th, 2015 | Vance Harwood

It’s not obvious (at least to me) that volatility theoretically scales with the square root of time (sqrt[t]).  For example if the market’s daily volatility is 0.5%, then the volatility for two days should be the square root of 2 times the daily volatility (0.5% * 1.414 = 0.707%), or for a 5 day stretch 0.5% * sqrt(5) = 1.118%.
This relationship holds for ATM option prices too.  With the Black and Scholes model if an option due to expire in 30 days has a price of $1, then the 60 day option with the same strike price and implied volatility should be priced at sqrt (60/30) = $1 * 1.4142 = $1.4142  (assuming zero interest rates and no dividends).