I was wondering what the most common, or most popular, ways - in both academia, and industry - there were to model the fat tails of volatility in asset prices changes.
I am presuming a basic Brownian motion random walk, is not what is used, because it will not replicate fat tails. Is that correct? Or am I wrong, and in most cases, a basic Brownian motion is "good enough"? What are more advanced methods that are used, whether it be in terms of stochastic calculus, statistical methods, etc.?