Since Mandelbrot, Fama and others have performed seminal work on the topic, it has been suspected that stock price fluctuations can be more appropriately modeled using Lévy alpha-stable distrbutions other than the normal distribution law. Yet, the subject is somewhat controversial, there is a lot of literature in defense of the normal law and criticizing distributions without bounded variation. Moreover, precisely because of the the unbounded variation, the whole standard framework of quantitative analysis can not be simply copy/pasted to deal with these more "exotic" distributions.

Yet, I think there should be something to say about how to value risk of fluctuations. After all, the approaches using the variance are just shortcuts, what one really has in mind is the probability of a fluctuation of a certain size. So I was wondering if there is any literature investigating that in particular.

In other words: what is the current status of financial theories based on Lévy alpha-stable distributions? What are good review papers of the field?