Models for fat tails I am trying to model stock returns and I thought it would be interesting to make a comparison between some well-known models. 
Could anyone name some well-known models that are used in the presence of fat tails? I was wondering if someone could provide me with an overview of alternatives. I am especially interested in somewhat more sophisticated models.
 A: Initial answer 
An example of a model with heavy-tailed errors: a simple regression model $y=\beta_0+\beta_1 x+\varepsilon$ where $\varepsilon$ is assumed to follow some heavy-tailed distribution. The model coefficients and perhaps the tail-heaviness parameter could be estimated by maximum likelihood. It could be estimated by OLS as well but OLS estimates would not coincide with ML estimates. 
Answer specific to stock return modelling 
A natural competitor of GARCH-type models is stochastic volatility (SV) model class; I think GARCH is a special case of SV models. I suspect the estimation of SV models is quite involved and the models look kind of fancy (but this is just an opinion).
Some literature on SV models:


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*Asai et al. "Multivariate stochastic volatility: A review" (2006)

*Broto & Ruiz "Estimation methods for stochastic volatility models: A survey" (2004)


I cannot give a good recommendation for any applied paper because I never worked on stochastic volatility myself. However, I am pretty sure that stochastic volatility models have not only been applied for option pricing but also for modelling returns on futures and stocks as well. Hopefully, you will find some relevant references in the two survey papers above.
