Most standard models assume stock returns are normally distributed even though everyone agrees that real-world returns have fat tails. We've all heard stories of hedge funds that went bankrupt cause something happened that their model called a "10 sigma event that should only happen once every billion years" and that's obviously a flawed model. My textbooks point this out but hand-wave it away like it's an unavoidable simplification.
But why do we have to simplify? We know the real-world historical kurtosis of stock returns, and it's easy to define a distribution that matches that kurtosis. Why can't we simply use that fat tail distribution in all standard models instead of the normal distribution? Is it simply that the PDF of the normal distribution has analytical solutions that are easier to work with?