# Summary

Normal distributions are commonly used in many modeling techniques to describe reality yet in many cases, reality is closer to a fat-tailed distribution than a normal one. This critical assumption of normality results in potentially inaccurate models, especially when modeling risk and options prices. The aim of this project is to develop a fat-tailed distribution model that fits the S&P500’s daily percent returns to be applied in standard financial calculations.

The project uses whole data models of t-distribution and mixed normal distributions as well as extreme scenario models of Generalized Extreme Value distribution (GEV) and Generalized Pareto Distribution (GPD) to calculate Value at Risk (VaR) and options prices. The results show that the models chosen exhibit fatter tails than the data and by a larger difference than that between the data and the normal distribution. Of all the models, the closest to the data is the mixed normal model. This leads to the recommendation of the project being to use the mixed normal model as the distribution in estimating VaR and options prices so that investment decisions include the expectation of more extreme scenarios. For VaR, this can also be achieved by using the normal model with a base value for VaR or by using a multiplier.