An approach to approximate the multivariate distribution of time-aggregated stock returns in the GARCH context is developed here. The approach yields a one time-step simulation procedure as opposed to a multiple time-step simulation required in such a context. For this purpose, the exact moment...

We propose a non-parametric method based on a model-free formula to evaluate the tails of a risk-neutral distribution using the full cross-section of option prices at a fixed horizon. The method leads to the joint estimation of risk-neutral tail probabilities and tail expectations beyond the...