Tracking Value at Risk

The focus of this work is on the problem of tracking parameters describing both the Risk Neutral and the Objective measures over time with the aim of monitoring Value at Risk. The methodology presented incorporates information both from contingent claim prices over time as well as from the underlying instrument’s price over time in order to perform parameter estimation. The models adopted for the Objective measure are based on Variance Gamma and Normal Inverse Gaussian processes, and new Monte Carlo based methods for pricing options under Risk Neutral versions of these measures are presented. A parametric model of the Stochastic Discount Factor is introduced based on empirical results in the literature. This, combined with the Objective measure discussed results in option prices which are found through a novel Monte Carlo based method. A Sequential Monte Carlo approach to tracking the parameters of these models, while considering prices as imperfect observations is developed, incorporating optimal updates in the sense of minimum variance where possible. This results in the successful tracking of these parameters and comparative results for the two models are presented. These show that both models appear to track realistically, however the Variance Gamma model showed far more variability in its estimation of expected daily price moves than the Normal Inverse Gaussian model. Conversely the Normal Inverse Gaussian model shows more variability in its estimation of the probability of large daily moves.