Order Statistics for Value at Risk Estimation and Option Pricing

The theory of order statistics has various applications in mathematical finance. Frederik Herzberg and Christoph Bennemann use it to calculate various properties of the quantile measured in Historical Simulations (and Monte Carlo simulations)

We apply order statistics to the setting of VaR estimation. Here techniques like historical and Monte Carlo simulation rely on using the k-th heaviest loss to estimate the quantile of the profit and loss distribution of a portfolio of assets. We show that when the k-th heaviest loss is used the expected quantile and its error will be independent of the portfolio composition and the return  functions of the assets in the portfolio. This is not the case when a linear combination of simulated losses is used. Furthermore, we briefly demonstrate how order statistics can be applied to pricing options depending on the quantile of a distribution.