Market Value-At-Risk: ROM Simulation, Cornish-Fisher Var and Chebyshev-Markov Var Bound

Abstract

We apply the recently developed sampling algorithm, called random orthogonal matrix (ROM) simulation by Ledermann et al. [3], to compute VaR of a market risk portfolio. Typically, the covariance matrix has a large influence on ROM VaR. But VaR, being a lower quantile of the portfolio return distribution, is also much impacted by the skewness and kurtosis of the risk factor returns. With ROM VaR it is possible to stress test risk factors under adverse market conditions by targeting other sample moments that are consistent with periods of financial crisis. In particular, the important effects of skewness or kurtosis in the tail of the portfolio returns can be incorporated in ROM VaR. In a simulation study, we integrate ROM VaR into other methods that take into account skewness and kurtosis, namely the Cornish-Fisher VaR approximation and a robust approximation to the Chebyshev-Markov VaR upper bound in Hürlimann [7].

Targeting Kollo skewness with random orthogonal matrix simulation

“…For instance, Geyer et al (2014) implements ROM simulation in an arbitrage-free setting, for option pricing and hedging. Hürlimann (2014) integrates ROM simulation into two semi-parametric methods that take into account skewness and kurtosis, in order to improve the computation of quantiles for linear combinations of the system variables, and Alexander and Ledermann (2012) investigate the use of target skewness and kurtosis via ROM simulation for stress testing. Further theoretical developments focus on the higher-moment characteristics of ROM simulation: Hürlimann (2015) introduces a new, recursive method for targeting the skewness and kurtosis metrics of Mardia (1970) using ROM simulation with GHL matrices, and Hanke et al (2017) improve on the skewness metric that ROM simulation can target, using the vector-valued moment of Kollo (2008) in place of the Mardia skewness.…”