Alternatives to Log-Normal and Normal Models in Market Risk: The Displaced Historical Simulation and the Mixed Model

The historical simulation is a standard technique in market risk estimation, in which the key choice to be made is whether to use absolute or relative shifts for the observed returns of the risk factors. To avoid this ambiguity, Fries et al. develop an approach called displaced historical simulation, which dynamically interpolates between a normal and a log-normal model. In the estimation of value-at-risk, the parameter governing this interpolation fluctuates strongly over time, which could be considered an obstacle in using this approach in practical applications. However, in this paper we show that the fluctuations do not impact the resulting shift scenarios significantly for the time series examined. Additionally, we present an alternative approach which sheds light on the origin of these fluctuations and allows us to assess the impact of some further assumptions made in the displaced historical simulation.