Distortion Risk Measures of Increasing Rearrangements

The increasing rearrangement is a rewarding instrument in financial risk management. In practice, risk must be managed in different perspectives. A common example is portfolio risk which often can be seen from at least two perspectives: market value and book value. Different perspectives that is different distributions can be coupled by increasing rearrangement. One distribution is regarded as underlying, and the other distribution can be expressed as increasing rearrangement of the underlying distribution. Then, the risk measure for the latter can be expressed in terms of the underlying distribution. Our first objective is to introduce the increasing rearrangement for application in financial risk management and apply the increasing rearrangement to the class of distortion risk measures. We derive formulas to compute risk measures in terms of the underlying distribution. Afterwards, we apply our results to a series of special distortion risk measures, namely Value at Risk, Expected Shortfall, Range Value at Risk, Conditional Value at Risk, Wang's risk measure. Finally, we present the connection of increasing rearrangement to inverse transform sampling, Monte Carlo simulation, and cost-efficient strategies. Butterfly options serve as illustrative example for the method.