Value-at-risk estimation with the Carr-Geman-Madan-Yor process: an empirical study on foreign exchange rates

To discover whether the Carr-Geman-Madan-Yor (CGMY) distribution, a Levy process that can have an infinite number of jumps in a finite time interval, is appropriate for describing the fat tails of the distribution of foreign exchange (FX) rate returns, we investigate its performance in estimating the risk of FX rates. We estimate value-at- risk (VaR), one of the most popular concepts in the area of risk management, for FX rate returns. To enhance the robustness of the estimation results, we use other VaR models, such as historical simulation, the generalized autoregressive conditional heteroscedasticity (GARCH) model and conditional extreme value theory (EVT). Our four methodologies are applied to three major foreign exchange rates (EUR/JPY, EUR/USD and USD/JPY) by using a different size estimation windows for each model to forecast one-day-ahead VaR. We propose a new procedure to find the best size of estimation window for each VaR model, with which we compare the respective VaR estimates. The conditional EVT and CGMY distributions provide superior forecasting performance for all three rates's daily returns, respectively. Furthermore, the performance of the other methods can be significantly improved by adjusting the size of the estimation window.