Application of asymmetric Laplace laws in financial risk measures and time series analysis

Asymmetric Laplace (AL) laws are applied in financial risk measurement and time series analysis. Traditional methods on financial risk measures and time series analysis are based on the assumption of normality. Recent studies on financial data suggest that the normality assumption is usually violated.; Explicit expressions are derived for maximum likelihood estimators (MLEs) and nonparametric estimators (NPEs) of financial risk measures, Value-at-Risk (VaR) and Conditional Value-at-Risk (CVaR), under random sampling from the Asymmetric Laplace distribution. Asymptotic distributions are established under very general conditions. Finite sample distributions are investigated by means of saddlepoint approximations. An application of the methodology in modeling currency exchange rates suggests that the AL distribution is successful in capturing the peakedness, leptokurticity and skewness, inherent in such data.; Time series autoregressive moving average (ARMA) models driven by Asymmetric Laplace noise are considered for modeling dependent data. Assuming AL noise, the model marginal distribution is derived analytically. Conditional maximum likelihood estimation is applied to fit ARMA models driven by AL noise and AL general autoregressive conditional heteroscedasticity (GARCH) noise. Daily returns of real estate mutual fund data are fitted by four methods. Models under AL noise have substantially lower Bias-corrected Akaike Information Criterion (AICc), indicating much better fit for the real nancial data.