Fourier Expansion Method on Cumulative Distribution Function and Value-at-Risk Estimate

Abstract

This paper proposes a Fourier-series based method to estimate the cumulative distribution function (cdf) of random variables that follow leÌvy process. Since most cdf(s) of leÌvy process do not have explicit forms but their characteristic functions are always available. Our method approaches a cdf by the Fourier-series and the coefficients are only determined by the characteristic function. Therefore, our method can be widely used in the leÌvy process without any difficulty. Finally, we apply this method to calculate the Value-at-Risk (VaR) of a portfolio that the constituents follow the stable distribution with or without identical characteristic exponents (α).