In this thesis, we study three different problems in financial risk management. The first one is to prove that the sample autocovariance with lag 1 (ACV(1)) of the increments of a mixed Poisson process converges to 0 almost surely as the sample size goes to infinity. Pólya processes, as a good candidate for modeling the arrival of events that cluster in time, are also mixed Poisson processes. The sample ACV(1) of the increments of a Pólya process converges to 0, instead of the theoretical population ACV(1) of the increments, almost surely as the sample size goes to infinity, implying that one cannot use sample ACV in parameter estimation of Pólya process by methods of moments. The second problem is about comparing the power of backtest of value-at-risk at level 1 % and ex...[ Read more ]

In this thesis, we study three different problems in financial risk management. The first one is to prove that the sample autocovariance with lag 1 (ACV(1)) of the increments of a mixed Poisson process converges to 0 almost surely as the sample size goes to infinity. Pólya processes, as a good candidate for modeling the arrival of events that cluster in time, are also mixed Poisson processes. The sample ACV(1) of the increments of a Pólya process converges to 0, instead of the theoretical population ACV(1) of the increments, almost surely as the sample size goes to infinity, implying that one cannot use sample ACV in parameter estimation of Pólya process by methods of moments. The second problem is about comparing the power of backtest of value-at-risk at level 1 % and expected shortfall at level 2%. With simulation, we show that 2% expected shortfall performs better than 1 % value-at-risk in detecting the misspecification of the forecast model. The third problem is about representation of risk measures incorporating uncertainty in probability measures. We propose a new set of axioms for risk measures and obtain representation of the risk measures that satisfy these axioms. The representation of the new class of risk measures, called natural risk measures, provide a theoretical framework that includes the Basel II and the Basel III risk measures as special cases.