The focus of my research has been computationally efficient means of computing measures of risk for portfolios of nonlinear financial instruments when the risk factors might be possibly non-Gaussian. In particular, the measures of risk chosen have been Value-at-Risk (VaR) and conditional Value-at-Risk (CVaR). I have studied the problem of computation of risk in two types of financial portfolios with nonlinear instruments which depend on possibly non-Gaussian risk factors: 1. Portfolios of European stock options when the stock return distribution may not be Gaussian; 2. Portfolios of sovereign bonds (which are nonlinear in the underlying risk factor, i.e. the short rate) when the risk factor may or may not be Gaussian. Addressing both these problems need a wide array of mathematical tools both from the field of applied statistics (Delta-Gamma-Normal models, characteristic function inversion, probability conserving transformation) and systems theory (Vasicek stochastic differential equation model, Kalman filter). A new heuristic is proposed for addressing the first problem, while an empirical study is presented to support the use of filter-based models for addressing the second problem. In addition to presenting a discussion of these underlying mathematical tools, the dissertation also presents comprehensive numerical experiments in both cases, with simulated as well as real financial market data. Backtesting is used to confirm the validity of the proposed methods.