Efficient simulations in finance

Abstract

Measuring the risk of a credit portfolio is a challenge for financial institutions because of the regulations brought by the Basel Committee. In recent years lots of models and state-of-the-art methods, which utilize Monte Carlo simulation, were proposed to solve this problem. In most of the models factors are used to account for the correlations between obligors. We concentrate on the the normal copula model, which assumes multivariate normality of the factors. Computation of value at risk (VaR) and expected shortfall (ES) for realistic credit portfolio models is subtle, since, (i) there is dependency throughout the portfolio; (ii) an efficient method is required to compute tail loss probabilities and conditional expectations at multiple points simultaneously. This is why Monte Carlo simulation must be improved by variance reduction techniques such as importance sampling (IS). Optimal IS probabilities are computed and compared with the “asymptotically optimal” probabilities for credit portfolios consisting of groups of independent obligors. Then, a new method is developed for simulating tail loss probabilities and conditional expectations for a standard credit risk portfolio. The new method is an integration of IS with inner replications using geometric shortcut for dependent obligors in a normal copula framework. Numerical results show that the new method is better than naive simulation for computing tail loss probabilities and conditional expectations at a single x and VaR value. Furthermore, it is clearly better than twostep IS in a single simulation to compute tail loss probabilities and conditional expectations at multiple x and VaR values. Then, the performance of outer IS strategies, which consider only shifting the mean of the systematic risk factors of realistic credit risk portfolios are evaluated. Finally, it is shown that compared to the standard t statistic a skewness-correction method of Peter Hall is a simple and more accurate alternative for constructing confidence intervals.