The sparse method of simulated quantiles: An application to portfolio optimization

The sparse multivariate method of simulated quantiles (S‐MMSQ) is applied to solve a portfolio optimization problem under value‐at‐risk constraints where the joint returns follow a multivariate skew‐elliptical stable distribution. The S‐MMSQ is a simulation‐based method that is particularly useful for making parametric inference in some pathological situations where the maximum likelihood estimator is difficult to compute. The method estimates parameters by minimizing the distance between quantile‐based statistics evaluated on true and synthetic data, simulated from the postulated model, penalized by adding the smoothly clipped absolute deviation ℓ1‐penalty in order to achieve sparsity. The S‐MMSQ aims to efficiently handle the problem of estimating large‐dimensional distributions with intractable likelihood, such as the stable distributions that have been widely applied in finance to model financial returns.