Approximating Expectation Functionals for Financial Optimization

Abstract

Numerical evaluation of the expectation of a function of a random vector is often difficult because either the knowledge of the underlying probability distribution is not complete, or the probability space is continuous and each function evaluation is expensive. Such difficulties often arise in financial optimization where a risk measure is expressed as an expectation functional of random (asset) returns. Not only does the latter expectation depends on investment positions created in the underlying assets, but also it requires the solution of a mathematical program. First, the basic results from generalized moment problems are presented to establish tightness properties of approximations. Then, first and second moment approximations are presented for the expectation. These results are applied within a financial optimization problem to illustrate the efficiency of the approximations for determining optimal positions in a portfolio of the Standard and Poors 100 stocks.