Optimal approximations for risk measures of sums of lognormals based on conditional expectations
The distribution function (d.f.) of the sum \(S=\sum_{i=1}^n\alpha_i e^{Z_i}\), where \(\alpha_i>0\) and the vector \((Z_1,Z_2,\ldots,Z_n)\) has a multivariate normal distribution, is studied. D.f. of \(S\) is approximated by the d.f. of conditional expectation \(E[S/\Lambda]\) with respect to a conditioning random variable \(\Lambda\). An appropriate choice of \(\Lambda\) leads to a comonotonicity of conditional vector and this helps to approximate risk measures related to the d.f. of \(S\) by the corresponding risk measures of \(E[S/\Lambda]\). Globally optimal choces of \(\Lambda\) connected with ``Taylor-based'' and ``maximal variance'' approximations are considered. Locally optimal choces of \(\Lambda\) are studied as well, they are connected with ``CTE\(_{p}\)-based'' and an ``asymptotically optimal'' approximations. Applications to discounting, compounding and to the pricing of Asian options are presented.
scientific article; zbMATH DE number 5355644