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Optimal approximations for risk measures of sums of lognormals based on conditional expectations. (English) Zbl 1154.91021

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.

MSC:

91B26 Auctions, bargaining, bidding and selling, and other market models
91B30 Risk theory, insurance (MSC2010)
60Exx Distribution theory

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