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.


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


Full Text: DOI


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