## 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

QRM
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### References:

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