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If I have understood your question correctly then you might be interested in the following paper: Weber, A. E. Annual risk measures and related statistics. Applied paper Ortec Finance Research Center (8 2017). You should be able to download a copy here.


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For an simple approximation (which is not particularly useful if you want values to be asymptotically close around $N$), $$\ln\left(\frac{\max\left(L,H\right)}{0.01}\right)\tanh\left(k\left(x-N\right)\right)$$ or $$\ln\left(\frac{\max\left(L,H\right)}{0.01}\right)\operatorname{erf}\left(k\left(x-N\right)\right)$$ are suitable. If you want an asymptotic ...


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