by Brückner, K..

**Series:** 2006-53, Preprints

- MSC:
- 60E15 Inequalities; stochastic orderings

**Abstract:**

The concepts of convex order and comonotonicity have become quite popular in risk theory, essentially since Kaaset al. (2000) constructed bounds in the convex order sense for a sum S of random variables without imposing any dependence structure upon it. Those bounds are especially helpful, if the distribution of S cannot be calculated explicitly or is too cumbersome to work with. This will be the case for sums of lognormally distributed random variables, which frequently appear in the context of insurance and finance.

In this article we quantify the maximal error in terms of truncated first moments, when S is approximated by a lower or an upper convex order bound to it. We make use of geometrical arguments; from the unknown distribution of S only its variance is involved in the computation of the error bounds. The results are illustrated by pricing an Asian option. It is shown, that under certain circumstances our error bounds outperform other known error bounds, e.g. the bound proposed by Nielsen and Sandmann (2003).

**Keywords:**

convex order, truncated first moments, stop-loss-premiums, sums of random variables, Asian options

**This paper was published in:**

Insurance: Mathematics and Economics 42 (2008), pp. 261-270.