Approximation of stop-loss premiums involving sums of lognormals by conditioning on two variables

In literature, lower and upper bounds were obtained for arithmetic Asian and basket options based on comonotonicity results and by conditioning upon one variable. In this paper, we derive analytical expressions for the comonotonic bounds of stop-loss premiums of sums of dependent random variables by conditioning upon two variables. We also use the idea of several conditioning variables to develop an approximation for cases for which it is cumbersome to obtain a comonotonic lower bound. The numerical analysis shows that conditioning on two variables leads to very sharp results.