## Abstract

In this paper we investigate approximations for the distribution function of

a sum S of lognormal random variables. These approximations are obtained

by considering the conditional expectation E[S | ] of S with respect to a

conditioning random variable .

The choice for is crucial in order to obtain accurate approximations. The

different alternatives for that have been proposed in literature to date are

'global' in the sense that is chosen such that the entire distribution of the

approximation E[S | ] is 'close' to the corresponding distribution of the original

sum S.

In an actuarial or a financial context one is often only interested in a particular

tail of the distribution of S. Therefore in this paper we propose approximations

E[S | ] which are only locally optimal, in the sense that the relevant tail of

the distribution of E[S | ] is an accurate approximation for the corresponding

tail of the distribution of S. Numerical illustrations reveal that local optimal

choices for can improve the quality of the approximations in the relevant tail

significantly.

We also explore asymptotic properties of the approximations E[S | ] and

investigate links with results from Asmussen & Royas-Nandayapa (2005). Finally,

we briefly adress the sub-optimality of Asian options from the point of view of

risk averse decision makers with a fixed investment horizon.

a sum S of lognormal random variables. These approximations are obtained

by considering the conditional expectation E[S | ] of S with respect to a

conditioning random variable .

The choice for is crucial in order to obtain accurate approximations. The

different alternatives for that have been proposed in literature to date are

'global' in the sense that is chosen such that the entire distribution of the

approximation E[S | ] is 'close' to the corresponding distribution of the original

sum S.

In an actuarial or a financial context one is often only interested in a particular

tail of the distribution of S. Therefore in this paper we propose approximations

E[S | ] which are only locally optimal, in the sense that the relevant tail of

the distribution of E[S | ] is an accurate approximation for the corresponding

tail of the distribution of S. Numerical illustrations reveal that local optimal

choices for can improve the quality of the approximations in the relevant tail

significantly.

We also explore asymptotic properties of the approximations E[S | ] and

investigate links with results from Asmussen & Royas-Nandayapa (2005). Finally,

we briefly adress the sub-optimality of Asian options from the point of view of

risk averse decision makers with a fixed investment horizon.

Original language | English |
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Pages (from-to) | 202-218 |

Number of pages | 17 |

Journal | Journal of Computational and Applied Mathematics |

Volume | 221 |

Issue number | 1 |

Publication status | Published - 2008 |

## Keywords

- comonotonicity