Abstract
Many attempts have been made to derive a simple expression for the characteristic function of the lognormal distribution. This would be useful for computing the sum of lognormal variables, either with each other, or with other statistical variables. In this paper, we provide a simple formula for the characteristic function, which is exact, closed and computable. An extension to the sum of correlated lognormals, used in the pricing of Asian options, is a consequence of this approach.
Highlights
IntroductionThe usefulness of finding a simple expression for the characteristic function (CF, a.k.a Fourier transform FT) of the lognormal distribution has been made clear in a variety of contexts: from insurance and finance to engineering to biology
Many attempts have been made to derive a simple expression for the characteristic function of the lognormal distribution
We provide a simple formula for the characteristic function, which is exact, closed and computable
Summary
The usefulness of finding a simple expression for the characteristic function (CF, a.k.a Fourier transform FT) of the lognormal distribution has been made clear in a variety of contexts: from insurance and finance to engineering to biology. In the context of Asian options, we find a similar view from [11]: “Contrary to the case of geometric averaging, the distribution for the arithmetic average is not known This implies that no exact closed form solution is available for European-style Asian options based on the arithmetic average. We provide and prove an exact expression for the lognormal CF, which can be verified numerically by the inverse Fourier transform. It is not the purpose of this paper to assess how it relates to the approximations, asymptotic or otherwise, in the literature. That is a task for the authors of those approximations
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