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My understanding is that Turnbull & Wakeman derived an approximation formula for continous arithmetic Asian option using Edgeworth series by matching the first two moments.

However, in the book Mathematical Models of Financial Derivatives, after a few pages of proof on the approximation, it writes "Besides the Edgeworth expansion method, .... Turnbull and Wakeman (1991) ..."

So the idea of Edgeworth approximation didn't come from Turnbull and Wakeman and possibly somebody else?

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Yes, AFAIK, they used Edgeworth expansions

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The objective of this paper is to describe a quick way to price European average options. While it is very difficult to determine the probability distribution for the average, all of its moments can be readily determined. Thus, an Edgeworth series expansion can be used to approximate the distribution

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