An affine property of the reciprocal Asian option process

This note describes a property of the Asian option process, which neatly links the process at with the one at -, where is the drift of the geometric Brownian motion.The proof is based on (i) a known result due to Yor, on the law of the Asian option process taken at an exponential time, and (ii) a recent result on beta and gamma distributions.Suppose is one-dimensional standard Brownian motion starting at the origin, and define what this author calls the Asian option process, for want of a better name:Asian options have payoffs such as ( () -) + , and have been studied by numerous authors in Finance and Mathematics; reciprocal Asian options have payoffs such as ( -1/ () ) + , and have not received much attention so far; for more details and references, the reader is referred to [3] and [4].