Functionals of exponential Brownian motion and divided differences

We provide a surprising new application of classical approximation theory to a fundamental asset-pricing model of mathematical finance. Specifically, we calculate an analytic value for the correlation coefficient between the exponential Brownian motion and its time average, and we find that the use of divided differences greatly elucidates formulae, providing a path to several new results. As applications, we find that this correlation coefficient is always at least 1/2 and, via the Hermite–Genocchi integral relation, demonstrate that all moments of the time average are certain divided differences of the exponential function. We also prove that these moments agree with the somewhat more complex formulae obtained by Oshanin and Yor.

► A surprising new application of approximation theory to exponential Brownian motion is provided.
► Correlation coefficient for exponential Brownian motion and time average is calculated.
► We demonstrate that the moments of the time average are divided differences of the exponential function.
► We also prove that the moments agree with the more complex formulae obtained by Oshanin and Yor.