Implementing fisher Black's simple discounting rule

We propose a simple implementation of Black’s (1988) elegant rule for discounting uncertain future cash flows. Black’s rule avoids the thorny problem of estimating an appropriate risk-adjusted discount rate. Instead, the rule calls for discounting conditional mean cash flows at appropriate riskless interest rates. Our contribution in this article is to describe and illustrate a method of estimating the conditional mean cash flows called for in Black’s rule. The method is quite flexible with respect to the types of information available concerning the distributions of future cash flows. We argue that this approach to computing present values offers a theoretically sound and generally feasible addition to the toolbox of financial managers.

The von Mises-Fisher distribution of the first exit point from the hypersphere of the drifted Brownian motion and the density of the first exit time

A characterization is provided for the von Mises–Fisher random variable, in terms of first exit point from the unit hypersphere of the drifted Wiener process. Laplace transform formulae for the first exit time from the unit hypersphere of the drifted Wiener process are provided. Post representations in terms of Bell polynomials are provided for the densities of the first exit times from the circle and from the sphere.