On the Black's equation for the risk tolerance function

TL;DR

This paper investigates Black's nonlinear equation for the risk tolerance function in a log-normal model, establishing mathematical properties like existence, uniqueness, and shape characteristics for general utility functions.

Contribution

It provides new theoretical results on the properties of Black's equation, especially for utilities with inverse marginal functions that are completely monotonic.

Findings

Existence and uniqueness of solutions for general utility functions.

Regularity and shape properties such as monotonicity and convexity.

Stronger results for utilities with completely monotonic inverse marginal functions.

Abstract

We analyze a nonlinear equation proposed by F. Black (1968) for the optimal portfolio function in a log-normal model. We cast it in terms of the risk tolerance function and provide, for general utility functions, existence, uniqueness and regularity results, and we also examine various monotonicity, concavity/convexity and S-shape properties. Stronger results are derived for utilities whose inverse marginal belongs to a class of completely monotonic functions.