Convergence of Optimal Expected Utility for a Sequence of Discrete-Time Markets

TL;DR

This paper investigates whether the optimal expected utility in a continuous-time Black–Scholes–Merton economy can be approximated by a sequence of discrete-time models, confirming the conjecture under certain utility conditions.

Contribution

It proves Kreps' conjecture for utility functions with asymptotic elasticity less than one and provides a counterexample when asymptotic elasticity equals one.

Findings

Conjecture holds for utility with asymptotic elasticity < 1.

Counterexample shows conjecture fails when asymptotic elasticity = 1.

Results depend on the third moment of the underlying random variable.

Abstract

We examine Kreps' (2019) conjecture that optimal expected utility in the classic Black--Scholes--Merton (BSM) economy is the limit of optimal expected utility for a sequence of discrete-time economies that "approach" the BSM economy in a natural sense: The $n$th discrete-time economy is generated by a scaled $n$-step random walk, based on an unscaled random variable $\zeta$ with mean zero, variance one, and bounded support. We confirm Kreps' conjecture if the consumer's utility function $U$ has asymptotic elasticity strictly less than one, and we provide a counterexample to the conjecture for a utility function $U$ with asymptotic elasticity equal to 1, for $\zeta$ such that $E[\zeta^3] > 0.$