Convergence of Optimal Expected Utility for a Sequence of Binomial Models

TL;DR

This paper investigates the convergence of expected utility in binomial models approximating the Black-Scholes model, providing positive results especially in symmetric and certain asymmetric cases, using advanced probabilistic estimates.

Contribution

It establishes convergence of expected utility in binomial models under conditions previously unresolved, notably in symmetric and specific asymmetric scenarios.

Findings

Convergence holds in symmetric binomial models.

Positive convergence results for certain asymmetric models.

Utilizes refined estimates from the Central Limit Theorem.

Abstract

We analyze the convergence of expected utility under the approximation of the Black-Scholes model by binomial models. In a recent paper by D. Kreps and W. Schachermayer a surprising and somewhat counter-intuitive example was given: such a convergence may, in general, fail to hold true. This counterexample is based on a binomial model where the i.i.d. logarithmic one-step increments have strictly positive third moments. This is the case, when the up-tick of the log-price is larger than the down-tick. In the paper by D. Kreps and W. Schachermayer it was left as an open question how things behave in the case when the down-tick is larger than the up-tick and -- most importantly -- in the case of the symmetric binomial model where the up-tick equals the down-tick. Is there a general positive result of convergence of expected utility in this setting? In the present note we provide a positive answer to this question. It is based on some rather fine estimates of the convergence arising in the Central Limit Theorem.