Convergence of Optimal Expected Utility for a Sequence of Binomial Models
Abstract
We analyze the convergence of expected utility under the approximation of the BlackScholes model by binomial models. In a recent paper by D. Kreps and W. Schachermayer a surprising and somewhat counterintuitive 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 onestep increments have strictly positive third moments. This is the case, when the uptick of the logprice is larger than the downtick. In the paper by D. Kreps and W. Schachermayer it was left as an open question how things behave in the case when the downtick is larger than the uptick and  most importantly  in the case of the symmetric binomial model where the uptick equals the downtick. 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.
 Publication:

arXiv eprints
 Pub Date:
 September 2020
 arXiv:
 arXiv:2009.09751
 Bibcode:
 2020arXiv200909751H
 Keywords:

 Mathematics  Probability;
 Quantitative Finance  Mathematical Finance;
 60G42;
 60G44;
 91G10;
 91G20
 EPrint:
 10 pages