2020/09/21 by Friedrich Hubalek, Hubalek, Friedrich, Walter Schachermayer +1 · 1 citation
Economics, Econometrics and Finance · Mathematics · #60G42 #60G44 #91G10 #91G20 #Economic theories and models #FOS: Economics and business #FOS: Mathematics #Mathematical Finance (q-fin.MF) #Monetary Policy and Economic Impact #Probability (math.PR) #Stochastic processes and financial applications #math.PR #msc:60G42 #msc:60G44 #msc:91G10 #msc:91G20 #q-fin.MF
paper · pdf · doi:10.48550/arxiv.2009.09751
10 pages
arxiv created 2020/09/21 · openalex publication_date 2020/09/21 · arxiv updated 2020/09/22 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28
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.