  Adachi and Ryu introduced a category Prob of probability spaces whose objects
are all probability spaces and whose arrows correspond to measurable functions
satisfying an absolutely continuous requirement in [Adachi and Ryu, 2019]. In
this paper, we develop a binomial asset pricing model based on Prob. We
introduce generalized filtrations with which we can represent situations such
as some agents forget information at some specific time. We investigate the
valuations of financial claims along this type of non-standard filtrations.
