We describe various models, which can be approximated by binomial models of a market of securities [\textit{J. C. Cox}, \textit{S. A. Ross} and \textit{M. Rubinstein}, J. Financ. Econ. 7, No. 3, 229--263 (1979; Zbl 1131.91333)] and introduce the corresponding approximation formulas for the value of options. In order to obtain more general and more realistic limit models, we introduce additional randomizations into the binomial model. This leads, in particular, to models, which behave in a very diverse way, having ``heavy'' tails, strongly expressed peaks of densities, nonsymmetric densities u.s.w. In correspondence to the above-mentioned article approximation formulas for the cost of options are derived, examples are presented.