<p>In the presented work, models under control of piecewise constant semimartingales are studied. These processes are constructed using a two-dimensional random sequence with independent components. The first component is a passage times regarding the level of the module of the standard Wiener process, and the second is a Rademacher random variable. The first semimartingale is a non-decreasing, non-negative process and is considered a random time process. The second process is a subordinated Wiener process by random time process. The appearance of random time significantly expands the possibilities of discrete time binary models as a means of modeling and as a means of approximating solutions to stochastic differential equations. These random processes replace Wiener processes in Ito processes and in stochastic differential equations. Their use of the process in the classical Black and Scholes model allowed us to obtain a new formula for calculating the fair price of a payoff.</p>