The author considers the approximation of a random variable \(f(W_1)\) with a Borel function \(f\). This variable is approximated by stochastic integrals with respect to the standard Brownian motion and the standard geometric Brownian motion, where the integrands are piecewise constant within deterministic time intervals (time-nets). This approximation problem originates from stochastic finances, where the pricing of European type options is often based on continuous time models like the Black-Scholes model. The main result states that there exist random variables \(f(W_1)\) such that approximation error tends as slowly to zero as one wishes. The optimization methods and dynamic programming type argument are used for the solution of this problem.