Binomial models, which describe the asset price dynamics of the continuous-time model in the limit, serve for approximate valuation of options, especially where formulas cannot be derived analytically due to properties of the considered option type. To evaluate results, one inevitably must understand the convergence properties. In the literature we find various contributions proving convergence of option prices. We examine convergence behaviour and convergence speed. Unfortunately, even in the case of European call options, distorted results occur when calculating prices along the iteration of tree refinements. These convergence patterns are examined and order of convergence one is proven for the Cox-Ross-Rubinstein model as well as for two alternative tree parameter selections from the literature. Furthermore, we define new binomial models, where the calculated option prices converge smoothly to the Black-Scholes solution, and we achieve order of convergence two with much smaller initial error. Notably, only the formulas to determine the up- and down-factors change. Finally, following a recent approach from the literature, all tree approaches are compared with respect to speed and accuracy, calculating the relative root-mean-squared error of approximate option values for a sample of randomly selected parameters across a set of refinements. Here, on average, the same degree of accuracy is achieved 1400 times faster with the new binomial models. We also give some insights into the peculiarities in the valuation of the American put option. Inspecting the numerical results, the approximation of American-type options with the new models exhibits order of convergence one, but with a smaller initial error than with previously existing binomial models, giving the same accuracy on average ten-times faster than previous binomial methods.
