One of the most important event in the financial world is definitely the invention of the revolutionary option pricing formula by Black, Scholes and Merton in 1973.
Trading with options has thus become very widespread, as there was finally an analytical formula for calculating option premiums. However, due to market needs, more sophisticated options were formed that did not have an analytical solution. Therefore, a couple of years later, a numerical method with binomial trees for calculating options was developed. The method was designed in such a way that the price of a financial instrument could rise or fall at each time step and that by reducing the step size, the solution approached the analytical solution. Due to the desire for faster convergence of the binomial model, a trinomial model was created, which has three decisions in each step, in addition to the rise and fall, we also add the possibility that the price does not change. Numerous parameterizations for these methods have also emerged, which are supposed to eliminate certain problems of the original models and, of course, ensure faster convergence.A similar class of method is also the finite difference method, where we reduce the domain of the partial differential equation of the Black-Scholes-Merton model to a finite set of points and calculate the value of the premium with the help of finite differences. Of course with the increase of the number of points, it approaches the analytical solution, but with the increased time complexity.
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