Radial basis functions method for valuing options: A multinomial tree approach

From the view point of probability, this study presents a theoretical framework to show the convergence of the RBFs method for valuing options. It will be proved to be equivalent to a multinomial tree approach, in which the underlying variable can move from its initial value to an infinity of possible values of the next time step. Specially, the probability of a move in a short period time follows the normal distribution when using the Gaussian basis kernel, it is a precise simulation of the behavior of the underlying variable, which provides a more reasonable explanation of high-accuracy of the RBFs method. This helps open a new area of research in developing the expected numerical method for derivative securities (in which the underlying asset follows other stochastic process) by using corresponding radial basis kernel. The paper also illustrates the approach by using it to value stock options and its Greek letters.