Narrow Results

Developing a novel and practical method based on the radial-base grid network (RBF) as well as the third-order polynomial of the exponential function $(f(t)=e^t)$ in order to solve the first-order differential equations based on $Z$-numbers is the aim of this paper. It is worth mentioning that the advantage of the proposed RBF is that sufficient information is not required. The RBF contains three distinct layers as follows: the input layer including the elementary nodes; the second layer including the hidden layers via high dimensions; and the output layer for responding and activating the patterns of the input layer. The obtained results revealed that this method could solve and approximate such problems under acceptable confidence.

The Radial Basis Functions (RBF) interpolation is a popular approximation technique used to smooth scattered data in various dimensions. This study uses RBF interpolation to interpolate the volatility skew of the S&P500 index options. The interpolated skews are used to construct the risk-neutral densities of the index and its local volatility surface. The RBF interpolation is contrasted throughout the study with the cubic spline interpolation. An analysis of the densities and the local volatility shows that RBF are an effective and practical tool for interpolating the implied volatility surface.