Dolomites Research Notes on Approximation

Option prices are usually represented by a three-dimensional set of volatility values, implied by the Black-Scholes formula or a stochastic model like the Heston model. Given such volatility points, it is required to reconstruct the corresponding volatility surface via an interpolation method. From the point of view of interpolation, it is intriguing to work with a particular data-set such as the one shown in the current work. This set is characterized by unscattered or non-specifically distributed data: they are rather arranged along lines. In this paper, we present a computational study based on radial basis function (RBF) methods. Initially, a reconstruction of the surface has been made globally, then the obtained output has been tested by removing points and evaluating errors. Furthermore, local methods such as RBF-partition of unity method have been adopted with variable sizes of subdomains and shape parameters. To improve the interpolant accuracy we propose a strategy consisting in adding points which were computed through the least square method. One of the issues of the financial world is to extrapolate the option volatility surface for unknown tenors and strikes, and therefore the study further develops these methods in order to cover extrapolated regions.