A radial basis function approach to reconstructing the local volatility surface of European options

A key problem in financial mathematics is modelling the volatility skew observed in options markets.Local volatility methods, which is one approach to modelling skew, requires the construction of a volatilitysurface to reconcile discretely observed market data and dynamics. In this thesis we propose a newmethod to construct this surface using radial basis functions. Our results show that this approach istractable and yields good results. When used in a local volatility context these results replicate the observedmarket prices. Testing against a skew model with known analytical solution shows that both pricesand hedging parameters are acurately reconstructed, with best case average relative errors in pricing of0.0012. While the accuracy of these results exceeds those reported by spline interpolation methods, thesolution is critically dependent upon the quality of the numerical solution of the resultant local volatilityPDE’s, heuristic parameter choices and data filtering.