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 volatility surface to reconcile discretely observed market data and dynamics. In this thesis we propose a new method to construct this surface using radial basis functions. Our results show that this approach is tractable and yields good results. When used in a local volatility context these results replicate the observed market prices. Testing against a skew model with known analytical solution shows that both prices and hedging parameters are acurately reconstructed, with best case average relative errors in pricing of 0.0012. While the accuracy of these results exceeds those reported by spline interpolation methods, the solution is critically dependent upon the quality of the numerical solution of the resultant local volatility PDE’s, heuristic parameter choices and data filtering.