We present a robust spectral method for solving partial differential equations arising from pricing the European and American options under local volatility. We consider these options on both dividend and non-dividend paying assets. In the case of European options, the PDE is discretised directly in space, while a penalty approach is used to solve the partial differential complementarity problem that arises from American options. The free boundary problem arising from American options is first reformulated as a variational inequality. The variational inequality is transformed into a nonlinear partial differential equation on fixed boundaries by adding a penalty term. Then, the resulting nonlinear partial differential equation is discretised in asset direction by means of a rational spectral collocation approximation. Illustrative numerical results will be presented at the conference.
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