Lie Symmetry Methods for Local Volatility Models

We investigate PDEs of the form ut = 1/2 s^2 (t, x)u_xx - g(x)u which are associated with the calculation of expectations for a large class of local volatility models. We find nontrivial symmetry groups that can be used to obtain standard integral transforms of fundamental solutions of the PDE. We detail explicit computations in the separable volatility case when s(t, x) = h(t)(a + sx + ?x^2), g = 0, corresponding to the so called Quadratic Normal Volatility Model. We also consider choices of g for which we can obtain exact fundamental solutions that are also positive and continuous probability densities.