In this paper, we prove the existence of an integral closed-form solution for pricing barrier options in both Heston and Bates frameworks. The option value depends on time, on the price and on the volatility of the underlying asset and it can be computed as the solution of a two dimensional pricing partial integro-differential equation. The integral representation formula of the solution is derived by projection of the differential equation and exploiting the properties of the adjoint operator. We derive the expression of the fundamental solution (Green's function) necessary for the integral representation formula. The computation is based on the interpretation of the fundamental solution as the joint transition probability density function of the underlying asset price and variance and is obtained through Fourier inverse transform of a suitable conditional characteristic function. We propose a numerical scheme to approximate the option price based on the classical Boundary Element Method and we provide two numerical examples showing the computational efficiency and accuracy of the proposed new method. The algorithm can be modified to compute greeks as well.
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