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This paper deals with pricing of European and American options, when theunderlying asset price follows Heston model, via the interior penaltydiscontinuous Galerkin finite element method (dGFEM). The advantages of dGFEMspace discretization with Rannacher smoothing as time integrator with nonsmoothinitial and boundary conditions are illustrated for European vanilla options,digital call and American put options. The convection dominated Heston modelfor vanishing volatility is efficiently solved utilizing the adaptive dGFEM.For fast solution of the linear complementary problem of the American options,a projected successive over relaxation (PSOR) method is developed with the normpreconditioned dGFEM. We show the efficiency and accuracy of dGFEM for optionpricing by conducting comparison analysis with other methods and numericalexperiments.

Pricing European and American Options under Heston Model using Discontinuous Galerkin Finite Elements