SEMINAR: Applied Mathematics Seminar

We investigate the numerical methods for the solution of the Hamilton-Jacobi-Bellman (HJB) equations arising in European and American option pricing with proportional transaction costs. We propose a penalty approach combined with a finite difference scheme to solve the HJB equations. We first present a penalty formulation for the HJB equation. We then develop a numerical scheme based on the finite differencing in both space and time for solving the penalized equation. We show that the viscosity solution to the penalized equation converges to that of the original HJB equation as penalty parameters tend to infinity. We also prove that the solution of the finite difference scheme converges to the viscosity solution of the penalized equation. Numerical results are given to demonstrate the effectiveness of the proposed method.