Analysis of quantization error in financial pricing 1 via finite difference methods

8 In this paper, we study the error of a second order finite difference scheme for the one-dimensional 9 convection-diffusion equation. We consider non-smooth initial conditions commonly encountered in 10 financial pricing applications. For these initial conditions, we establish the explicit expression of the 11 quantization error, which is loosely defined as the error of the numerical solution due to the placement 12 of the point of non-smoothness on the numerical grid. Based on our analysis, we study the issue of 13 optimal placement of such non-smoothness points on the grid, and the effect of smoothing operators 14 on quantization errors. 15