Unstructured meshing for two asset barrier options

Discretely observed barriers introduce discontinuities in the solution of two asset option pricing partial differential equations (PDEs) at barrier observation dates. Consequently, an accurate solution of the pricing PDE requires a fine mesh spacing near the barriers. Non-rectangular barriers pose difficulties for finite difference methods using structured meshes. It is shown that the finite element method (FEM) with standard unstructured meshing techniques can lead to significant efficiency gains over structured meshes with a comparable number of vertices. The greater accuracy achieved with unstructured meshes is shown to more than compensate for a greater solve time due to an increase in sparse matrix condition number. Results are presented for a variety of barrier shapes, including rectangles, ellipses, and rotations of these shapes. It is claimed that ellipses best represent constant (risk neutral) probability regions of underlying asset price-point movement, and are thus natural two-dimensional barrier shapes.