Localization of the Black-Scholes equation using transparent boundary conditions

Résumé

Numerical solution of the Black-Scholes type pricing equation requires a localization on a bounded computational domain and, in consequence, a choice of boundary conditions. The standard approach usually proposed in the textbooks on financial engineering and academic literature is to take Dirichlet or Neumann boundary conditions deduced from the shape of the payoff function. Equivalently, one first subtracts the payoff from the solution, and then takes zero boundary conditions on the excess to payoff. It can be shown that the localization error in these cases converges to zero when the computational domain increases. However, these conditions are only asymptotic, and the localization error may be large for a fixed finite domaine. In computational physics, some authors have developed the so called transparent boundary conditions in order to bypass this difficulty. In this talk, we adapt this approach to computational finance.