## A robust finite difference scheme for pricing American put options with singularity-separating method.(English)Zbl 1192.91190

The determination of the value of an American option is more complicated than for a European option. The former is governed by a linear complementarity problem involving the Black-Scholes differential operator and a constraint on the value of the option. Since analytical solutions can rarely be found for practical problems, numerical methods needed to be developed. There are several methods in the literature for the valuation of European and American options: the lattice technique, classical difference methods, the upwind numerical approach, linearized methods based on differential quadrature methods etc. This paper presents a stable numerical method which is based on a hybrid finite difference spatial discretization on a piecewise uniform mesh and an implicit time stepping technique. Error estimates are derived for the direct application of the finite difference method to the American option pricing model. The scheme is shown to be second-order convergent with respect to the spatial variable. To overcome the lack of smoothness of the American put option pricing, the singularity-separating method is used.

### MSC:

 91G60 Numerical methods (including Monte Carlo methods) 91G20 Derivative securities (option pricing, hedging, etc.) 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
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### References:

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