## Penalty methods for the numerical solution of American multi-asset option problems.(English)Zbl 1152.91542

Summary: We derive and analyze a penalty method for solving American multi-asset option problems. A small, non-linear penalty term is added to the Black-Scholes equation. This approach gives a fixed solution domain, removing the free and moving boundary imposed by the early exercise feature of the contract. Explicit, implicit and semi-implicit finite difference schemes are derived, and in the case of independent assets, we prove that the approximate option prices satisfy some basic properties of the American option problem. Several numerical experiments are carried out in order to investigate the performance of the schemes. We give examples indicating that our results are sharp. Finally, the experiments indicate that in the case of correlated underlying assets, the same properties are valid as in the independent case.

### MSC:

 91G60 Numerical methods (including Monte Carlo methods) 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs 65Y20 Complexity and performance of numerical algorithms 91G20 Derivative securities (option pricing, hedging, etc.) 60G40 Stopping times; optimal stopping problems; gambling theory

### Keywords:

American options; penalty methods; numerical solution
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### References:

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