# A Penalty Scheme for Solving American Option Problems

• B. F. Nielsen
• O. Skavhaug
• A. Tveito
Conference paper
Part of the Mathematics in Industry book series (MATHINDUSTRY, volume 1)

## Abstract

The value of American options is modeled by a parabolic differential equation with boundary conditions specified at free and moving boundaries, cf. i.e. [1,4]. These problems are nonlinear and analytical solutions are in general not available. Hence, such derivatives must be priced by numerical techniques. The basic idea of the penalty method [2,3,5] is to remove the free and moving boundary from the problem by adding a small and continuous penalty term to the Black-Scholes equation. Then the problem can be solved on a fixed domain and thus removing the difficulties associated with a moving boundary. For explicit, semi-implicit and fully implicit numerical schemes, we prove that the numerical option values generated by the penalty method mimics the basic properties of the analytical solution of the American option problem. Further details can be found in our papers [2,3].

### Keywords

Assure Volatility Gridding

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### References

1. 1.
Kwok, Y. K. (1998) Mathematical models of financial derivatives. Springer-Verlag.Google Scholar
2. 2.
Nielsen, B. F., Skavhaug, O. and Tveito, A. (2000) Penalty and front-fixing methods for the numerical solution of American option problems. Preprint 20005, Department of Informatics, University of Oslo.Google Scholar
3. 3.
Nielsen, B. F., Skavhaug, O. and Tveito, A. (2000) Penalty methods for the numerical solution of American multi-asset option problems. Preprint 2000–289, Department of Informatics, University of Oslo.Google Scholar
4. 4.
Wilmott, P., Dewynne, J. and Howison, S. (1993) Option Pricing. Mathematical models and computation. Oxford Financial Press.Google Scholar
5. 5.
Zvan, R., Forsyth, P. A. and Vetzal (1998) Penalty methods for american options with stochastic volatility. Journal of Computational and Applied Mathematics 91, 199–218.