Geodesic
A semilinear Black and Scholes partial differential equation for valuing American options: approximate solutions and convergence
F. E. Benth1 ; Kenneth Hvistendahl Karlsen1 ; K. Reikvam1
1University of Oslo, Norway
Interfaces and free boundaries, Tome 6 (2004) no. 4, pp. 379-404

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DOI

In [7], we proved that the American (call/put) option valuation problem can be stated in terms of one single semilinear Black and Scholes partial differential equation set in a fixed domain. The semilinear Black and Scholes equation constitutes a starting point for designing and analyzing a variety of “easy to implement” numerical schemes for computing the value of an American option. To demonstrate this feature, we propose and analyze an upwind finite difference scheme of “predictor-corrector type” for the semilinear Black and Scholes equation. We prove that the approximate solutions generated by the predictor-corrector scheme respect the early exercise constraint and that they converge uniformly to the the American option value. A numerical example is also presented. Besides the predictor-corrector schemes, other methods for constructing approximate solution sequences are discussed and analyzed as well.
DOI : 10.4171/ifb/106
Classification : 35-XX, 65-XX, 76-XX, 92-XX
Mots-clés : American option, semilinear Black and Scholes partial differential equation, viscosity solution, approximate solutions, numerical schemes, convergence

F. E. Benth  1   ; Kenneth Hvistendahl Karlsen  1   ; K. Reikvam  1

1 University of Oslo, Norway 
F. E. Benth; Kenneth Hvistendahl Karlsen; K. Reikvam. A semilinear Black and Scholes partial differential equation for valuing American options: approximate solutions and convergence. Interfaces and free boundaries, Tome 6 (2004) no. 4, pp. 379-404. doi: 10.4171/ifb/106
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     title = {A semilinear {Black} and {Scholes} partial differential equation for valuing {American} options: approximate solutions and convergence},
     journal = {Interfaces and free boundaries},
     pages = {379--404},
     year = {2004},
     volume = {6},
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     doi = {10.4171/ifb/106},
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