MPS-RR 2001-29

September 2001

# On a Semilinear Black and Scholes Partial Differential Equation for Valuing American Options.

## Part I: Viscosity Solutions and Well-Posedness

by:

### Fred E. Benth, Kenneth H. Karlsen, and Kristin Reikvam

Using the dynamic programming principle in optimal stopping theory, we
derive a semilinear Black and Scholes type partial differential equation set in
a fixed domain for the value of an American (call/put) option. The
nonlinearity in the semilinear Black and Scholes equation depends
discontinuously on the American option value, so that standard theory for
partial differential equation does not apply. In fact, it is not clear what one
should mean by a solution to the semilinear Black and Scholes equation.
Guided by the dynamic programming principle, we suggest an appropriate
definition of a viscosity solution. Our main results imply that there exists
exactly one such viscosity solution of a semilinear Black and Scholes
equation, namely the American option value. In other words, we provide
herein a new formulation of the American option valuation problem. Our
formulation constitutes a starting point for designing and analyzing "easy
to implement" numerical algorithms for computing the value of an
American option. The numerical aspects of the semilinear Black and Scholes
equation are addressed in [7].

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This paper has now been published in *Finance & Stochastics 7, 277-298 (2003)*