A semilinear Black and Scholes partial differential equation for valuing American options

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 the 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].