A semilinear Black and Scholes partial differential equation for valuing American options
Based on the dynamic programming principle in optimal stopping theory the authors investigate a semilinear Black and Scholes type partial differential equation in a fixed domain for the value of an American (call/put) option. This equation is of the form \[ \partial_tv(t,x)+(r-d)x\partial_xv(t,x)+\frac{1}{2}\sigma^2x^2 \partial_x^2v(t,x)-rv(t,x)=-q(x,v(t,x)), \quad v(T,x)=g(x), \] \(x\geq0\), \(t\in[0,T]\), where \(r,d,\sigma\) are given constants and the nonlinear reaction term \(q(x,v)\) is of the form \[ q(x,v)=\begin{cases}0,&g(x)-v<0,\\ c(x),&g(x)-v\geq0,\end{cases}\quad c(x)=\begin{cases} (dx-rK)^+,&\text{call options},\\ (rK-dx)^+,&\text{put options}. \end{cases} \] The nonlinearity \(v\to q(x,v)\) in the equation depends discontinuously on the American option value, so that standard theory for partial differential equation does not apply and it is not clear what one should mean by a solution to the semilinear Black and Scholes equation. Guided by the dynamic programming principle the authors suggest a ``correct'' notion of weak solution to the equation, which is still called a viscosity solution. This notion can be viewed as an adaptation of definition of viscosity solution proposed by \textit{H. Ishii} [in: 100th Annivers. Bull. Chuo Univ., Tokyo 1985, 1--23 (1985; Zbl 0593.35062)] for a class of Hamilton-Jacobi equations with discontinuous Hamiltonians. The main results of the paper imply that there exists exactly one such viscosity solution of the semilinear Black and Scholes equation, namely the American option value.
semilinear Black and Scholes partial differential equation
scientific article; zbMATH DE number 2169530