A semilinear Black and Scholes partial differential equation for valuing American options
type: Item
instance of: scholarly article
title: A semilinear Black and Scholes partial differential equation for valuing American options
zbMATH Open document ID: 1064.60080
author: Kristin Reikvam
author: Fred Espen Benth
author: Kenneth H. Karlsen
published in: Finance and Stochastics
publication date: 2005-05-20T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 91B28
Mathematics Subject Classification ID: 60G40
Mathematics Subject Classification ID: 49L25
Mathematics Subject Classification ID: 91B24
zbMATH DE Number: 2169530
zbMATH Keywords: American options
zbMATH Keywords: semilinear Black and Scholes partial differential equation
zbMATH Keywords: viscosity solution
zbMATH Keywords: existence
zbMATH Keywords: comparison result
zbMATH Keywords: uniqueness
reviewed by: M. P. Moklyachuk
MaRDI profile type: Publication
full work available at URL: s007800200091
OpenAlex ID: W2020885938
Wikidata QID: Q115385487
DOI: 10.1007/S007800200091
description: scientific article; zbMATH DE number 2169530