IJPAM: Volume 61, No. 2 (2010)

ANALYTIC SOLUTION OF A NONLINEAR
BLACK-SCHOLES PARTIAL DIFFERENTIAL EQUATION

Joseph Esekon$^1$, Silas Onyango$^2$, Naftali Omolo-Ongati$^3$
$^{1,3}$Department of Mathematics and Applied Statistics
Maseno University
P.O. Box 333, Maseno, KENYA
$^1$e-mail: [email protected]
$^3$e-mail: [email protected]
$^2$Research Services Office
Strath more University
P. O. Box 59857, City Square Nairobi, 00200, KENYA
e-mail: [email protected] more.edu


Abstract.We study a nonlinear Black-Scholes partial differential equation whose nonlinearity is as a result of a feedback effect. This is an illiquid market effect arising from transaction costs. An analytic solution to the nonlinear Black-Scholes equation via a solitary wave solution is currently unknown. After transforming the equation into a parabolic nonlinear porous medium equation, we find that the assumption of a traveling wave profile to the later equation reduces it to ordinary differential equations. This together with the use of localizing boundary conditions facilitate a twice continuously differentiable nontrivial analytic solution by integrating directly.

Received: May 10, 2010

AMS Subject Classification: 35K10, 35K61

Key Words and Phrases: nonlinear black-scholes equation, option hedging, volatility, illiquid markets, transaction cost, analytic solution

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2010
Volume: 61
Issue: 2