IJPAM: Volume 76, No. 2 (2012)

THE BLACK-SCHOLES OPERATOR AS THE GENERATOR
OF A $C_0$-SEMIGROUP AND APPLICATIONS

O. González-Gaxiola$^1$, José A. Santiago$^2$
$^{1,2}$Department of Applied Mathematics and Systems
UAM-Cuajimalpa
Artificios, 40, México, D.F. 01120, MEXICO


Abstract. Considering the Black-Scholes equation as a Cauchy problem for different options: european, asian and vanilla, in this work we see that the second order differential operator

\begin{displaymath}\hat{A}u=ax^{2}D^{2}u+bxDu-bu, \quad a,b\in\mathbb{R}\end{displaymath}

on the Schwartz space $S_{0}(0,\infty)$, generates a $C_{0}$-semigroup. We also find the corresponding solution as some class of evolution equation.

Received: August 24, 2011

AMS Subject Classification: 47D06, 47D60, 91G80

Key Words and Phrases: black-scholes equation, differential operator, $C_{0}$-semigroups, Vanilla option, Asian option.

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Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2012
Volume: 76
Issue: 2