Journal of Applied Mathematics
Volume 2 (2002), Issue 3, Pages 131-139
doi:10.1155/S1110757X02111041
Semigroup theory applied to options
Department of Applied Economics, University of La Laguna, Campus de Guajara, s/n, Edificio de Económicas - Empresariales, (Tenerife), La Laguna 38071, Spain
Received 5 November 2001; Revised 5 March 2002
Copyright © 2002 D. I. Cruz-Báez and J. M. González-Rodríguez. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
Black and Scholes (1973) proved that under certain assumptions about the market place, the value of a European option, as a
function of the current value of the underlying asset and time, verifies a Cauchy problem. We give new conditions for the existence and uniqueness of the value of a European option by using semigroup theory. For this, we choose a suitable space that verifies some conditions, what allows us that the operator that appears in the Cauchy problem is the infinitesimal generator of a C0-semigroup T(t). Then we are able to guarantee the existence and uniqueness of the value of a European option and we also achieve an explicit expression of that value.