International Journal of Mathematics and Mathematical Sciences
Volume 27 (2001), Issue 3, Pages 177-188
Asymptotic analysis of American call options
Department of Applied Mathematics, University of Western Ontario, Ontario, London N6A 5B7, Canada
Received 7 August 2000
Copyright © 2001 Ghada Alobaidi and Roland Mallier. 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.
American call options are financial derivatives that give the holder the right but not the obligation to buy an underlying security at a pre-determined price. They differ from European
options in that they may be exercised at any time prior to their expiration, rather than only at expiration. Their value is described by the Black-Scholes PDE together with a constraint that
arises from the possibility of early exercise. This leads to a free boundary problem for the optimal exercise boundary, which determines whether or not it is beneficial for the holder to
exercise the option prior to expiration. However, an exact solution cannot be found, and therefore by using asymptotic techniques employed in the study of boundary layers in fluid
mechanics, we find an asymptotic expression for the location of the optimal exercise boundary and the value of the option near to expiration.