DIFFERENtIAl & DIFFERENCE EqUAtIONS ANd APPlICAtIONS

THE SEMILINEAR EVOLUTION EQUATION FOR UNIVERSAL
CONTINGENT CLAIMS: EXAMPLES AND APPLICATIONS
VALERY A. KHOLODNYI
We present and further develop the concept of a universal contingent claim introduced by
the author in 1995. This concept provides a unified framework for the analysis of a wide
class of financial derivatives. Those, for example, include Bermudan and American contingent
claims. We also show that the value of a universal contingent claim is determined,
generally speaking, by an impulsive semilinear evolution equation also introduced by the
author in 1995.
Copyright © 2006 Valery A. Kholodnyi. 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.
1. Introduction
We present and further develop the concept of a universal contingent claim introduced
by the author in [4, 5, 8, 9]. This concept provides a unified framework for the analysis of
a wide class of financial derivatives in a general market environment. Those, for example,
include Bermudan and American contingent claims. We also show that the value of a
universal contingent claim is determined, generally speaking, by an impulsive semilinear
evolution equation introduced by the author in [4–6, 8, 9].
2. Market environment
In this section we present the framework of a market environment in which contingent
claims are being priced that was introduced by the author in [4, 5, 8]. For the sake of
financial clarity, but without loss of generality, we only consider the case of a single underlying
security throughout the article. Consider an economy without transaction costs
in which trading is allowed at any time in a trading time set ,anarbitrarysubsetofthe
real numbers R. Denote by s t > 0 the unit price of the (only) underlying security at time t
in . Whenever ambiguity is unlikely, we will write s in place of s t . Denote by Π the vector
space of all real-valued functions on the set of positive real numbers R ++ . Equipped with
the partial order generated by the nonnegative cone Π + consisting of all nonnegative realvalued
functions on R ++ , Π is a partially ordered vector space. Moreover, equipped with
Hindawi Publishing Corporation
Proceedings of the Conference on Differential & Difference Equations and Applications, pp. 519–528