Nonlinear Volterra Integral equation for the Early Exercise Boundary

Author: Malkhaz Shashiashvili
Co-authors: Besarion Dochviri, Giorgi Lominashvili
Keywords: Volterra Integral Equation, The Early Exercise Boundary for the American Option
Annotation:

Nonlinear Volterra Integral Equation for the Early Exercise Boundary Malkhaz Shashiashvili ^a, Besarion Dochviri ^b, and Giorgi Lominashvili ^c, e-mail: malkhaz.shashiashvili.tsu.ge ^a, Department of Mathematics, Faculty of Exact and Natural Sciences, ^b, Ivane Javakhishvili Tbilisi State University, Chavchavadze av.1, Tbilisi. ^c, Department of Mathematics, Faculty of Exact and Natural Sciences, Akaki Tsereteli State University, 59 Tamar Mepe Street, Kutaisi. Resume. Peskir [1] has proved the uniqueness of the solution of the nonlinear Volterra integral equation satisfied by the early exercise boundary of the American put option in the class of continuous increasing functions. He considered only the classical case of Black-Sholes model of asset prices. In this report we consider the general case of one-dimensional diffusion model with local volatility depending on the current time and the current stock price. We establish the uniqueness of the solution of the corresponding nonlinear Volterra integral equation in the class of all continuous functions ( and not only in the class of continuous functions of bounded variation). References [1] G. Peskir, On the American Option Problem, Math.Finance, Vol.15, No.1,pp. 169-181, 2005.

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