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Convergence with respect to the parameter of a series and the differentiability of barrier option prices with respect to the barrier


Authors: O. M. Kulik, Yu. S. Mishura and O. M. Soloveĭko
Translated by: S. Kvasko
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 81 (2010).
Journal: Theor. Probability and Math. Statist. 81 (2010), 117-130
MSC (2000): Primary 91B28; Secondary 60F17, 60G15, 60H07
DOI: https://doi.org/10.1090/S0094-9000-2011-00814-9
Published electronically: January 20, 2011
MathSciNet review: 2667314
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Abstract | References | Similar Articles | Additional Information

Abstract: We obtain the weak convergence of measures generated by the price process and prove the continuity of the price of a barrier call option with respect to the parameter of a series for the Black-Scholes model of a complete market. The explicit form of the price of the barrier option is not required. The result obtained allows one to prove the continuity of a solution of the corresponding boundary-value problem for the parabolic partial differential equation with respect to the parameter of a series. Applying the Malliavin calculus, we establish the existence of a bounded continuous density of the distribution of a Wiener integral with shift restricted to an arbitrary ``positive'' ray and prove the differentiability of the fair price with respect to the barrier (the differentiability with respect to other parameters is a classical result).


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Additional Information

O. M. Kulik
Affiliation: Institute of Mathematics, National Academy of Sciences of Ukraine, Tereshchenkivs’ka Street 3, Kyiv 01601, Ukraine
Email: kulik@imath.kiev.ua

Yu. S. Mishura
Affiliation: Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue 2, Kiev 03127, Ukraine
Email: myus@univ.kiev.ua

O. M. Soloveĭko
Affiliation: Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue 2, Kiev 03127, Ukraine
Email: osoloveyko@univ.kiev.ua

DOI: https://doi.org/10.1090/S0094-9000-2011-00814-9
Keywords: Barrier call option, Black–Scholes model, weak convergence of measures, boundary-value problem for a parabolic equation, Malliavin calculus, differentiability of the price with respect to the barrier
Received by editor(s): September 1, 2009
Published electronically: January 20, 2011
Article copyright: © Copyright 2011 American Mathematical Society

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