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), 117130
MSC (2000):
Primary 91B28; Secondary 60F17, 60G15, 60H07
Published electronically:
January 20, 2011
MathSciNet review:
2667314
Fulltext PDF
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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 BlackScholes 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 boundaryvalue 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:
http://dx.doi.org/10.1090/S009490002011008149
Keywords:
Barrier call option,
Black–Scholes model,
weak convergence of measures,
boundaryvalue 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
