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
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
Full-text PDF Free Access
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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).
References
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References
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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
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
