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

Full-text PDF Free Access

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