Limit behavior of the prices of a barrier option in the Black-Scholes model with random drift and volatility

Authors:
Yu. S. Mishura and Yu. V. Yukhnovs’kii

Translated by:
N. Semenov

Original publication:
Teoriya Imovirnostei ta Matematichna Statistika, tom **84** (2011).

Journal:
Theor. Probability and Math. Statist. **84** (2012), 99-106

MSC (2010):
Primary 60G44, 60F05, 60B12

Published electronically:
July 31, 2012

MathSciNet review:
2857420

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A generalized Black-Scholes model with random drift and volatility dependent on a parameter is studied in the paper. Sufficient conditions for the convergence of a sequence of prices of a European barrier option are established.

**1.**O. M. Kulik, Yu. S. Mīshura, and O. M. Soloveĭko,*Convergence with respect to the parameter of a series and the differentiability of barrier option prices with respect to the barrier*, Teor. Ĭmovīr. Mat. Stat.**81**(2009), 102–113 (Ukrainian, with English, Russian and Ukrainian summaries); English transl., Theory Probab. Math. Statist.**81**(2010), 117–130. MR**2667314**, 10.1090/S0094-9000-2011-00814-9**2.**Yu. S. Mīshura, G. M. Shevchenko, and Yu. V. Yukhnovs′kiĭ,*Functional limit theorems for stochastic integrals with applications to risk processes and to capital of self-financing strategies in a multidimensional market. I*, Teor. Ĭmovīr. Mat. Stat.**81**(2009), 114–127 (Ukrainian, with English, Russian and Ukrainian summaries); English transl., Theory Probab. Math. Statist.**81**(2010), 131–146. MR**2667315**, 10.1090/S0094-9000-2011-00815-0**3.**Yu. S. Mīshura and Yu. V. Yukhnovs′kiĭ,*Functional limit theorems for stochastic integrals with applications to risk processes and to capital of self-financing strategies in a multidimensional market. II*, Teor. Ĭmovīr. Mat. Stat.**82**(2010), 92–103 (Ukrainian, with English, Russian and Ukrainian summaries); English transl., Theory Probab. Math. Statist.**82**(2011), 87–101. MR**2790485**, 10.1090/S0094-9000-2011-00829-0**4.**Patrick Billingsley,*Convergence of probability measures*, John Wiley & Sons, Inc., New York-London-Sydney, 1968. MR**0233396**

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Additional Information

**Yu. S. Mishura**

Affiliation:
Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue 4E, Kiev 03127, Ukraine

Email:
myus{@}univ.kiev.ua

**Yu. V. Yukhnovs’kii**

Affiliation:
Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue 4E, Kiev 03127, Ukraine

Email:
Yuhnovskiy{@}hq.eximb.com

DOI:
http://dx.doi.org/10.1090/S0094-9000-2012-00863-6

Keywords:
Stochastic integrals,
functional limit theorems,
weak convergence,
semimartingales,
barrier options

Received by editor(s):
January 10, 2011

Published electronically:
July 31, 2012

Article copyright:
© Copyright 2012
American Mathematical Society