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
Journal:
Theor. Probability and Math. Statist. 84 (2012), 99-106
MSC (2010):
Primary 60G44, 60F05, 60B12
DOI:
https://doi.org/10.1090/S0094-9000-2012-00863-6
Published electronically:
July 31, 2012
MathSciNet review:
2857420
Full-text PDF Free Access
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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.
References
- 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, DOI https://doi.org/10.1090/S0094-9000-2011-00814-9
- 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, DOI https://doi.org/10.1090/S0094-9000-2011-00815-0
- 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, DOI https://doi.org/10.1090/S0094-9000-2011-00829-0
- Patrick Billingsley, Convergence of probability measures, John Wiley & Sons, Inc., New York-London-Sydney, 1968. MR 0233396
References
- O. M. Kulik, Yu. S. Mishura, and O. M. Soloveiko, Convergence with respect to the parameter of a series and the differentiability of barrier option prices with respect to the barrier, Teor. Imovir. ta Matem. Statyst. 81 (2009), 102–113; English transl. in Theor. Probability and Math. Statist. 81 (2010), 117–130. MR 2667314 (2011f:91175)
- Yu. S. Mishura, G. M. Shevchenko, and Yu. V. Yukhnovs’kii, Functional limit theorems for stochastic integrals with applications to risk processes and to self-financing strategies in a multidimensional market. I, Teor. Imovir. ta Matem. Statyst. 81 (2009), 114–127; English transl. in Theor. Probability and Math. Statist. 81 (2010), 131–146. MR 2667315 (2011e:60069)
- Yu. S. Mishura and Yu. V. Yukhnovs’kii, Functional limit theorems for stochastic integrals with applications to risk processes and to value processes of self-financing strategies in a multidimensional market. II, Teor. Imovir. ta Matem. Statyst. 82 (2010), 92–103; English transl. in Theor. Probability and Math. Statist. 82 (2011), 87–101. MR 2790485 (2011m:60095)
- P. Billingsley, Convergence of Probability Measures, John Wiley & Sons, 1968. MR 0233396 (38:1718)
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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
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
