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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
DOI: https://doi.org/10.1090/S0094-9000-2012-00863-6
Published electronically: July 31, 2012
MathSciNet review: 2857420
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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.


References [Enhancements On Off] (What's this?)

  • 1. 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)
  • 2. 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)
  • 3. 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)
  • 4. 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

DOI: https://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

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