Convergence of option rewards for Markov type price processes modulated by stochastic indices. II

Silvestrov, D.; Jönsson, H.; Stenberg, F.

doi:10.1090/S0094-9000-2010-00802-7

Convergence of option rewards for Markov type price processes modulated by stochastic indices. II

Authors: D. S. Silvestrov, H. Jönsson and F. Stenberg
Journal: Theor. Probability and Math. Statist. 80 (2010), 153-172
MSC (2000): Primary 60J05, 60H10; Secondary 91B28, 91B70
DOI: https://doi.org/10.1090/S0094-9000-2010-00802-7
Published electronically: August 20, 2010
MathSciNet review: 2541960
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A general price process represented by a two-component Markov process is considered. Its first component is interpreted as a price process and the second one as a stochastic index modulating the price component. American type options with pay-off functions, which admit upper bounds of a power type, are studied. Both the transition characteristics of the price processes and the pay-off functions are assumed to depend on a perturbation parameter $\delta \geq 0$ and to converge to the corresponding limit characteristics as $\delta \to 0$. In the first part of the paper, asymptotically uniform skeleton approximations connecting reward functionals for continuous and discrete time models are given. In the second part of the paper, these skeleton approximations are used for getting results about the convergence of reward functionals for American type options for perturbed price processes with discrete and continuous time. Examples related to modulated exponential price processes with independent increments are given.

References

D. S. Sīl′vestrov, Kh. Ĭonson, and F. Stenberg, Convergence of option rewards for Markov-type price processes modulated by stochastic indices. I, Teor. Ĭmovīr. Mat. Stat. 79 (2008), 138–154 (Ukrainian, with Ukrainian summary); English transl., Theory Probab. Math. Statist. 79 (2009), 153–170. MR 2494545, DOI https://doi.org/10.1090/S0094-9000-09-00776-5
Y. S. Chow, Herbert Robbins, and David Siegmund, The theory of optimal stopping, Dover Publications, Inc., New York, 1991. Corrected reprint of the 1971 original. MR 1105325

H. Jönsson, Monte Carlo studies of American type options with discrete time, Theory Stoch. Process. 7(23) (2001), no. 1–2, 163–188.

H. Jönsson, Optimal Stopping Domains and Reward Functions for Discrete Time American Type Options, Ph.D. Thesis, vol. 22, Mälardalen University, 2005.

H. Jönsson, A. G. Kukush, and D. S. Silvestrov, Threshold structure of optimal stopping strategies for American type option. I, Teor. Ĭmovīr. Mat. Stat. 71 (2004), 82–92; English transl., Theory Probab. Math. Statist. 71 (2005), 93–103. MR 2144323, DOI https://doi.org/10.1090/S0094-9000-06-00650-8

H. Jönsson, A. G. Kukush, and D. S. Silvestrov, Threshold structure of optimal stopping strategies for American type option. II, Teor. Ĭmovīr. Mat. Stat. 72 (2005), 42–53; English transl., Theory Probab. Math. Statist. 72 (2006), 47–58. MR 2168135, DOI https://doi.org/10.1090/S0094-9000-06-00663-6

R. Lundgren, D. S. Silvestrov, and A. G. Kukush, Reselling of options and convergence of option rewards, J. Numer. Appl. Math. 1(96) (2008), 149–172.

A. G. Kukush and D. S. Silvestrov, Skeleton approximation of optimal stopping strategies for American type options with continuous time, Theory Stoch. Process. 7(23) (2001), no. 1–2, 215–230.

Alexander G. Kukush and Dmitrii S. Silvestrov, Optimal pricing for American type options with discrete time, Theory Stoch. Process. 10 (2004), no. 1-2, 72–96. MR 2327852

A. N. Shiryaev, Statisticheskiĭ posledovatel′nyĭ analiz. Optimal′nye pravila ostanovki, Izdat. “Nauka”, Moscow, 1976 (Russian). Second edition, revised. MR 0445744

Dmitrii S. Silvestrov, Limit theorems for randomly stopped stochastic processes, Probability and its Applications (New York), Springer-Verlag London, Ltd., London, 2004. MR 2030998

D. Silvestrov, V. Galochkin, and A. Malyarenko, OPTAN — a pilot program system for analysis of options, Theory Stoch. Process. 7(23) (2001), no. 1–2, 291–300.

D. S. Silvestrov, V. G. Galochkin, and V. G. Sibirtsev, Algorithms and programs for optimal Monte Carlo pricing of American type options, Theory Stoch. Process. 5(21) (1999), no. 1–2, 175–187.

A. V. Skorohod, Limit theorems for stochastic processes, Teor. Veroyatnost. i Primenen. 1 (1956), 289–319 (Russian, with English summary). MR 0084897

A. V. Skorokhod, Sluchaĭ nye protsessy s nezavisimymi prirashcheniyami, 2nd ed., Teoriya Veroyatnosteĭ i Matematicheskaya Statistika. [Probability Theory and Mathematical Statistics], “Nauka”, Moscow, 1986 (Russian). MR 860563

F. Stenberg, Semi-Markov Models for Insurance and Option Rewards, Ph.D. Thesis, vol. 38, Mälardalen University, 2006.