Connection between discrete financial models and continuous models with Wiener and Poisson processes
Электронный научный архив УРФУ
Информация об архиве | Просмотр оригинала| Поле | Значение | |
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Connection between discrete financial models and continuous models with Wiener and Poisson processes
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| Автор |
Melnikova, I. V.
Bovkun, V. A. |
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| Тематика |
BINOMIAL MODEL
BROWNIAN MOTION DISCOUNTED PRICE MARTINGALE POISSON PROCESS STOCHASTIC EQUATION |
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| Описание |
The paper is devoted to the study of relationships between discrete and continuous models financial processes and their probabilistic characteristics. First, a connection is established between the price processes of stocks, hedging portfolio and options in the models conditioned by binomial perturbations and their limit perturbations of the Brownian motion type. Secondly, analogues in the coefficients of stochastic equations with various random processes, continuous and jumpwise, and in the coefficients corresponding deterministic equations for their probabilistic characteristics. Statement of the results on the connections and finding analogies, obtained in this paper, led to the need for an adequate presentation of preliminary information and results from financial mathematics, as well as descriptions of related objects of stochastic analysis. In this paper, partially new and known results are presented in an accessible form for those who are not specialists in financial mathematics and stochastic analysis, and for whom these results are important from the point of view of applications. Specifically, the following sections are presented. • In one- and n-period binomial models, it is proposed a unified approach to determining on the probability space a risk-neutral measure with which the discounted option price becomes a martingale. The resulting martingale formula for the option price is suitable for numerical simulation. In the following sections, the risk-neutral measures approach is applied to study financial processes in continuous-time models. • In continuous time, models of the price of shares, hedging portfolios and options are considered in the form of stochastic equations with the Ito integral over Brownian motion and over a compensated Poisson process. The study of the properties of these processes in this section is based on one of the central objects of stochastic analysis — the Ito formula. Special attention is given to the methods of its application. • The famous Black – Scholes formula is presented, which gives a solution to the partial differential equation for the function v(t, x), which, when x = S (t) is substituted, where S (t) is the stock price at the moment time t, gives the price of the option in the model with continuous perturbation by Brownian motion. • The analogue of the Black – Scholes formula for the case of the model with a jump-like perturbation by the Poisson process is suggested. The derivation of this formula is based on the technique of risk-neutral measures and the independence lemma. © 2023 Irina V. Melnikova, Vadim A. Bovkun This work is licensed under the Creative Commons Attribution-NoDerivs 3.0 Unported License.
Russian Science Foundation, RSF: 23–21–00199 This work was supported by Russian Science Foundation, project No. 23–21–00199. |
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| Дата |
2024-04-05T16:30:38Z
2024-04-05T16:30:38Z 2023 |
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| Тип |
Article
Journal article (info:eu-repo/semantics/article) |info:eu-repo/semantics/publishedVersion |
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| Идентификатор |
Melnikova, IV & Bovkun, VA 2023, 'Связь между дискретными финансовыми моделями и непрерывными моделями с процессами Винера и Пуассона', Computer Research and Modeling, Том. 15, № 3, стр. 781-795. https://doi.org/10.20537/2076-7633-2023-15-3-781-795
Melnikova, I. V., & Bovkun, V. A. (2023). Связь между дискретными финансовыми моделями и непрерывными моделями с процессами Винера и Пуассона. Computer Research and Modeling, 15(3), 781-795. https://doi.org/10.20537/2076-7633-2023-15-3-781-795 2076-7633 Final All Open Access, Gold https://www.scopus.com/inward/record.uri?eid=2-s2.0-85166775703&doi=10.20537%2f2076-7633-2023-15-3-781-795&partnerID=40&md5=2d58cefdb4e5ceaa1cead420da57e18e http://crm.ics.org.ru/uploads/crmissues/crm_2023_03/52_melnikova.pdf http://elar.urfu.ru/handle/10995/130698 54284817 10.20537/2076-7633-2023-15-3-781-795 85166775703 |
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| Язык |
ru
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| Связанные ресурсы |
info:eu-repo/grantAgreement/RSF//23-21-00199
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| Права |
Open access (info:eu-repo/semantics/openAccess)
cc-by-nd https://creativecommons.org/licenses/by-nd/4.0/ |
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| Формат |
application/pdf
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| Издатель |
Institute of Computer Science Izhevsk
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| Источник |
Computer Research and Modeling
Computer Research and Modeling |
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