Bulletin of Belgorod State Technological University named after. V. G. Shukhov Bulletin of Belgorod State Technological University named after. V. G. Shukhov Вестник Белгородского государственного технологического университета им. В.Г. Шухова 2071-7318 15726 10.12737/article_58e24de420f6a0.19667564 Экономические науки Economic science Экономические науки EVOLUTION OF AMERICAN OPTION VALUE FUNCTION ON A DIVIDEND PAYING STOCK UNDER JUMP-DIFFUSION PROCESSES ЭВОЛЮЦИЯ ФУНКЦИИ ЦЕНЫ АМЕРИКАНСКОГО ОПЦИОНА НА АКЦИИ С ВЫПЛАТОЙ ДИВИДЕНДОВ В МОДЕЛИ ДИФФУЗИИ СО СКАЧКАМИ Рехман Назир Rekhman Nazir Хуссейн Закир Khusseyn Zakir Али Файха Ali Faykha Бендерская О.Б. Borisovna Benderskaya Olga Borisovna

кандидат экономических наук;

candidate of economic sciences;

 Хуссейн Султан Khusseyn Sultan Зуев Сергей Валентинович Zuev Sergey Валентинович Белгородский государственный технологический университет им В.Г. Шухова Belgorod State Technological University named after V.G. Shukhov 13 03 2017 13 03 2017 2 3 212 221 https://zh-szf.ru/en/nauka/article/15726/view

Настоящая работа посвящена анализу и изменениям функции цены (премии) американского опциона на акции с выплатой дивиденда, построенной по модели диффузии со скачками. Получен и исследован эквивалентный вид функции. Кроме того, исследуются вариационные неравенства, удовлетворяющие этой функции. Полученные результаты могут быть использованы для нахождения оптимальной стратегии хеджирования и определения оптимальных границ торговли связанными опционами.

This work is devoted to the analysis and evolution of the value function of American type options on a dividend paying stock under jump diffusion processes. An equivalent form of the value function is obtained and analyzed. Moreover, variational inequalities satisfied by this function are investigated. These results can be used to investigate the optimal hedging strategies and optimal exercise boundaries of the corresponding options.

 американский опцион модель диффузии со скачками пуассоновский процесс локальная непрерывность Липшица слабые производные american option jump-diffusion model poisson process locally lipschitz continuity weak derivatives.

Black F., Scholes M. The pricing of op-tions and corporate liabilities // The Journal of Political Economy. 1973. V. 1. P. 637-654. Black F., Scholes M. The pricing of op-tions and corporate liabilities. The Journal of Political Economy. 1973. V. 1. P. 637-654. Chiarella C., Kang B. The evaluation of American compound option prices under stochastic volatility and stochastic interest rates // The Journal of Computational Finance. 2011. V.14. № 9. P. 1-21. Chiarella C., Kang B. The evaluation of American compound option prices under stochastic volatility and stochastic interest rates. The Journal of Computational Finance. 2011. V.14. № 9. P. 1-21. El Karoui N.E., Jeanblanc Picqu M., Shreve S.E. Robustness of the Black and Scholes formula // Mathematical Finance. 1998. V. 8. № 2. P. 93-126. El Karoui N.E., Jeanblanc Picqu M., Shreve S.E. Robustness of the Black and Scholes formula. Mathematical Finance. 1998. V. 8. № 2. P. 93-126. Elliot R., Kopp P. Option Pricing and Hedge Portfolio for Poisson Process // Stochastic Analysis and Applications. 1990. V. 8. P. 157-167. Elliot R., Kopp P. Option Pricing and Hedge Portfolio for Poisson Process. Stochastic Analysis and Applications. 1990. V. 8. P. 157-167. Glowinsky R., Lions J.L., Treamoliyeres R. Numerical Analysis of Variational Inequalities. Amsterdam: North-Holland Publishing Company, New York, 2011. Glowinsky R., Lions J.L., Treamoliyeres R. Numerical Analysis of Variational Inequalities. Amsterdam: North-Holland Publishing Company, New York, 2011. Hussain S., Shashiashvili M. Discrete time hedging of the American option // Mathematical Finance. 2010. V. 20. № 4. P. 647-670. Hussain S., Shashiashvili M. Discrete time hedging of the American option. Mathematical Finance. 2010. V. 20. № 4. P. 647-670. Hussain S., Rehman N. Estimate for the discrete time hedging error of the American option on a dividend paying stock // Math. Inequal. Appl. 2012. V. 15. P. 137-163. Hussain S., Rehman N. Estimate for the discrete time hedging error of the American option on a dividend paying stock. Math. Inequal. Appl. 2012. V. 15. P. 137-163. Hussain S., Rehman N. Regularity of the American Option Value Function in Jump-Diffusion Model // Journal of Computational Analysis and Applications. 2017. V. 22. P. 286-297. Hussain S., Rehman N. Regularity of the American Option Value Function in Jump-Diffusion Model. Journal of Computational Analysis and Applications. 2017. V. 22. P. 286-297. Israel V.P., Rincon M.A. Variational ine-qualities applied to option market problem // Ap-plied Mathematics and Computation. 2008. V. 201. № 1. P. 384-397. Israel V.P., Rincon M.A. Variational ine-qualities applied to option market problem. Ap-plied Mathematics and Computation. 2008. V. 201. № 1. P. 384-397. Jaillet P., Lamberton D., Lapeyre B. Variational inequalities and the pricing of American options // Acta Applicandae Mathematica. 1990. V. 21. № 3. P. 263-289. Jaillet P., Lamberton D., Lapeyre B. Variational inequalities and the pricing of American options. Acta Applicandae Mathematica. 1990. V. 21. № 3. P. 263-289. Karatzas I., Shreve S.E. Methods of mathematical finance. Springer Science and Business Media, 1998. Karatzas I., Shreve S.E. Methods of mathematical finance. Springer Science and Business Media, 1998. Lamberton D., Lapeyre B. Stochastic Calculus Applied to Finance. UK: Chapman and Hall, 1997. Lamberton D., Lapeyre B. Stochastic Calculus Applied to Finance. UK: Chapman and Hall, 1997. Leduc G. Can High-Order Convergence of European Option Prices be Achieved with Common CRR-Type Binomial Trees? // Bulletin of the Malaysian Mathematical Sciences Society. 2015. DOI 10.1007/s40840-015-0221-2. P. 1-14. Leduc G. Can High-Order Convergence of European Option Prices be Achieved with Common CRR-Type Binomial Trees?. Bulletin of the Malaysian Mathematical Sciences Society. 2015. DOI 10.1007/s40840-015-0221-2. P. 1-14. Pham H. Optimal stopping, free boundary, and American option in a jump diffusion model // Applied Mathematics and Optimization. 1997. V. 35. № 2. P.145-164. Pham H. Optimal stopping, free boundary, and American option in a jump diffusion model. Applied Mathematics and Optimization. 1997. V. 35. № 2. P.145-164. Rehman N., Hussain S., Wasim Ul-Haq. Sensitivity analysis of the optimal exercise boundary of the American put option // Georgian Mathematical Journal. 2016. V. 23. № 3. P.429-433. Rehman N., Hussain S., Wasim Ul-Haq. Sensitivity analysis of the optimal exercise boundary of the American put option. Georgian Mathematical Journal. 2016. V. 23. № 3. P.429-433. Rehman N., Shashiashvili M. The American Foreign Exchange option in Time-Dependent One-Dimensional Diffusion Model // Appl. Math Optim. 2016. V. 59. № 3. P. 329-363. Rehman N., Shashiashvili M. The American Foreign Exchange option in Time-Dependent One-Dimensional Diffusion Model. Appl. Math Optim. 2016. V. 59. № 3. P. 329-363. Royden H.L. Real analysis. New Delhi: Macmillan, Prentice-Hall of India, 1997. Royden H.L. Real analysis. New Delhi: Macmillan, Prentice-Hall of India, 1997. Shreve, S.E. Stochastic Calculus for Fi-nance. V. I. New York: Springer, 2004. Shreve, S.E. Stochastic Calculus for Fi-nance. V. I. New York: Springer, 2004. Shreve S.E. Stochastic Calculus for Fi-nance. V. II. New York: Springer, 2004. Shreve S.E. Stochastic Calculus for Fi-nance. V. II. New York: Springer, 2004. Situ R. Theory of Stochastic Differential Equation with Jumps and Applications. New York: Springer, 2012. Situ R. Theory of Stochastic Differential Equation with Jumps and Applications. New York: Springer, 2012. Zhang X.L. Numerical analysis of American option pricing in a jump diffusion model // Mathematics of Operations Research. 1997. V.22. № 3. P. 668-690. Zhang X.L. Numerical analysis of American option pricing in a jump diffusion model. Mathematics of Operations Research. 1997. V.22. № 3. P. 668-690.