МОДЕЛЬ ГЕСТОНА ІЗ СТОХАСТИЧНОЮ ВІДСОТКОВОЮ СТАВКОЮ
Ключові слова:
стохастичні рівняння, броунівський рух, процес Кокса-Інгерсолла-Росса, ціна опціону, модель Блека-Шоулза, модель Гестона, модель Гестона-КІР
Анотація
Розглянуто гібридну модель Гестона, що враховує стохастичну ди-наміку відсоткової ставки, яка задана процесом Кокса-Інгерсолла-Росса (КІР). Проаналізовано випадок точного розв’язку моделі Гестона-КІР, а саме за відсутності кореляції процесу Вінера відсоткової ставки з процеса-ми Вінера класичної моделі Гестона. Використано розв’язок моделі Гесто-на-КІР для ціни європейського опціону кол, який за структурою відповідає класичній моделі Гестона. Наведено формулу ціни опціону у формі зруч-ній для аналітичного та чисельного аналізу. Здійснено чисельний аналіз впливу стохастичної динаміки відсоткової ставки на ціну європейського опціону кол порівняно з класичною моделлю Гестона. Для заданого набо-ру параметрів показано, що в моделі Гестона-КІР для тривалих опціонів ціна помітно відрізняється від такої ж знайденою за класичною моделлю Гестона.
Посилання
1. S. L. Heston. A closed-form solution for options with stochastic volatility with applications to bond and currency options. Review of Financial Studies. 1993. Vol. 6, No. 2, P. 327–343.
2. R. Schöbel, J. Zhu. Stochastic volatility with an Ornstein–Uhlenbeck process: An extension. European Finance Review. 1999. Vol. 3 (1), P. 23–46.
3. Black F., Scholes M. The pricing of options and corporate liabilities. Journal of Political Economy. 1973. Vol. 81, No. 3, P. 637-654.
4. Hull J., White A. The pricing of options on assets with stochastic volatilities. Journal of Finance. 1987. Vol. 42, No. 2, P. 281-300.
5. Long Teng, Matthias Ehrhardt, Michael Günther. On the Heston model with stochastic correlation. International Journal of Theoretical and Applied Finance. 2016. Vol. 19, No. 6, P. 1-25.
6. Rebonato R. Volatility and Correlation. The Perfect Hedger and the Fox. John Wiley & Sons, 2nd Edition, 2004. 864 p.
7. Stein E. M., Stein J. C. Stock price distributions with stochastic volatility: An analytic approach. Review of Financial Studies. 1991. Vol. 4, No. 4, P. 727-752.
8. Grzelak L. A., Oosterlee C.W., Van Weeren S. The affine Heston model with correlated Gaussian interest rates for pricing hybrid derivatives. Quantitative Finance. 2011. Vol. 11, P. 1647-1663.
9. Grzelak L. A., Oosterlee C. W., Van Weeren S. Extension of stochastic volatility equity models with the Hull-White interest rate process. Quantitative Finance. 2012. Vol. 12, P. 89-105.
10. Guo S., Grzelak L. A., Oosterlee C. W. Analysis of an affine version of the Heston-Hull-White option pricing partial differential equation. Applied Numerical Mathematics. 2013. Vol. 72, P.143-159.
11. Van Haastrecht A., Lord R., Pelsser A., Schrager D. Pricing long-dated insurance contracts with stochastic interest rates and stochastic volatility. Insurance: Mathematics and Economics. 2009. Vol. 45(3), P. 436-448.
12. Cox J.C., Ingersoll J.E., Ross S.A. An intertemporal general equilibrium model of asset prices. Econometrica. 1985. Vol. 53, No. 2, P. 363-384.
13. Hull J., White A. Pricing interest-rate-derivative securities. Review of Financial Studies. 1990. Vol. 3, No. 4, P. 573-592.
14. Vasicek O. An equilibrium characterization of the term structure. Journal of Financial Economics. 1977. Vol. 5 (2), P. 177-188.
15. Sippel J., Ohkoshi S. All power to PRDC notes. Risk Magazine. 2002. Vol. 15(11), P. 1-3.
16. Cox J. C., Ingersoll J. E., Ross S. A. A re-examination of traditional hypotheses about the term structure of interest rates. Journal of Finance. 1981. Vol. 36 (4), P. 769–799.
17. Grzelak L. A., Oosterlee C. W. On the Heston model with stochastic interest rates. SIAM Journal on Financial Mathematics. 2011. Vol. 2 (1), P. 255-286.
18. Ahlip R., Rutkowski M. Pricing of foreign exchange options under the Heston stochastic volatility model and CIR interest rates. Quantitative Finance. 2013. Vol. 13 (6), P. 955-966.
19. Grzelak L. A., Oosterlee C. W. An equity-interest rate hybrid model with stochastic volatility and the interest rate smile. The Journal of Computational Finance. 2012. Vol. 15, No. 4, P. 12-32.
20. Grzelak L. A., C. W. Oosterlee. On cross-currency models with stochastic volatility and correlated interest rates. Applied Mathematical Finance. 2012. Vol. 19 (1), P. 1-35.
21. A. van Haastrecht, A. Pelsser. Generic pricing of FX, inflation and stock options under stochastic interest rates and stochastic volatility. Quantitative Finance. 2011. Vol. 11(5), P. 665-691.
22. S. Fallah, A. Najafi, F. Mehrdoust. A fractional version of the Cox-Ingersoll-Ross interest rate model and pricing double barrier option with Hurst index. Communications in Statistics – Theory and Methods. 2018. Vol. 48(9), P. 1-16.
23. M. Abudy, Y. Izhakian. Pricing stock options with stochastic interest rate. International Journal of Portfolio Analysis and Manage-ment. 2013. Vol. 1(3), P. 250–277.
24. K. Rindell. Pricing of index options when interest rates are stochastic: An empirical test. Journal of Banking and Finance. 1995. Vol. 19 (5), P. 785–802.
25. J. C. Hull, A. D. White. Using Hull-White interest rate trees. Journal of Derivatives. 1996. Vol. 3 (3), P. 26–36.
26. Yanishevskyi V. S. Path Integral Solutions for Extended Heston Models. Mathematical Modeling and Computing. 2025. Vol. 12, No. 4, P. 1341–1356.
27. Xin-Jiang He, Song-Ping Zhu. A closed-form pricing formula for European options under the Heston model with stochastic interest rate. Journal of Computational and Applied Mathematics. 2018. Vol. 335, P. 323-333.
28. Янішевський В.С. Ціна опціону в розширеній моделі Гестона. Економіка та суспільство. 2025. No 79.
URL: https://economyandsociety.in.ua/index.php/journal/article/view/6796, DOI: https://doi.org/10.32782/2524-0072/2025-79-139
1. S. L. Heston. A closed-form solution for options with stochastic volatility with applications to bond and currency options. Review of Financial Studies. 1993. Vol. 6, No. 2, P. 327–343.
2. R. Schöbel, J. Zhu. Stochastic volatility with an Ornstein–Uhlenbeck process: An extension. European Finance Review. 1999. Vol. 3 (1), P. 23–46.
3. Black F., Scholes M. The pricing of options and corporate liabilities. Journal of Political Economy. 1973. Vol. 81, No. 3, P. 637-654.
4. Hull J., White A. The pricing of options on assets with stochastic volatilities. Journal of Finance. 1987. Vol. 42, No. 2, P. 281-300.
5. Long Teng, Matthias Ehrhardt, Michael Günther. On the Heston model with stochastic correlation. International Journal of Theoretical and Applied Finance. 2016. Vol. 19, No. 6, P. 1-25.
6. Rebonato R. Volatility and Correlation. The Perfect Hedger and the Fox. John Wiley & Sons, 2nd Edition, 2004. 864 p.
7. Stein E. M., Stein J. C. Stock price distributions with stochastic volatility: An analytic approach. Review of Financial Studies. 1991. Vol. 4, No. 4, P. 727-752.
8. Grzelak L. A., Oosterlee C.W., Van Weeren S. The affine Heston model with correlated Gaussian interest rates for pricing hybrid derivatives. Quantitative Finance. 2011. Vol. 11, P. 1647-1663.
9. Grzelak L. A., Oosterlee C. W., Van Weeren S. Extension of stochastic volatility equity models with the Hull-White interest rate process. Quantitative Finance. 2012. Vol. 12, P. 89-105.
10. Guo S., Grzelak L. A., Oosterlee C. W. Analysis of an affine version of the Heston-Hull-White option pricing partial differential equation. Applied Numerical Mathematics. 2013. Vol. 72, P.143-159.
11. Van Haastrecht A., Lord R., Pelsser A., Schrager D. Pricing long-dated insurance contracts with stochastic interest rates and stochastic volatility. Insurance: Mathematics and Economics. 2009. Vol. 45 (3), P. 436-448.
12. Cox J.C., Ingersoll J.E., Ross S.A. An intertemporal general equilibrium model of asset prices. Econometrica. 1985. Vol. 53, No. 2, P. 363-384.
13. Hull J., White A. Pricing interest-rate-derivative securities. Review of Financial Studies. 1990. Vol. 3, No. 4, P. 573-592.
14. Vasicek O. An equilibrium characterization of the term structure. Journal of Financial Economics. 1977. Vol. 5(2), P. 177-188.
15. Sippel J., Ohkoshi S. All power to PRDC notes. Risk Magazine. 2002. Vol. 15 (11), P. 1-3.
16. Cox J. C., Ingersoll J. E., Ross S. A. A re-examination of traditional hypotheses about the term structure of interest rates. Journal of Finance. 1981. Vol. 36 (4), P. 769–799.
17. Grzelak L. A., Oosterlee C. W. On the Heston model with stochastic interest rates. SIAM Journal on Financial Mathematics. 2011. Vol. 2 (1), P. 255-286.
18. Ahlip R., Rutkowski M. Pricing of foreign exchange options under the Heston stochastic volatility model and CIR interest rates. Quantitative Finance. 2013. Vol. 13(6), P. 955-966.
19. Grzelak L. A., Oosterlee C. W. An equity-interest rate hybrid model with stochastic volatility and the interest rate smile. The Journal of Computational Finance. 2012. Vol. 15, No. 4, P. 12-32.
20. Grzelak L. A., C. W. Oosterlee. On cross-currency models with stochastic volatility and correlated interest rates. Applied Mathematical Finance. 2012. Vol. 19 (1), P. 1-35.
21. A. van Haastrecht, A. Pelsser. Generic pricing of FX, inflation and stock options under stochastic interest rates and stochastic volatility. Quantitative Finance. 2011. Vol. 11(5), P. 665-691.
22. S. Fallah, A. Najafi, F. Mehrdoust. A fractional version of the Cox-Ingersoll-Ross interest rate model and pricing double barrier option with Hurst index. Communications in Statistics – Theory and Methods. 2018. Vol. 48 (9), P. 1-16.
23. M. Abudy, Y. Izhakian. Pricing stock options with stochastic interest rate. International Journal of Portfolio Analysis and Manage-ment. 2013. Vol. 1 (3), P. 250–277.
24. K. Rindell. Pricing of index options when interest rates are stochastic: An empirical test. Journal of Banking and Finance. 1995. Vol. 19 (5), P. 785–802.
25. J. C. Hull, A. D. White. Using Hull-White interest rate trees. Journal of Derivatives. 1996. Vol. 3 (3), P. 26–36.
26. Yanishevskyi V. S. Path Integral Solutions for Extended Heston Models. Mathematical Modeling and Computing. 2025. Vol. 12, No. 4, P. 1341–1356.
27. Xin-Jiang He, Song-Ping Zhu. A closed-form pricing formula for European options under the Heston model with stochastic interest rate. Journal of Computational and Applied Mathematics. 2018. Vol. 335, P. 323-333.
28. Yanishevsʹkyy V.S. (2025) Tsina optsionu v rozshyreniy modeli Hestona. [Option price in the extended Heston model]. Ekonomika ta suspilʹstvo, no 79.
URL:https://economyandsociety.in.ua/index.php/journal/article/view/6796, DOI: https://doi.org/10.32782/2524-0072/2025-79-139
2. R. Schöbel, J. Zhu. Stochastic volatility with an Ornstein–Uhlenbeck process: An extension. European Finance Review. 1999. Vol. 3 (1), P. 23–46.
3. Black F., Scholes M. The pricing of options and corporate liabilities. Journal of Political Economy. 1973. Vol. 81, No. 3, P. 637-654.
4. Hull J., White A. The pricing of options on assets with stochastic volatilities. Journal of Finance. 1987. Vol. 42, No. 2, P. 281-300.
5. Long Teng, Matthias Ehrhardt, Michael Günther. On the Heston model with stochastic correlation. International Journal of Theoretical and Applied Finance. 2016. Vol. 19, No. 6, P. 1-25.
6. Rebonato R. Volatility and Correlation. The Perfect Hedger and the Fox. John Wiley & Sons, 2nd Edition, 2004. 864 p.
7. Stein E. M., Stein J. C. Stock price distributions with stochastic volatility: An analytic approach. Review of Financial Studies. 1991. Vol. 4, No. 4, P. 727-752.
8. Grzelak L. A., Oosterlee C.W., Van Weeren S. The affine Heston model with correlated Gaussian interest rates for pricing hybrid derivatives. Quantitative Finance. 2011. Vol. 11, P. 1647-1663.
9. Grzelak L. A., Oosterlee C. W., Van Weeren S. Extension of stochastic volatility equity models with the Hull-White interest rate process. Quantitative Finance. 2012. Vol. 12, P. 89-105.
10. Guo S., Grzelak L. A., Oosterlee C. W. Analysis of an affine version of the Heston-Hull-White option pricing partial differential equation. Applied Numerical Mathematics. 2013. Vol. 72, P.143-159.
11. Van Haastrecht A., Lord R., Pelsser A., Schrager D. Pricing long-dated insurance contracts with stochastic interest rates and stochastic volatility. Insurance: Mathematics and Economics. 2009. Vol. 45(3), P. 436-448.
12. Cox J.C., Ingersoll J.E., Ross S.A. An intertemporal general equilibrium model of asset prices. Econometrica. 1985. Vol. 53, No. 2, P. 363-384.
13. Hull J., White A. Pricing interest-rate-derivative securities. Review of Financial Studies. 1990. Vol. 3, No. 4, P. 573-592.
14. Vasicek O. An equilibrium characterization of the term structure. Journal of Financial Economics. 1977. Vol. 5 (2), P. 177-188.
15. Sippel J., Ohkoshi S. All power to PRDC notes. Risk Magazine. 2002. Vol. 15(11), P. 1-3.
16. Cox J. C., Ingersoll J. E., Ross S. A. A re-examination of traditional hypotheses about the term structure of interest rates. Journal of Finance. 1981. Vol. 36 (4), P. 769–799.
17. Grzelak L. A., Oosterlee C. W. On the Heston model with stochastic interest rates. SIAM Journal on Financial Mathematics. 2011. Vol. 2 (1), P. 255-286.
18. Ahlip R., Rutkowski M. Pricing of foreign exchange options under the Heston stochastic volatility model and CIR interest rates. Quantitative Finance. 2013. Vol. 13 (6), P. 955-966.
19. Grzelak L. A., Oosterlee C. W. An equity-interest rate hybrid model with stochastic volatility and the interest rate smile. The Journal of Computational Finance. 2012. Vol. 15, No. 4, P. 12-32.
20. Grzelak L. A., C. W. Oosterlee. On cross-currency models with stochastic volatility and correlated interest rates. Applied Mathematical Finance. 2012. Vol. 19 (1), P. 1-35.
21. A. van Haastrecht, A. Pelsser. Generic pricing of FX, inflation and stock options under stochastic interest rates and stochastic volatility. Quantitative Finance. 2011. Vol. 11(5), P. 665-691.
22. S. Fallah, A. Najafi, F. Mehrdoust. A fractional version of the Cox-Ingersoll-Ross interest rate model and pricing double barrier option with Hurst index. Communications in Statistics – Theory and Methods. 2018. Vol. 48(9), P. 1-16.
23. M. Abudy, Y. Izhakian. Pricing stock options with stochastic interest rate. International Journal of Portfolio Analysis and Manage-ment. 2013. Vol. 1(3), P. 250–277.
24. K. Rindell. Pricing of index options when interest rates are stochastic: An empirical test. Journal of Banking and Finance. 1995. Vol. 19 (5), P. 785–802.
25. J. C. Hull, A. D. White. Using Hull-White interest rate trees. Journal of Derivatives. 1996. Vol. 3 (3), P. 26–36.
26. Yanishevskyi V. S. Path Integral Solutions for Extended Heston Models. Mathematical Modeling and Computing. 2025. Vol. 12, No. 4, P. 1341–1356.
27. Xin-Jiang He, Song-Ping Zhu. A closed-form pricing formula for European options under the Heston model with stochastic interest rate. Journal of Computational and Applied Mathematics. 2018. Vol. 335, P. 323-333.
28. Янішевський В.С. Ціна опціону в розширеній моделі Гестона. Економіка та суспільство. 2025. No 79.
URL: https://economyandsociety.in.ua/index.php/journal/article/view/6796, DOI: https://doi.org/10.32782/2524-0072/2025-79-139
1. S. L. Heston. A closed-form solution for options with stochastic volatility with applications to bond and currency options. Review of Financial Studies. 1993. Vol. 6, No. 2, P. 327–343.
2. R. Schöbel, J. Zhu. Stochastic volatility with an Ornstein–Uhlenbeck process: An extension. European Finance Review. 1999. Vol. 3 (1), P. 23–46.
3. Black F., Scholes M. The pricing of options and corporate liabilities. Journal of Political Economy. 1973. Vol. 81, No. 3, P. 637-654.
4. Hull J., White A. The pricing of options on assets with stochastic volatilities. Journal of Finance. 1987. Vol. 42, No. 2, P. 281-300.
5. Long Teng, Matthias Ehrhardt, Michael Günther. On the Heston model with stochastic correlation. International Journal of Theoretical and Applied Finance. 2016. Vol. 19, No. 6, P. 1-25.
6. Rebonato R. Volatility and Correlation. The Perfect Hedger and the Fox. John Wiley & Sons, 2nd Edition, 2004. 864 p.
7. Stein E. M., Stein J. C. Stock price distributions with stochastic volatility: An analytic approach. Review of Financial Studies. 1991. Vol. 4, No. 4, P. 727-752.
8. Grzelak L. A., Oosterlee C.W., Van Weeren S. The affine Heston model with correlated Gaussian interest rates for pricing hybrid derivatives. Quantitative Finance. 2011. Vol. 11, P. 1647-1663.
9. Grzelak L. A., Oosterlee C. W., Van Weeren S. Extension of stochastic volatility equity models with the Hull-White interest rate process. Quantitative Finance. 2012. Vol. 12, P. 89-105.
10. Guo S., Grzelak L. A., Oosterlee C. W. Analysis of an affine version of the Heston-Hull-White option pricing partial differential equation. Applied Numerical Mathematics. 2013. Vol. 72, P.143-159.
11. Van Haastrecht A., Lord R., Pelsser A., Schrager D. Pricing long-dated insurance contracts with stochastic interest rates and stochastic volatility. Insurance: Mathematics and Economics. 2009. Vol. 45 (3), P. 436-448.
12. Cox J.C., Ingersoll J.E., Ross S.A. An intertemporal general equilibrium model of asset prices. Econometrica. 1985. Vol. 53, No. 2, P. 363-384.
13. Hull J., White A. Pricing interest-rate-derivative securities. Review of Financial Studies. 1990. Vol. 3, No. 4, P. 573-592.
14. Vasicek O. An equilibrium characterization of the term structure. Journal of Financial Economics. 1977. Vol. 5(2), P. 177-188.
15. Sippel J., Ohkoshi S. All power to PRDC notes. Risk Magazine. 2002. Vol. 15 (11), P. 1-3.
16. Cox J. C., Ingersoll J. E., Ross S. A. A re-examination of traditional hypotheses about the term structure of interest rates. Journal of Finance. 1981. Vol. 36 (4), P. 769–799.
17. Grzelak L. A., Oosterlee C. W. On the Heston model with stochastic interest rates. SIAM Journal on Financial Mathematics. 2011. Vol. 2 (1), P. 255-286.
18. Ahlip R., Rutkowski M. Pricing of foreign exchange options under the Heston stochastic volatility model and CIR interest rates. Quantitative Finance. 2013. Vol. 13(6), P. 955-966.
19. Grzelak L. A., Oosterlee C. W. An equity-interest rate hybrid model with stochastic volatility and the interest rate smile. The Journal of Computational Finance. 2012. Vol. 15, No. 4, P. 12-32.
20. Grzelak L. A., C. W. Oosterlee. On cross-currency models with stochastic volatility and correlated interest rates. Applied Mathematical Finance. 2012. Vol. 19 (1), P. 1-35.
21. A. van Haastrecht, A. Pelsser. Generic pricing of FX, inflation and stock options under stochastic interest rates and stochastic volatility. Quantitative Finance. 2011. Vol. 11(5), P. 665-691.
22. S. Fallah, A. Najafi, F. Mehrdoust. A fractional version of the Cox-Ingersoll-Ross interest rate model and pricing double barrier option with Hurst index. Communications in Statistics – Theory and Methods. 2018. Vol. 48 (9), P. 1-16.
23. M. Abudy, Y. Izhakian. Pricing stock options with stochastic interest rate. International Journal of Portfolio Analysis and Manage-ment. 2013. Vol. 1 (3), P. 250–277.
24. K. Rindell. Pricing of index options when interest rates are stochastic: An empirical test. Journal of Banking and Finance. 1995. Vol. 19 (5), P. 785–802.
25. J. C. Hull, A. D. White. Using Hull-White interest rate trees. Journal of Derivatives. 1996. Vol. 3 (3), P. 26–36.
26. Yanishevskyi V. S. Path Integral Solutions for Extended Heston Models. Mathematical Modeling and Computing. 2025. Vol. 12, No. 4, P. 1341–1356.
27. Xin-Jiang He, Song-Ping Zhu. A closed-form pricing formula for European options under the Heston model with stochastic interest rate. Journal of Computational and Applied Mathematics. 2018. Vol. 335, P. 323-333.
28. Yanishevsʹkyy V.S. (2025) Tsina optsionu v rozshyreniy modeli Hestona. [Option price in the extended Heston model]. Ekonomika ta suspilʹstvo, no 79.
URL:https://economyandsociety.in.ua/index.php/journal/article/view/6796, DOI: https://doi.org/10.32782/2524-0072/2025-79-139
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Опубліковано
2025-11-24
Як цитувати
Янішевський, В., & Михайлинин, М. (2025). МОДЕЛЬ ГЕСТОНА ІЗ СТОХАСТИЧНОЮ ВІДСОТКОВОЮ СТАВКОЮ. Економіка та суспільство, (81). https://doi.org/10.32782/2524-0072/2025-81-16
Розділ
ФІНАНСИ, БАНКІВСЬКА СПРАВА ТА СТРАХУВАННЯ
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