HESTON MODEL WITH A STOCHASTIC INTEREST RATE
Keywords:
stochastic equations, Brownian motion, process Cox-Ingersoll-Ross, option pricing, Black-Scholes model, Heston model, Heston-CIR model
Abstract
The well-known Black–Scholes model is based on the assumptions of constant asset-price volatility and a constant interest rate, which contradicts the dynamics observed in modern financial markets. In comparison with the Black–Scholes framework, the classical Heston model incorporates stochastic asset-price volatility; however, it still relies on the assumption of a constant interest rate. The stochastic volatility model developed by Heston removes one of the main limitations of the Black–Scholes model. Therefore, constructing models that account for the stochastic dynamics of interest rates is essential. The CIR stochastic process has become a benchmark in modeling interest rates. A distinctive feature of the CIR model is that interest rates never become negative, unlike in the Vasicek model. Given the advantages of both the Heston and CIR frameworks, a so-called hybrid version – the Heston-CIR model – has been developed. In this work, we consider the Heston-CIR model, where both the asset-price volatility and the dynamics of the interest rate are driven by a CIR stochastic process. The CIR model possesses important properties such as mean reversion and the non-negativity of the stochastic variable. Further-more, under certain conditions, the Heston-CIR model admits analytical solu-tions, which simplifies model calibration using market data and enhances its practical applicability. The probability distribution of the asset price and a closed-form option pricing formula for this model were derived in the author’s previous work. The resulting formula for the price of a European option is presented in a form convenient for numerical computation. We also examine the impact of the stochastic interest rate on option prices by comparing the re-sults with the standard Heston model. Exact solutions for the Heston-CIR model are employed, which apply in the case of zero correlation between the Wiener process in the interest-rate equation and the other Wiener processes in the model. Based on the exact solution, a pricing formula for a European call option is constructed. A numerical study is carried out to investigate the de-pendence of the option price on its maturity for a given set of model parame-ters, along with a comparison to the ordinary Heston model. A clear deviation of the option price in this model from the classical Heston model is observed as the option maturity increases.
References
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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Published
2025-11-24
How to Cite
Yanishevskyi, V., & Mykhailynyn, M. (2025). HESTON MODEL WITH A STOCHASTIC INTEREST RATE. Economy and Society, (81). https://doi.org/10.32782/2524-0072/2025-81-16
Section
FINANCE, BANKING AND INSURANCE
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