OPTION PRICE IN THE EXTENDED HESTON MODEL
Abstract
Analytical methods for studying option pricing become significantly more complicated as the dimensionality of models increases. In this work, we consider an extension of the Heston model that accounts for the stochastic dynamics of the interest rate. The interest rate is modeled as a Vasicek stochastic process. It is known that this extended Heston model admits exact solutions in the absence of correlations between the Wiener process of the interest rate and the other processes of the model. However, there are certain difficulties in practical application, since the solutions involve complex integrals that can only be evaluated numerically. Therefore, the development of analytical methods for modeling the stochastic dynamics of option pricing within the extended Heston model is important. In this study, we derive a formula for determining the option price in the extended Heston model, in which the term structure of the interest rate is explicitly incorporated, and averaging is performed with the conditional probability density of the Heston model discount factor corresponding to a constant interest rate is replaced by the term structure factor for a stochastic interest rate in the Vasicek model, and the payoff function is replaced by an effective payoff function. A qualitative analysis of the option price obtained from the above formula, compared with the standard Heston model, shows an increase in the option price. Compared to the Heston model, where the interest rate r_0 is constant, the stochastic dynamics cause its expected value to decrease. As a result, the discount factor increases, and consequently, the option price also increases. A comparative numerical analysis of option prices in the Heston model and the extended Heston model is presented. Although the problem is two-dimensional, performing the numerical computations does not present significant difficulties, since the infinite integration limits can be replaced with finite ones. The obtained structure of the option pricing formula corresponds to a decomposition of the extended Heston model into subsystems: the Heston model and the Vasicek interest rate model. The proposed analytical method can be effective for studying stochastic multifactor models in financial engineering that can be decomposed into subsystems with known solutions. The resulting formulas may be useful for analyzing stock market statistical data and calibrating models.
References
C. Gardiner. Handbook of Stochastic Methods - For Physics, Chem, Nat. Sciences. Springer, 2004. 417 p.
Lyuu Y.-D. Financial Engineering and Computation: Principles, Mathemat-ics, and Algorithms. Cambridge University Press, 2004. 648 p.
Grigorios A. Pavliotis. Stochastic Processes and Applications. Diffusion Processes, the Fokker-Planck and Langevin Equations. Springer New York Heidelberg Dordrecht London, 2013. 339 p.
Paul W., Baschnagel J. Stochastic Processes. From Physics to Finance. Springer Cham, 2013. 231 p.
M. Scott. Applied Stochastic Processes in science and engineering. Springer New York Heidelberg Dordrecht London, 2013. 316 p.
Vadim Linetsky. The Path Integral Approach to Financial Modeling and Options Pricing. Computational Economics. 1998. 11, P. 129-163.
Marc Goovaertsa, Ann De Schepperb, Marc Decampsa. Closed-form ap-proximations for diffusion densities: a path integral approach. Journal of Computational and Applied Mathematics. 2004. 164-165, P. 337-364.
Yanishevskyi V. Solution to the Fokker-Plank equation in the path integral method. Mathematical Modeling and Computing. 11(4), 1046-1057 (2024).
F. D. Rouah. The Heston Model and its Extensions in Matlab and C. John Wiley & Sons, Hoboken, NJ, USA. 2013. 432 p.
Adrian A. Druágulescu and Victor M. Yakovenko. Probability distribution of returns in the Heston model with stochastic volatility. Quantitative fi-nance. 2002. Vol. 2, P. 443-453.
Sebastian del Bano Rollin, Albert Ferreiro-Castilla, Frederic Utzet. A new look at the Heston characteristic function. Preprint, arXiv:0902.2154v1 [math.PR]. 2009. P. 30.
Grzelak L. A. and Oosterlee C. W. On the Heston model with stochastic interest rates. SIAM Journal on Financial Mathematics. 2011. 2 (1), P. 255-286.
Van Haastrecht A. and Pelsser A. Generic pricing of FX, inflation and stock options under stochastic interest rates and stochastic volatility. Quantitative Finance. 2011. 5, P. 665-691.
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. 2, P. 323-333.
Chao Zheng, Jiangtao Pan. Unbiased estimators for the Heston model with stochastic interest rates. Preprint, arXiv:2301.12072v2 [q-fin.CP]. 2023. P. 22.
Giacomo Ascione, Farshid Mehrdoust, Giuseppe Orlando, Oldouz Samimi. Foreign Exchange Options on Heston-CIR Model Under Levy Process Framework. Applied Mathematics and Computation. 2022. Vol 446, P. 31.
Bakshi, S., Cao, C., Chen, Z. Pricing and hedging long-term options. Jour-nal of Econometrics. 2000. 94, P. 2003-2049.
Long Teng, Matthias Ehrhardt and Michael ünther. On the Heston model with stochastic correlation. International Journal of Theoretical and Applied Finance. 2016.Vol. 9, No 6, P. 1650-1680.
Farshid Mehrdoust, Idin Noorani, Abdelouahed Hamdi. Two-factor Heston model equipped with regime-switching: American option pricing and model calibration by Levenberg-Marquardt optimization algorithm. Mathematics and Computers in Simulation. 2023. Vol. 204, P. 660-678.
Javier de Frutos, Victor Gatón. An extension of Heston's SV model to sto-chastic interest rates. Journal of Computational and Applied Mathematics. 2019. 2, P. 174-182.
Yanishevskyi V. Solution to extended Heston models in the path integral method. Mathematical Modeling and Computing. (2025) (in print).
Янішевський В.С, Пелех О. В. Чисельне моделювання цінової динаміки в розширеній моделі Гестона. Економіка та суспільство. 2024. no 69. URL: https://economyandsociety.in.ua/index.php/journal/article/view/5090
DOI: 10.32782/2524-0072/2024-69-10
C. Gardiner. Handbook of Stochastic Methods - For Physics, Chem, Nat. Sciences. Springer, 2004. 417 p.
Lyuu Y.-D. Financial Engineering and Computation: Principles, Mathemat-ics, and Algorithms. Cambridge University Press, 2004. 648 p.
Grigorios A. Pavliotis. Stochastic Processes and Applications. Diffusion Processes, the Fokker-Planck and Langevin Equations. Springer New York Heidelberg Dordrecht London, 2013. 339 p.
Paul W., Baschnagel J. Stochastic Processes. From Physics to Finance. Springer Cham, 2013. 231 p.
M. Scott. Applied Stochastic Processes in science and engineering. Springer New York Heidelberg Dordrecht London, 2013. 316 p.
Vadim Linetsky. The Path Integral Approach to Financial Modeling and Options Pricing. Computational Economics. 1998. 11, P. 129-163.
Marc Goovaertsa, Ann De Schepperb, Marc Decampsa. Closed-form ap-proximations for diffusion densities: a path integral approach. Journal of Computational and Applied Mathematics. 2004. 164-165, P. 337-364.
Yanishevskyi V. Solution to the Fokker-Plank equation in the path integral method. Mathematical Modeling and Computing. 11(4), 1046-1057 (2024).
F. D. Rouah. The Heston Model and its Extensions in Matlab and C. John Wiley & Sons, Hoboken, NJ, USA. 2013. 432 p.
Adrian A. Druágulescu and Victor M. Yakovenko. Probability distribution of returns in the Heston model with stochastic volatility. Quantitative fi-nance. 2002. Vol. 2, P. 443-453.
Sebastian del Bano Rollin, Albert Ferreiro-Castilla, Frederic Utzet. A new look at the Heston characteristic function. Preprint, arXiv:0902.2154v1 [math.PR]. 2009. P. 30.
Grzelak L. A. and Oosterlee C. W. On the Heston model with stochastic interest rates. SIAM Journal on Financial Mathematics. 2011. 2 (1), P. 255-286.
Van Haastrecht A. and Pelsser A. Generic pricing of FX, inflation and stock options under stochastic interest rates and stochastic volatility. Quantitative Finance. 2011. 5, P. 665-691.
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. 2, P. 323-333.
Chao Zheng, Jiangtao Pan. Unbiased estimators for the Heston model with stochastic interest rates. Preprint, arXiv:2301.12072v2 [q-fin.CP]. 2023. P. 22.
Giacomo Ascione, Farshid Mehrdoust, Giuseppe Orlando, Oldouz Samimi. Foreign Exchange Options on Heston-CIR Model Under Levy Process Framework. Applied Mathematics and Computation. 2022. Vol 446, P. 31.
Bakshi, S., Cao, C., Chen, Z. Pricing and hedging long-term options. Jour-nal of Econometrics. 2000. 94, P. 2003-2049.
Long Teng, Matthias Ehrhardt and Michael ünther. On the Heston model with stochastic correlation. International Journal of Theoretical and Applied Finance. 2016.Vol. 9, No 6, P. 1650-1680.
Farshid Mehrdoust, Idin Noorani, Abdelouahed Hamdi. Two-factor Heston model equipped with regime-switching: American option pricing and model calibration by Levenberg-Marquardt optimization algorithm. Mathematics and Computers in Simulation. 2023. Vol. 204, P. 660-678.
Javier de Frutos, Victor Gatón. An extension of Heston's SV model to sto-chastic interest rates. Journal of Computational and Applied Mathematics. 2019. 2, P. 174-182.
Yanishevskyi V. Solution to extended Heston models in the path integral method. Mathematical Modeling and Computing. (2025) (in print).
Yanishevsʹkyy V.S, Pelekh O. V. (2024) Chyselʹne modelyuvannya tsinovoyi dynamiky v rozshyreniy modeli Hestona [Numerical modeling of price dynamics in the extended Heston model]. Ekonomika ta suspilʹstvo, no 69. URL: https://economyandsociety.in.ua/index.php/journal/article/view/5090 DOI: 10.32782/2524-0072/2024-69-10
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