This paper proposes a correlated binomial tree approach to model the Heston stochastic volatility process. The construction preserves the instantaneous correlation between the asset and volatility processes within a strictly recombining binomial framework. The Heston model is utilized for its ability to capture the dynamics of stochastic volatility, offering a more realistic representation of market behavior. The proposed approach improves upon existing tree-based models by incorporating the intrinsic correlation between the asset price and volatility processes, which is a key aspect of the original Heston model. Additionally, unlike Monte Carlo methods, which rely on random sampling and yield random outcomes, this approach is nonstochastic and yields convergent results with respect to the discretization of the tree. This allows for more stable and consistent parameter fitting when applied to real-world data, providing a reliable estimation of model parameters. An additional assumption that volatility remains constant during each transition is introduced, under which it is proven that the tree approach converges in distribution to the continuous Heston model under this assumption. Numerical experiments and empirical evidence from the China Securities Index 300 option market further demonstrate the robustness and practical applicability of the proposed approach, showing that it can serve as a reliable tool for both option pricing and volatility modeling in practice.
Citation: Pu-Ern Kow, You-Beng Koh, Kok-Haur Ng, Hailiang Yang. A correlated Heston's stochastic volatility model: A binomial tree approach[J]. Journal of Industrial and Management Optimization, 2026, 22(5): 2181-2207. doi: 10.3934/jimo.2026080
This paper proposes a correlated binomial tree approach to model the Heston stochastic volatility process. The construction preserves the instantaneous correlation between the asset and volatility processes within a strictly recombining binomial framework. The Heston model is utilized for its ability to capture the dynamics of stochastic volatility, offering a more realistic representation of market behavior. The proposed approach improves upon existing tree-based models by incorporating the intrinsic correlation between the asset price and volatility processes, which is a key aspect of the original Heston model. Additionally, unlike Monte Carlo methods, which rely on random sampling and yield random outcomes, this approach is nonstochastic and yields convergent results with respect to the discretization of the tree. This allows for more stable and consistent parameter fitting when applied to real-world data, providing a reliable estimation of model parameters. An additional assumption that volatility remains constant during each transition is introduced, under which it is proven that the tree approach converges in distribution to the continuous Heston model under this assumption. Numerical experiments and empirical evidence from the China Securities Index 300 option market further demonstrate the robustness and practical applicability of the proposed approach, showing that it can serve as a reliable tool for both option pricing and volatility modeling in practice.
| [1] |
F. Black, M. Scholes, The pricing of options and corporate liabilities, J. Political Econ., 81 (1973), 637–654. https://doi.org/10.1086/260062 doi: 10.1086/260062
|
| [2] | E. Derman, M. B. Miller, The Volatility Smile, John Wiley & Sons, 2016. https://doi.org/10.1002/9781119289258 |
| [3] | E. Derman, I. Kani, The volatility smile and its implied tree, Goldman Sachs Quantitative Strategies Research Notes, 2 (1994), 45–60. |
| [4] | B. Dupire, Pricing with a smile, Risk, 7 (1994), 18–20. |
| [5] |
S. L. Heston, A closed-form solution for options with stochastic volatility with applications to bond and currency options, Rev. Financ. Stud., 6 (1993), 327–343. https://doi.org/10.1093/rfs/6.2.327 doi: 10.1093/rfs/6.2.327
|
| [6] | W. F. Sharpe, Investments, Prentice-Hall, 1978. |
| [7] |
J. C. Cox, S. A. Ross, M. Rubinstein, Option pricing: A simplified approach, J. Financ. Econ., 7 (1979), 229–263. https://doi.org/10.1016/0304-405X(79)90015-1 doi: 10.1016/0304-405X(79)90015-1
|
| [8] |
J. A. Pareja-Vasseur, F. H. Marín-Sánchez, Quadrinomial trees to value options in stochastic volatility models, J. Deriv., 27 (2019), 49–66. https://doi.org/10.3905/jod.2019.1.076 doi: 10.3905/jod.2019.1.076
|
| [9] |
X. Y. Wu, C. Q. Ma, S. Y. Wang, Warrant pricing under GARCH diffusion model, Econ. Model., 29 (2012), 2237–2244. https://doi.org/10.1016/j.econmod.2012.06.020 doi: 10.1016/j.econmod.2012.06.020
|
| [10] |
U. H. Lok, Y. D. Lyuu, A valid and efficient trinomial tree for general local-volatility models, Comput. Econ., 60 (2022), 817–832. https://doi.org/10.1007/s10614-021-10166-x doi: 10.1007/s10614-021-10166-x
|
| [11] |
S. K. Nawalkha, N. A. Beliaeva, Efficient trees for CIR and CEV short rate models, J. Altern. Invest., 10 (2007), 71–90. https://doi.org/10.3905/jai.2007.688995 doi: 10.3905/jai.2007.688995
|
| [12] |
N. A. Beliaeva, S. K. Nawalkha, A simple approach to pricing American options under the Heston stochastic volatility model, J. Deriv., 17 (2010), 25. https://doi.org/10.3905/jod.2010.17.4.025 doi: 10.3905/jod.2010.17.4.025
|
| [13] |
D. B. Nelson, K. Ramaswamy, Simple binomial processes as diffusion approximations in financial models, Rev. Financ. Stud., 3 (1990), 393–430. https://doi.org/10.1093/rfs/3.3.393 doi: 10.1093/rfs/3.3.393
|
| [14] | Y. D. Lyuu, Financial Engineering and Computation: Principles, Mathematics, Algorithms, Cambridge University Press, 2002. |
| [15] | W. H. Fleming, H. M. Soner, Controlled Markov Processes and Viscosity Solutions, Springer, 2006,321–346. https://doi.org/10.1007/0-387-31071-1 |
| [16] |
J. C. Cox, J. E. Ingersoll, S. A. Ross, A theory of the term structure of interest rates, Econometrica, 53 (1985), 385–407. https://doi.org/10.2307/1911242 doi: 10.2307/1911242
|
| [17] |
J. Hull, A. White, Numerical procedures for implementing term structure models Ⅰ: Single-factor models, J. Deriv., 2 (1994), 7–16. https://doi.org/10.3905/jod.1994.407902 doi: 10.3905/jod.1994.407902
|
| [18] |
M. Mrázek, J. Pospíšil, Calibration and simulation of Heston model, Open Math., 15 (2017), 679–704. https://doi.org/10.1515/math-2017-0058 doi: 10.1515/math-2017-0058
|
| [19] | A. Ortiz-Ramírez, F. Venegas-Martínez, M. T. V. Martínez-Palacios, Parameter calibration of stochastic volatility Heston's model: Constrained optimization vs. differential evolution, Contaduría y Administración, 67 (2022). https://doi.org/10.22201/fca.24488410e.2022.2789 |
| [20] |
Y. Cui, S. del Baño Rollin, G. Germano, Full and fast calibration of the Heston stochastic volatility model, Eur. J. Oper. Res., 263 (2017), 625–638. https://doi.org/10.1016/j.ejor.2017.05.018 doi: 10.1016/j.ejor.2017.05.018
|
| [21] |
F. A. Longstaff, E. S. Schwartz, Valuing American options by simulation: A simple least-squares approach, Rev. Financ. Stud., 14 (2001), 113–147. https://doi.org/10.1093/rfs/14.1.113 doi: 10.1093/rfs/14.1.113
|
| [22] |
A. A. Christie, The stochastic behavior of common stock variances: Value, leverage and interest rate effects, J. Financ. Econ., 10 (1982), 407–432. https://doi.org/10.1016/0304-405X(82)90018-6 doi: 10.1016/0304-405X(82)90018-6
|
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Pu-Ern Kow, You-Beng Koh, Kok-Haur Ng, Hailiang Yang. A correlated Heston's stochastic volatility model: A binomial tree approach[J]. Journal of Industrial and Management Optimization, 2026, 22(5): 2181-2207. doi: 10.3934/jimo.2026080
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