Estimation of Heston model parameters and its application in long-term memory data

Subject Areas : Statistics

Parviz Nasiri ¹ , Amir Haj Salmani ² , Masou Yarmohammadi ³

1 -
2 -
3 -

Received: 2023-02-26 Accepted : 2024-05-13 Published : 2025-09-23

Keywords: Models with long-term memory, Heston&#039, s model, Mean square error, Lasso estimation, fractional differential equations.,

Abstract :

Models with long-term memory with fractional differentiation parameter are widely used in fractional stochastic differential models. In this article , the Heston's model as one of fractional random differential models is introduced and the closed formula for trading options is presented using the characteristic function. The rule has been investigated while obtaining the characteristic functions of its parameter estimation methods. In this article, In meantime while calculating the rule of the characteristic functions, their parameters are obtained by using two methods of maximum correctness parameter and Lasso calculation. Finally in this paper the accuracy of the two estimation methods are compared with the simulated data and the stock data of Iran Khodro (iran) company in the stock market (car symbol). By using the criterion of mean square error and goodness of fit of the model, it has been shown that the Lasso method performs better compared to the maximum likelihood method.

References:


[1].Durham, G., Geweke, J., Porter‐Hudak, S. & Sowell, F., (2019). Bayesian inference for ARFIMA models. Journal of Time Series Analysis,40(4), 388-410
[2].Delgado-Vences, F., & Ornelas, A., (2019). Modelling Italian mortality rates with a geometric-type fractional Ornstein-Uhlenbeck process. arXiv preprint arXiv:1901.00795.
[3].Karlsson, P., & Nilsson, B. (2009). The Heston Model. Lund University,Bachelorarbeit.
[4].Bu, D., & Liao, Y. (2013). Structural credit risk model with stochastic volatility: a particle-filter approach. Available at SSRN 2336452.
[5].Björk, Tomas (2009). Arbitrage Theory in Continuous Time (3rd ed.). Oxford University Press. pp. 375-381.
[6].Zhang, Y., Pan, D., Zhou, S. W., & Han, M. (2014). Asian option pricing with transaction costs and dividends under the fractional Brownian motion model. Journal of Applied Mathematics, 2014.
[7].B.B. Mandelbrot and J.W. van Ness (1968), Fractional Brownian motions, fractional noises and applications, SIAM Review, vol. 10, no. 4, pp. 422–437.
[8].Dunn, R., Hauser, P., Seibold, T., & Gong, H. (2014). Estimating option prices with Heston’s stochastic volatility model
[9].Mishura, Y., & Yurchenko-Tytarenko, A. (2019). Option pricing in fractional Heston-type model. arXiv preprint arXiv:1907.01846.
[10].Chang, Ying, Wang, Yiming & Zhang, Sumei., (2021). Option Pricing under Double Heston Model with Approximative Fractional Stochastic Volatility. Mathematical Problems in Engineering. 1-12.
[11].Imkeller, P., & Schmalfuss, B., (2001). The conjugacy of stochastic and random differential equations and the existence of global attractors. Journal of Dynamics and Differential Equations, 13(2), 215-249.
[12].Schilling, R. L., & Partzsch, L., ((2014). In Brownian Motion. De Gruyter.272-273.
[13].Mishura, Y., & Yurchenko-Tytarenko, A. (2018). Fractional Cox–Ingersoll–Ross process with small Hurst indices. Modern Stochastics: Theory and Applications, 6(1), 13-39.
[14].Dunford, N., & Schwartz, J. T. (1958). Linear operators, vol. I.
[15].Oksendal, B., (2008). Stochastic partial differential equations driven by multi-parameter white noise of Lévy processes. Quarterly of applied mathematics, 66(3), 521-537.
[16].Capinski, M., & Zastawniak, T. J., (2013). Probability through problems. Springer Science & Business Media.
[17].Benhamou, E., Gobet, E., & Miri, M., (2010). Time dependent Heston model. SIAM Journal on Financial Mathematics, 1(1), 289-325.
[18].P Sattayatham and A. Intarasit., (2011). An approximate formula of European option for fractional stochastic volatility jump-diffusion model, Journal of Mathematics and Statistics, vol. 7, no. 3, pp. 230–238.
[19].Mrázek, M., Pospíšil, J., & Sobotka, T. (2016). On calibration of stochastic and fractional stochastic volatility models. European Journal of Operational Research, 254(3), 1036-1046.
[20].Pospíšil, J., & Sobotka, T. (2016). Market calibration under a long memory stochastic volatility model. Applied Mathematical Finance, 23(5), 323-343.
[21].Strogatz, S. H. (2001). Exploring complex networks. nature, 410(6825), 268-276.
[22].Athreya, K. B., & Lahiri, S. N. (2006). Measure theory and probability theory (Vol. 19). New York: Springer. - 115
[23].Schneider, G., Craigmile, P. F., & Herbei, R. (2014). Maximum likelihood estimation for stochastic differential equations using sequential kriging-based optimization. arXiv preprint arXiv:1408.2441.
[24].Iacus, S. M., & Yoshida, N. (2018). Simulation and inference for stochastic processes with YUIMA. A comprehensive R framework for SDEs and other stochastic processes. Use R.
[25].De Gregorio, A., & Iacus, S. M. (2012). Adaptive LASSO-type estimation for multivariate diffusion processes. Econometric Theory, 28(4), 838-860.
[26].Iacus, S. M., & de Gregorio, A. (2010). Adaptive LASSO-type estimation for ergodic diffusion processes (No. 2010-13).