The option pricing model can predict the future trend of the financial market. In order to more accurately describe the changing process of the financial market, the Hurst index which can describe the characteristics of long-term memory is introduced into the traditional Heston model. Under the assumption that the underlying asset price follows fractional Brownian motion, the fractional stochastic volatility pricing of European option pricing model (Hurst-Heston model) is constructed, and the closed solution of the model is obtained according to the partial differential equation satisfied by the model. By analyzing the relationship between Hurst index and asset price, it is found that the movement process of asset price under the hypothesis of this model is more consistent with the real market change law, which verifies the rationality of the model. In the process of empirical analysis of SSE 50ETF put option data, Self-adaptive Differential Evolution algorithm is used to estimate parameters. The results showed that the error of Hurst-Heston model is smaller than other models, and the prediction error for consecutive trading days is similar. It showed that the pricing results of Hurst-Heston model are more accurate and stable.
References
[1]
Qi, Y., Sun, J. and Li, F. (2019) Research Progress of Digital Option Theory. Economic Dynamics, No. 5, 119-134.
[2]
Black, F. and Myron, S. (1973) The Pricing of Options and Corporate Liabilities. Journal of Political Economy, 81, 639-654. https://doi.org/10.1086/260062
[3]
Zhang, J., Wang, Y. and Zhang, S. (2022) A New Homotopy Transformation Method for Solving the Fuzzy Fractional Black-Scholes European Option Pricing Equations under the Concept of Granular Differentiability. Fractal and Fractional, 6, Article No. 286. https://doi.org/10.3390/fractalfract6060286
[4]
He, X.-J. and Lin, S. (2021) A Fractional Black-Scholes Model with Stochastic Volatility and European Option Pricing. Expert Systems with Applications, 178, Artilce No. 114983. https://doi.org/10.1016/j.eswa.2021.114983
[5]
Merton, R.C. (1976) Option Pricing When Underlying Process of Stock Returns Are Discontinuous. Journal of Financial Economics, 3, 124-144. https://doi.org/10.1016/0304-405X(76)90022-2
[6]
Leland, H. (1985) Option Pricing and Replication with Transactions Costs. The Journal of Finance, 40, 1283-1301. https://doi.org/10.1111/j.1540-6261.1985.tb02383.x
[7]
Heston, S. (1993) A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options. The Review of Financial Studies, 6, 327-343. https://doi.org/10.1093/rfs/6.2.327
[8]
Li, P., Yang, J. and Lin, Y. (2020) Parameter Estimation and Application of Fast Mean Reversion Stochastic Volatility Model. Operations Research and Management Science, 29, 137-143.
[9]
He, J. and Wei, Z. (2019) The Pricing of Carry Options Based on Jump Diffusion Model of Non-Affine Stochastic Volatility. Mathematics in Practice and Theory, 49, 132-139.
[10]
Wu, X., Li, X. and Mam C, (2019) Research on SSE 50ETF Option Pricing Based on Stochastic Volatility Model. Journal of Applied Statistics and Management, 38, 115-131.
[11]
SenGupta, I. (2014) Option Pricing with Transaction Costs and Stochastic Interest Rate. Applied Mathematical Finance, 21, 399-416. https://doi.org/10.1080/1350486X.2014.881263
[12]
Zhou, R. (2018) Option Pricing Based on Volatility Decomposition. Systems Engineering Theory and Practice, 38, 1919-1929.
[13]
Li, D. (2017) Research on Long Memory of Financial Market Based on Fractal Method. Ph.D. Thesis, Foreign Economic and Trade University, China.
[14]
Yuan, Y., Zhuang, X. and Jin, X. (2016) Intraday Effect, Long Memory and Multifractality of Chinese Stock Market: Based on the Perspective of Price-Quantity Cross-Correlation. Journal of Systems Management, 25, 28-35.
[15]
Geng, K. and Zhang, S. (2008) Research on Long Memory of UHF Duration Sequence in Chinese Stock Market. Chinese Journal of Management Science, No. 2, 7-13.
[16]
Xue, C. and Li, X. (2008) Long-Memory Characteristics of China’s Stock Market Returns and Volatility. Statistics and Decision, No. 4, 117-119.
[17]
Yuan, Y. and Zhuang, X. (2008) The Long Memory and Market Development Status of China’s Stock Market. Journal of Applied Statistics and Management, No. 1, 156-163.
[18]
Li, Y., He, Y. and Zhu, H. (2003) An Empirical Study on China’s Stock Market Return and Volatility Long Memory. Systems Engineering Theory and Practice, No. 1, 9-15.
[19]
Shi, Y., Tao, X. and Zhang, S. (2011) On-Off Hurst Index Fractal Black-Scholes Market Pricing of European Options. Economic Mathematics, 28, 40-44.
[20]
Yang, Z. and Caselli, F. (2017) Optimal Exercise Boundary of American Fractional Lookback Option in a Mixed Jump-Diffusion Fractional Brownian Motion Environment. Mathematical Problems in Engineering, 2017, Article ID: 5904125. https://doi.org/10.1155/2017/5904125
[21]
Qin, X., Lin, X. and Wang, W. (2019) Research on Fuzzy Pricing of European Options Based on Long Memory Characteristics. Systems Engineering Theory and Practice, 39, 3073-3083.
[22]
Xue, D., Zhu, G., Zhu, Y. and Xiong, Y. (1995) Stationarity Analysis and Description of Fractional Brownian Motion. Journal of Huazhong University of Science and Technology, 10, 78-81.
[23]
Huang, Y. (2009) Time Axis Transformation Method for Option Pricing under Geometric Fractional Brownian Motion. Statistics and Decision, No. 5, 10-12.
[24]
Fadugba, S.E. and Nwozo, C.R. (2016) Valuation of European Call Options via the Fast Fourier Transform and the Improved Mellin Transform. Journal of Mathematical Finance, 6, 338-359. https://doi.org/10.4236/jmf.2016.62028
[25]
Ding, Q. and Yin, X. (2017) Overview of Differential Evolution Algorithms. Journal of Intelligent Systems, 12, 431-442.
[26]
Yang, Q., Cai, L. and Xue, Y. (2008) Overview of Differential Evolution Algorithms. Pattern Recognition and Artificial Intelligence, 21, 506-513.
[27]
Hu, F., Dong, Q. and Lu, L. (2021) Improved Differential Evolution Algorithm for Adaptive Quadratic Mutation and Its Application. Computer Applications and Software, 38, 271-280.
[28]
Bakshi, G., Cao, C. and Chen, Z. (1997) Empirical Performance of Alternative Option Pricing Models. The Journal of Finance, 52, 2003-2049. https://doi.org/10.1111/j.1540-6261.1997.tb02749.x
