Document Type : Research Article
Authors
1 Zagros Financial Research Group, Ayatollah Boroujerdi University, Boroujerd, Iran.
2 Zagros Financial Research Group, Ayatollah Boroujerdi University, Boroujerd, Iran
Abstract
In order to reduce the risk of financial markets, various tools have emerged, and option contracts are the most common tools in this regard. The Black-Scholes model is used to price a wide range of options contracts. The basic assumption in this model is to follow the normal distribution of returns. But the reality of the market indicates the skewness and kurtosis of the data, which reduces the accuracy of calculating the option price. The Gram-Charlie model has more flexibility than Black-Scholes model with abnormal skewness and kurtosis. The main purpose of this research is to determine the European call option price using non-normal data. In this regard, we present new models, fractional Gram-Charlier model and mixed fractional Gram-Charlier model, for option pricing. For this purpose, the data of Shasta and Khodro symbols have been selected from Iran Stock Exchange that Khodro in the period 2020-07-27 to 2023-11-1 and Shasta in the period 2022-7-25 to 2023-11-1 have been used. The results of this research show that Shasta has more abnormal skewness and kurtosis than Khodro. The option price calculated with the Gram-Charlier and extended models of Gram-Charlier are shown a smaller error compared to other models in the Shasta. Also, the results show that under abnormal skewness and kurtosis, our new models have more flexibility than the Black-Scholes model and fractional models.
Keywords
in financial markets, Physica A 391 (2012), no. 17, 4234–4251.
[2] C. Bender, Integration with respect to fractional Brownian motion and related market models,
Ph.D. thesis, University of Konstanz, Department of Mathematics and Statistics, 2003.
[3] F. J. Breidt, N. Crato, and P. de Lima, The detection and estimation of long memory in
stochastic volatility, J. Econom. 83 (1998), 325–348.
[4] F. Comte and E. Renault, Long memory in continuous-time stochastic volatility models,
Math. Finance 8 (1998), 291–323.
[5] A. Dasgupta and G. Kallianpur, Arbitrage opportunities for a class of Gladyshev processes,
Appl. Math. Optim. 41 (2000), 377–385.
[6] L. Della Ratta, F. Leccadito, and G. Urga, The Fractional Merton Model: A New Approach
to Credit Risk Pricing (Revised), Centre for Econometric Analysis London, Working Paper,
2008.
[7] A. Ghasemifard, S. Banihashemi, and A. Babaei, Option valuation in markets with finite
liquidity under fractional CEV assets, J. Math. Model. Finance 2 (2022), no. 2, 167–180.
[8] H. Gu, J. Liang, and Y. Zhang, Time-changed geometric fractional Brownian motion and
option pricing with transaction costs, Physica A 391 (2012), no. 15, 3971–3977.
[9] P. Cheridito, Arbitrage in fractional Brownian motion models, Finance Stoch. 7 (2003), no.
4, 533–553.
[10] S. Hung, H. Hsi, and S. Han, Option pricing under truncated Gram-Charlier expansion, N.
Am. J. Econ. Finance 32 (2015), 77–97.
[11] Y. Hu and B. Øksendal, Fractional white noise calculus and applications to finance, Infinite
Dimens. Anal. Quantum Probab. Relat. Top. 6 (2003), no. 1, 1–32.
[12] J. Hull, Options, Futures, and Other Derivative Securities, Prentice Hall, Englewood Cliffs,
NJ, 1993.
[13] S. N. Iqmal Ibrahim, M. Misiran, and M. F. Laham, Geometric fractional Brownian motion
model for commodity market simulation, Alex. Eng. J. 60 (2021), no. 1, 955–962.
[14] B. B. Mandelbrot and J. W. Van Ness, Fractional Brownian motions, fractional noises and
applications, SIAM Rev. 10 (1968), 422–437.
[15] L. Meng and M. Wang, Comparison of Black-Scholes formula with fractional Black-Scholes
formula in the foreign exchange option market with changing volatility, Asia-Pac. Financ.
Mark. 17 (2010), no. 2, 99–111.
[16] E. Mliki, Correlation structure of time-changed generalized mixed fractional Brownian motion, Fractal Fract. 7 (2023), 591.
[17] S. Natenberg, Option Volatility and Pricing, Probus Publishing, Chicago, 1994.
[18] T. Nasiri, A. Zakeri, and A. Aminataei, A numerical method for solving the underlying
price problem driven by a fractional Levy process, J. Math. Model. Finance 2 (2022), no. 1,
195–208.
[19] C. Necula, Option Pricing in a Fractional Brownian Motion Environment, Preprint,
Academy of Economic Studies, Bucharest, 2002.
[20] S. Rostek, Option Pricing in Fractional Brownian Markets, Springer, 2009.
[21] S. Rostek and R. Sch¨obel, A note on the use of fractional Brownian motion for financial
modeling, Econ. Model. 30 (2013), 30–35.
[22] T. Sottinen, Fractional Brownian motion, random walks and binary market models, Finance
Stoch. 5 (2001), 343–355.
[23] W. Xiao, W. Zhang, X. Zhang, and Y. Wang, Pricing currency options in a fractional
Brownian motion with jumps, Econ. Model. 27 (2010), no. 5, 935–942.
)