Time-changed generalized mixed fractional Brownian motion and application to arithmetic average Asian option pricing


  • ömer önalan Faculty of Business Administration, Marmara University, Turkey






Arithmetic Average Asian Option Pricing, Generalized Mixed Fractional Brownian motion, Inverse Gamma Subordination, Time-Chanced Processes.


In this paper we present a novel model to analyze the behavior of random asset price process under the assumption that the stock price pro-cess is governed by time-changed generalized mixed fractional Brownian motion with an inverse gamma subordinator. This model is con-structed by introducing random time changes into generalized mixed fractional Brownian motion process. In practice it has been observed that many different time series have long-range dependence property and constant time periods. Fractional Brownian motion provides a very general model for long-term dependent and anomalous diffusion regimes. Motivated by this facts in this paper we investigated the long-range dependence structure and trapping events (periods of prices stay motionless) of CSCO stock price return series. The constant time periods phenomena are modeled using an inverse gamma process as a subordinator. Proposed model include the jump behavior of price process because the gamma process is a pure jump Levy process and hence the subordinated process also has jumps so our model can be capture the random variations in volatility. To show the effectiveness of proposed model, we applied the model to calculate the price of an average arithmetic Asian call option that is written on Cisco stock. In this empirical study first the statistical properties of real financial time series is investigated and then the estimated model parameters from an observed data. The results of empirical study which is performed based on the real data indicated that the results of our model are more accuracy than the results based on traditional models.


Ait-Sahalia,Y.(1996), Nonparametric pricing of interest rate derivative securities, Econometrica, 64, 3,527-560. https://doi.org/10.2307/2171860.

Prakasa Rao BLS. (2016), Pricing Asian power options under mixed fractional Brownian motion environment. Physica A: Statistical Mechanics and its Applications,446 ,92-99. https://doi.org/10.1016/j.physa.2015.11.013.

Bates, D. (1996), Jumps and stochastic volatility: exchange rate processes implicit in deutsche mark options. Review of Financial Studies, 9, 69-107 https://doi.org/10.1093/rfs/9.1.69.

Cao G. & Wong Y. (2015), Risk-neutral pricing for geometric average Asian options with floating strike, Journal of university of Chinese academy of sciences,32,1,13-17.

Cheridito,P.(2001),Fixed fractional Brownian motion. Bernoulli, 7,913-934. https://doi.org/10.2307/3318626.

Cont,R & Tankov, P.(2004), Financial modeling with jump processes, Chapman & Hall/CRC Baton; Florida.

Filetova D.V. & Grzywaczewski, M. (2008), Mathematical modeling in selected biological systems with fractional Brownian motion. HIS, Krakow, Poland, May 25-27.

Gajda, J. & Magdziarz, M.(2010), Fractional Fokker-Planck equation with tempered alpha-stable waiting times: Langevin picture and computer simulation, Phys.Rev. E,82, doi:10.1103/ Phys Rev E. 82.011117 https://doi.org/10.1103/PhysRevE.82.011117.

Guo, Z. & Yuan, H. (2014), Pricing European option under the time-changed mixed Brownian-fractional Brownian model, Physica A, 406, 73-79. https://doi.org/10.1016/j.physa.2014.03.032.

Jian-Kai, L.,Wan-Qing, S.& Qing, L.(2015), Research on cutting tool wear based on fractional Brownian motion, International Journal of Mechanical Engineering and Applications, 3(1), 1-5. https://doi.org/10.11648/j.ijmea.20150301.11.

Kubiluis, K. & Melichov, D. (2010), Quadratic variations and estimation of the Hurst index of the solution of the SDE driven by a fractional Brownian motion. Lithuanian Mathematical Journal, 50(4), 401-417. https://doi.org/10.1007/s10986-010-9095-z.

Lo,A.W.(1991),Long term memory in stock market prices, Econometrica, 59,(5),1279-1313. https://doi.org/10.2307/2938368.

Magdziarz,M.(2009), Black-Scholes formula in sub-diffusive regime, J. Stat. Phys., 136, 553–564. https://doi.org/10.1007/s10955-009-9791-4.

Magdziarz,M.& Gajda,J.(2012),Anomalous dynamics of Black-Scholes model time-changed by inverse subordinators, Acta Physica Polonica B,43,1093-1109. https://doi.org/10.5506/APhysPolB.43.1093.

Mandelbrot, B.B. & Van Ness, J.W. (1968), Fractional Brownian motion, fractional noise and applications. SIAM Review, 10,422-437. https://doi.org/10.1137/1010093.

Shokrollahi,F.& Kılıçman, A.(2014),Pricing currency option in a mixed fractional Brownian motion with jumps environment, Mathematical Problems in Engineering.1-13. https://doi.org/10.1155/2014/858210.

Taqqu,M.S.(2013), Benoit Mandelbrot and fractional Brownian motion, Statistical Science,28, (1) https://doi.org/10.1214/12-STS389.

Villasenor J.A. & Gonzales-Estrada, E. (2015), A variance ratio test of fit for gamma distributions. Statistics and Probability Letters, 96,281-286. https://doi.org/10.1016/j.spl.2014.10.001.

Wyłomańska, A. (2012), Arithmetic Brownian motion subordinated by tempered stable and inverse tempered stable processes. Physica A, 391 (22), 5685-5696. https://doi.org/10.1016/j.physa.2012.05.072.

Xu,Y., Guo,R., Liu,D.& Zhang,H.(2014), Stochastic averaging principle for dynamical systems with fractional Brownian motion. Discrete and Continuous Dynamical Systems Series B., 19, (4), 1197-1212. https://doi.org/10.3934/dcdsb.2014.19.1197.

Wenting, C., Bowen, Y., Guanghua, L. & Ying, Z. (2016), Numerical pricing American options under the generalized mixed fractional Brownian motion model. Physica A, 451, 180-189. https://doi.org/10.1016/j.physa.2015.12.154.

Xiao, W., Zhang, W. & Zhang, X. (2015), Parameter identification for drift fractional Brownian motion with application to the Chinese stock markets. Communications in Statistics Simulation and Computation, 5, 44, 2117-2136 https://doi.org/10.1080/03610918.2013.849738.

Zhang, P.; Sun, Q. & Xiao, W.L. (2014), Parameter identification in mixed Brownian-fractional Brownian motions using Powell’s optimization algorithm. Econ. Model, 40,314-19. https://doi.org/10.1016/j.econmod.2014.04.026.

Mittinik,S.; Paolella, M.S.& Rachev, S.T.(2000), Diagnosing and treating the fat tails in financial return data. J. Empirical Finance, 7, 389-416. https://doi.org/10.1016/S0927-5398(00)00019-0.

Xiao, W., Weiguo, Z., Weijun, X. & Xili, Z. (2012), the valuation of equity warrants in a fractional Brownian environment. Physica A, 391,(4),1742-1752, https://doi.org/10.1016/j.physa.2011.10.024.

Xiao, W., Weiguo, Z. & Xili, Z. (2015), Parameter identification for the discretely observed geometric fractional Brownian motion. Journal of Statistical Computation and Simulation, 85, (2), 269-283. https://doi.org/10.1080/00949655.2013.814135.

Kleinert, H. & Korbel, J. (2016), Option pricing beyond Black-Scholes based on double-fractional diffusion. Physica A, 449, 200-214. https://doi.org/10.1016/j.physa.2015.12.125.

Zhijuan, M. & Zhian, L. (2014), Evaluation of geometric Asian power options under fractional Brownian motion. Journal of Mathematical Finance, 4, 1-9. https://doi.org/10.4236/jmf.2014.41001.

Stanley, H.E., Gabaix,X., Gopikrishnan,P.& Plerou,V.(2007), Quantifying extremely rare and less rare events in finance, Physica A, 382,286-301. https://doi.org/10.1016/j.physa.2007.02.023.

Kumar,A.,Wylomanska,A.,Poloczanski,R&Sundar,S.(2017), Fractional Brownian motion time-changed by gamma and inverse gamma process,Physica A. 468,648-667. https://doi.org/10.1016/j.physa.2016.10.060.

Wanging,S.,Ming,L.&Jian-Kai Liang(2016), Prediction of bearing fault using fractional Brownian motion and minimum entropy deconvolution,Entropy,MDPI,18,418,1-15.hysica A.

Lv,L,Ren,F.Y.&Qiu,W.Y.(2010),The application of fractional derivatives in stochastic models driven by fractional Brownian motion, Physica A,21,4809-4818.

Jeon, J. H. & Metzler,R.(2010),;Fractional Brownian motion and motion governed by the fractional Langevin equation in confined geometries. Phys.Rev.E,81,21-33. https://doi.org/10.1103/physreve.81.021103.

Thale,C.(2009),Further remarks on mixed fractional Brownian motion,Appl.Math.Sci.,3,38,1885-1901.

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