Journal of Applied Mathematics

Fractional Black-Scholes Model and Technical Analysis of Stock Price

Song Xu and Yujiao Yang

Full-text: Open access


In the stock market, some popular technical analysis indicators (e.g., Bollinger bands, RSI, ROC, etc.) are widely used to forecast the direction of prices. The validity is shown by observed relative frequency of certain statistics, using the daily (hourly, weekly, etc.) stock prices as samples. However, those samples are not independent. In earlier research, the stationary property and the law of large numbers related to those observations under Black-Scholes stock price model and stochastic volatility model have been discussed. Since the fitness of both Black-Scholes model and short-range dependent process has been questioned, we extend the above results to fractional Black-Scholes model with Hurst parameter H>1/2, under which the stock returns follow a kind of long-range dependent process. We also obtain the rate of convergence.

Article information

J. Appl. Math., Volume 2013 (2013), Article ID 631795, 7 pages.

First available in Project Euclid: 14 March 2014

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Xu, Song; Yang, Yujiao. Fractional Black-Scholes Model and Technical Analysis of Stock Price. J. Appl. Math. 2013 (2013), Article ID 631795, 7 pages. doi:10.1155/2013/631795.

Export citation


  • W. Liu, X. Huang, and W. Zheng, “Black-Scholes' model and Bollinger bands,” Physica A, vol. 371, no. 2, pp. 565–571, 2006.
  • W. Liu and W. A. Zheng, “Stochastic volatility model and technical analysis of stock price,” Acta Mathematica Sinica, vol. 27, no. 7, pp. 1283–1296, 2011.
  • X. Huang and W. Liu, “Properties of some statistics for AR-ARCH model with application to technical analysis,” Journal of Computational and Applied Mathematics, vol. 225, no. 2, pp. 522–530, 2009.
  • C. D, I. I. Kirkpatrick, and J. R. Dahlquist, Technical Analysis: The Complete Resource for Financial Market Technicians, Pearson Education, Upper Saddle River, NJ, USA, 2nd edition, 2011.
  • A. W. Lo, H. Mamaysky, and J. Wang, “Foundations of technical analysis: computational algorithms, statistical inference, and empirical implementation,” Journal of Finance, vol. 55, no. 4, pp. 1705–1770, 2000.
  • W. Willinger, M. S. Taqqu, and V. Teverovsky, “Stock market prices and long-range dependence,” Finance and Stochastics, vol. 3, no. 1, pp. 1–13, 1999.
  • A. W. Lo, “Fat tails, long memory, and the stock market since the 1960s,” Economic Notes, vol. 26, no. 2, pp. 219–252, 1997.
  • P. Robinson, Ed., Time Series with Long Memory, Advanced Texts in Econometrics, Oxford University Press, New York, NY, USA, 2003.
  • R. Cont, “Long range dependence in financial markets,” in Fractals in Engineering, E. Lutton and J. Vehel, Eds., Springer, London, UK, 2005.
  • D. O. Cajueiro and B. M. Tabak, “Testing for time-varying long-range dependence in real state equity returns,” Chaos, Solitons and Fractals, vol. 38, no. 1, pp. 293–307, 2008.
  • P. Cheridito, “Arbitrage in fractional Brownian motion models,” Finance and Stochastics, vol. 7, no. 4, pp. 533–553, 2003.
  • T. E. Duncan, Y. Hu, and B. Pasik-Duncan, “Stochastic calculus for fractional Brownian motion. I. Theory,” SIAM Journal on Control and Optimization, vol. 38, no. 2, pp. 582–612, 2000.
  • Y. Hu, B. ${\text{\O}}$ksendal, and A. Sulem, “Optimal consumption and portfolio in a Black-Scholes market driven by fractional Brownian motion,” Infinite Dimensional Analysis, Quantum Probability and Related Topics, vol. 6, no. 4, pp. 519–536, 2003.
  • J. Bollinger, Bollinger on Bollinger Bands, McGraw-Hill, New York, NY, USA, 2002.
  • J. W. Wilder, New Concepts in Technical Trading Systems, Trend Research, Greensboro, NC, USA, 1st edition, 1978.
  • W. Zhu, Statistical analysis of stock technical indicators [M.S. thesis], East China Normal University, Shanghai, China, 2006.
  • W. Li, A new standard of effectiveness of moving average technical indicators in stock markets [M.S. thesis], East China Normal University, Shanghai, China, 2006.
  • W. Deng and E. Barkai, “Ergodic properties of fractional Brownian-Langevin motion,” Physical Review E, vol. 79, no. 1, Article ID 011112, 7 pages, 2009.
  • D. S. Bernstein, Matrix Mathematics: Theory, Facts, and Formulas with Application to Linear Systems Theory, Princeton University Press, Princeton, NJ, USA, 2005.