Is the Driving Force of a Continuous Process a Brownian Motion or Fractional Brownian Motion?

Affiliation(s)

School of Management, Fudan University, Shanghai, China.

Department of Mathematics, Hong Kong University of Science and Technology, Hong Kong, China; School of Mathematics and Statistics, Lanzhou University, Lanzhou, China.

School of Management, Fudan University, Shanghai, China.

Department of Mathematics, Hong Kong University of Science and Technology, Hong Kong, China; School of Mathematics and Statistics, Lanzhou University, Lanzhou, China.

ABSTRACT

It?’s semimartingale driven by a Brownian motion is typically used in modeling the asset prices, interest rates and exchange rates, and so on. However, the assumption of Brownian motion as a driving force of the underlying asset price processes is rarely contested in practice. This naturally raises the question of whether this assumption is really appropriate. In the paper we propose a statistical test to answer the above question using high frequency data. The test can be used to validate the assumption of semimartingale framework and test for the existence of the long run dependence captured by the fractional Brownian motion in a parsimonious way. Asymptotic properties of the test statistics are investigated. Simulations justify the performance of the test. Real data sets are also analyzed.

Cite this paper

X. Kong, B. Jing and C. Li, "Is the Driving Force of a Continuous Process a Brownian Motion or Fractional Brownian Motion?,"*Journal of Mathematical Finance*, Vol. 3 No. 4, 2013, pp. 454-464. doi: 10.4236/jmf.2013.34048.

X. Kong, B. Jing and C. Li, "Is the Driving Force of a Continuous Process a Brownian Motion or Fractional Brownian Motion?,"

References

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http://dx.doi.org/10.1214/07-AOS568

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http://dx.doi.org/10.1214/09-AOS749

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http://dx.doi.org/10.1093/jjfinec/nbh001

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[15] B. Oksendal and Y. Hu, “Fractional White Noise and Applications to Finance,” Infinite Dimensional Analysis, Quantum Probability and Related Topics, Vol. 6, No. 1, 2000, pp. 1-32.

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[17] S. Si, “Two-Step Variations for Processes Driven by Fractional Brownian Motion with Applications in Testing for Jumps form the High Frequency Data,” Ph.D. Thesis, University of Tennessee, Knoxville, 2009.

[18] L. Zhang, P. A. Mykland and Y. Ait-Sahalia, “A Tale of Two Time Scales: Determining Integrated Volatility with Noisy High-Frequency Data,” Journal of the American Statistical Association, Vol. 100, No. 472, 2005, pp. 1394-1411.

http://dx.doi.org/10.1198/016214505000000169

[19] O. E. Barndorff-Nielsen, S. Graversen, J. Jacod and N. Shephard, “A Central Limit Theorem for Realized Power and Bipower Variations of Continuous Semi Martingales,” From Stochastic Calculus to Mathematics, 2006, pp. 33-68.

[20] D. Nualart, “Stochastic Calculus with Respect to the Fractional Brownian Motion and Applications,” Contemporary Mathematics, Vol. 336, 2003, pp. 3-39.

http://dx.doi.org/10.1090/conm/336/06025

[21] C. Mancini, “Estimating the Integrated Volatility in Stochastic Volatility Models with Lévy Type Jumps,” Technical Report, University di Firenze, Firenze, 2004.

[22] J. Jacod, Y. Li, P. A. Mykland, M. Podolskij and M. Vetter, “Microstructure Noise in the Continuous Case: The Pre-Averaging Approach,” Stochastic Processes and Their Applications, Vol. 119, No. 7, 2009, pp. 2249-2276.

http://dx.doi.org/10.1016/j.spa.2008.11.004

[1] F. Black and M. Scholes, “The Pricing of Options and Corporate Liabilities,” Journal of Political Economy, Vol. 81, No. 3, 1973, pp. 133-155.

[2] J. Hull and A. White, “Pricing Interest Rate Derivative Securities,” Review of Financial Studies, Vol. 3, No. 4, 1990, pp. 573-592. http://dx.doi.org/10.1093/rfs/3.4.573

[3] S. Heston, “A Closed-From Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options,” Review of Financial Studies, Vol. 6, No. 2, 1993, pp. 327-343.

http://dx.doi.org/10.1093/rfs/6.2.327

[4] Y. Ait-Sahalia and J. Jacod, “Testing for Jumps in a Discretely Observed Process,” The Annals of Statistics, Vol. 37, No. 1, 2009, pp. 184-222.

http://dx.doi.org/10.1214/07-AOS568

[5] Y. Ait-Sahalia and J. Jacod, “Is Brownian Motion Necessary to Model High Frequency Data?” The Annals of Statistics, Vol. 38, No. 5, 2010, pp. 3093-3128.

http://dx.doi.org/10.1214/09-AOS749

[6] O. E. Barndorff-Nielsen and N. Shephard, “Econometrics of Testing for Jumps in Financial Economics Using Bipower Variation,” Journal of Financial Econometrics, Vol. 2, No. 1, 2006, pp. 1-48.

http://dx.doi.org/10.1093/jjfinec/nbh001

[7] V. Todorov and G. Tanchen, “Activity Signature Functions for High-Frequency Data Analysis,” Journal of Econometrics, Vol. 154, No. 2, 2010, pp. 125-138.

http://dx.doi.org/10.1016/j.jeconom.2009.06.009

[8] M. T. Greene and B. D. Fielitz, “Long term Dependence in Common Stock Returns,” Journal of Financial Economics, Vol. 4, No. 3, 1977, pp. 339-349.

http://dx.doi.org/10.1016/0304-405X(77)90006-X

[9] B. B. Mandelbrot, “When Can Price Be Arbitraged Efficiently? A Limit to the Validity of the Random Walk and Martingale Models,” Review of Economic Statistics, Vol. 53, No. 3, 1971, pp. 225-236.

http://dx.doi.org/10.2307/1937966

[10] N. J. Cutland, P. E. Kopp and W. Willinger, “Stock Price Returns and the Joseph Effect: A Fractal Version of the Black-Scholes Model,” Progress in Probability, Vol. 36, 1995, pp. 327-351.

[11] E. Bayraktar, H. V. Poor and R. Sircar, “Estimating the fractal Dimension of the S&P 500 Index Using Wavelet Analysis,” International Journal of Theoretical and Applied Finance, Vol. 7, No. 5, 2004, pp. 615-643.

http://dx.doi.org/10.1142/S021902490400258X

[12] J. Corcuera, D. Nualart and J. Woerner, “Power Variation of Some Integral Fractional Processes,” Bernoulli, Vol. 12, No. 4, 2006, pp. 713-735.

http://dx.doi.org/10.3150/bj/1155735933

[13] L. Rogers, “Arbitrage with Fractional Brownian Motion,” Mathematical Finance, Vol. 7, No. 1, 1997, pp. 95-105.

http://dx.doi.org/10.1111/1467-9965.00025

[14] T. E. Duncan, Y. Hu and B. Pasik-Duncan, “Stochastic Calculus for Fractional Brownian Motion,” SIAM Journal on Control and Optimization, Vol. 38, No. 1, 2000, pp. 582-612.

[15] B. Oksendal and Y. Hu, “Fractional White Noise and Applications to Finance,” Infinite Dimensional Analysis, Quantum Probability and Related Topics, Vol. 6, No. 1, 2000, pp. 1-32.

[16] P. Cheridito, “Arbitrage in Fractional Brownian Motion Models,” Finance and Stochastics, Vol. 7, No. 4, 2003, pp. 533-553. http://dx.doi.org/10.1007/s007800300101

[17] S. Si, “Two-Step Variations for Processes Driven by Fractional Brownian Motion with Applications in Testing for Jumps form the High Frequency Data,” Ph.D. Thesis, University of Tennessee, Knoxville, 2009.

[18] L. Zhang, P. A. Mykland and Y. Ait-Sahalia, “A Tale of Two Time Scales: Determining Integrated Volatility with Noisy High-Frequency Data,” Journal of the American Statistical Association, Vol. 100, No. 472, 2005, pp. 1394-1411.

http://dx.doi.org/10.1198/016214505000000169

[19] O. E. Barndorff-Nielsen, S. Graversen, J. Jacod and N. Shephard, “A Central Limit Theorem for Realized Power and Bipower Variations of Continuous Semi Martingales,” From Stochastic Calculus to Mathematics, 2006, pp. 33-68.

[20] D. Nualart, “Stochastic Calculus with Respect to the Fractional Brownian Motion and Applications,” Contemporary Mathematics, Vol. 336, 2003, pp. 3-39.

http://dx.doi.org/10.1090/conm/336/06025

[21] C. Mancini, “Estimating the Integrated Volatility in Stochastic Volatility Models with Lévy Type Jumps,” Technical Report, University di Firenze, Firenze, 2004.

[22] J. Jacod, Y. Li, P. A. Mykland, M. Podolskij and M. Vetter, “Microstructure Noise in the Continuous Case: The Pre-Averaging Approach,” Stochastic Processes and Their Applications, Vol. 119, No. 7, 2009, pp. 2249-2276.

http://dx.doi.org/10.1016/j.spa.2008.11.004