Evaluation of Geometric Asian Power Options under Fractional Brownian Motion

Affiliation(s)

School of Finance, Shanghai University of Finance and Economics, Shanghai, China.

Department of Applied Mathematics, Shanghai University of Finance and Economics, Shanghai, China.

School of Finance, Shanghai University of Finance and Economics, Shanghai, China.

Department of Applied Mathematics, Shanghai University of Finance and Economics, Shanghai, China.

ABSTRACT

Modern option pricing techniques are often considered among the most mathematical complex of all applied areas of financial mathematics. In particular, the fractional Brownian motion is proper to model the stock dynamics for its long-range dependence. In this paper, we evaluate the price of geometric Asian options under fractional Brownian motion framework. Furthermore, the options are generalized to those with the added feature whose payoff is a power function. Based on the equivalent martingale theory, a closed form solution has been derived under the risk neutral probability.

Cite this paper

Z. Mao and Z. Liang, "Evaluation of Geometric Asian Power Options under Fractional Brownian Motion,"*Journal of Mathematical Finance*, Vol. 4 No. 1, 2014, pp. 1-9. doi: 10.4236/jmf.2014.41001.

Z. Mao and Z. Liang, "Evaluation of Geometric Asian Power Options under Fractional Brownian Motion,"

References

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[15] F. Biagini, Y. Hu, B. Oksendal and T. Zhang, “Stochastic Calculus for Fractional Brownian motion and Applications,” Springer, Berlin, 2008. http://dx.doi.org/10.1007/978-1-84628-797-8

[16] G. Fusai and A. Meucci, “Pricing Discretely Monitored Asian Options under Levy Processes,” Journal of Bank Finance, Vol. 32, No. 10, 2008, pp. 2076-2088.

http://dx.doi.org/10.1016/j.jbankfin.2007.12.027

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http://dx.doi.org/10.1137/S0363012999351826

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[19] Y. Xiao and J. Zhou, “Observed Information Based Option Pricing Model in Fractional B-S Market,” Mathematics in Economics, Vol. 25, No. 2, 2008, pp. 171-174.

[20] Y. J. Wang, Y. Zhang and S. Fan, “Pricing of Power-Asian Options,” Journal of Xuzhou Institute of Architectural Technology, Vol. 6, Vol. 2, 2006, pp. 39-41.

[21] D. Zhao, “The Pricing of European Power Options Based on FBM,” Mathematics in Economics, Vol. 24, No.1, 2007, pp. 22-26.

[22] Sh. Zhou and H. Liu, “Power Option Pricing under FBM,” College Mathematics, Vol. 25, No. 5, 2009, pp. 69-72.

[1] B. Rosenow, “Fluctuations and Market Friction in Financial Trading,” International Journal of Modern Physics, Vol. 13, No. 3, 2002, p. 419. http://dx.doi.org/10.1142/S012918310200322X

[2] R. P. Feynman, “Space-Time Approach to Non-Relativistic Quantum Mechanics,” Reviews of Modern Physics, Vol. 20, No. 2, 1948, pp. 367-387. http://dx.doi.org/10.1103/RevModPhys.20.367

[3] N. Wiener, “The Average of an Analytical Functional,” Proceedings of the National Academy of Sciences, Vol. 7, No. 9, 1921, pp. 253-260. http://dx.doi.org/10.1073/pnas.7.9.253

[4] R. Feynman and H. Kleinert, “Effective Classical Partition Functions,” Physical Review, Vol. 34, 1986, pp. 5080-5084.

[5] A. Kemna and A. Vorst, “A Pricing Method for Options Based on Average Values,” Journal of Banking and Finance, Vol. 14, No. 1, 1990, pp. 113-129. http://dx.doi.org/10.1016/0378-4266(90)90039-5

[6] L. C. G. Rogers and Z. Shi, “The Value of an Asian Option,” JAP, Vol. 32, 1995, pp. 1077-1088.

[7] L. Bachelier, “Theory of Speculation: The Origins of Modern Finance,” Princeton University Press, Princeton, 1900.

[8] F. Black and M. Scholes, “The Pricing of Options and Corporate Liabilities,” Journal of Political Economy, Vol. 81, No. 3, 1973, pp. 637-654.

[9] M. Li and M. Wang, “Comparison of Black-Scholes Formula with Fractional Black-Scholes Formula in the Foreign Exchange Option Market with Changing Volatility,” APFM, Vol.17, 2010, pp. 99-111.

[10] A. M. Yaglom, “Correlation Theory of Processes with Random Stationary nth Increments,” AMS Translation, Vol. 2, No. 8, 1958, pp. 87-141.

[11] C. Necula, “Option Pricing in a Fracitonal Brownian Motion Environment,” Bucharest University of Economics, Bucharest, 2008.

[12] B. Mandelbrot and H. Taylor, “On the Distribution of Stock Price Differences,” Operations Research, Vol. 15, No. 6, 1967, pp. 1057-1062. http://dx.doi.org/10.1287/opre.15.6.1057

[13] E. E. Peters, “Fractal Structure in the Capital Markets,” FAJ, Vol. 45, No. 4, 1989, pp. 32-37.

[14] Y. Hu and B. Oksendal, “Fractional White Noise Calculus and Applications to Finance,” Infinite Dimensional Analysis, Vol. 6, No. 1, 2003, pp. 1-32.

[15] F. Biagini, Y. Hu, B. Oksendal and T. Zhang, “Stochastic Calculus for Fractional Brownian motion and Applications,” Springer, Berlin, 2008. http://dx.doi.org/10.1007/978-1-84628-797-8

[16] G. Fusai and A. Meucci, “Pricing Discretely Monitored Asian Options under Levy Processes,” Journal of Bank Finance, Vol. 32, No. 10, 2008, pp. 2076-2088.

http://dx.doi.org/10.1016/j.jbankfin.2007.12.027

[17] T. E. Duncan, B. Maslowski and B. P. Duncan, “Adaptive Control for Semi-Linear Stochastic Systems,” Journal on Control and Optimization, Vol. 38, No. 6, 2000, pp. 1683-1706.

http://dx.doi.org/10.1137/S0363012999351826

[18] Y. J. Wang and S. W. Zhou, “The Pricing of European Power Options,” Journal of Gansu Science, Vol. 17, No. 2, 2005, pp. 21-23.

[19] Y. Xiao and J. Zhou, “Observed Information Based Option Pricing Model in Fractional B-S Market,” Mathematics in Economics, Vol. 25, No. 2, 2008, pp. 171-174.

[20] Y. J. Wang, Y. Zhang and S. Fan, “Pricing of Power-Asian Options,” Journal of Xuzhou Institute of Architectural Technology, Vol. 6, Vol. 2, 2006, pp. 39-41.

[21] D. Zhao, “The Pricing of European Power Options Based on FBM,” Mathematics in Economics, Vol. 24, No.1, 2007, pp. 22-26.

[22] Sh. Zhou and H. Liu, “Power Option Pricing under FBM,” College Mathematics, Vol. 25, No. 5, 2009, pp. 69-72.