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Analytical pricing of geometric Asian power options on an underlying driven by a mixed fractional Brownian motion

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  • Zhang, Wei-Guo
  • Li, Zhe
  • Liu, Yong-Jun

Abstract

In this paper, we study the pricing problem of the continuously monitored fixed and floating strike geometric Asian power options in a mixed fractional Brownian motion environment. First, we derive both closed-form solutions and mixed fractional partial differential equations for fixed and floating strike geometric Asian power options based on delta-hedging strategy and partial differential equation method. Second, we present the lower and upper bounds of the prices of fixed and floating strike geometric Asian power options under the assumption that both risk-free interest rate and volatility are interval numbers. Finally, numerical studies are performed to illustrate the performance of our proposed pricing model.

Suggested Citation

  • Zhang, Wei-Guo & Li, Zhe & Liu, Yong-Jun, 2018. "Analytical pricing of geometric Asian power options on an underlying driven by a mixed fractional Brownian motion," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 490(C), pages 402-418.
    • Handle: RePEc:eee:phsmap:v:490:y:2018:i:c:p:402-418
    DOI: 10.1016/j.physa.2017.08.070
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    as
    1. Xiao, Wei-Lin & Zhang, Wei-Guo & Zhang, Xili & Zhang, Xiaoli, 2012. "Pricing model for equity warrants in a mixed fractional Brownian environment and its algorithm," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(24), pages 6418-6431.
    2. Kim, Jerim & Kim, Bara & Moon, Kyoung-Sook & Wee, In-Suk, 2012. "Valuation of power options under Heston's stochastic volatility model," Journal of Economic Dynamics and Control, Elsevier, vol. 36(11), pages 1796-1813.
    3. Min Dai, 2003. "One-state variable binomial models for European-/American-style geometric Asian options," Quantitative Finance, Taylor & Francis Journals, vol. 3(4), pages 288-295.
    4. Fusai, Gianluca & Meucci, Attilio, 2008. "Pricing discretely monitored Asian options under Levy processes," Journal of Banking & Finance, Elsevier, vol. 32(10), pages 2076-2088, October.
    5. Wang, Xiao-Tian & Li, Zhe & Zhuang, Le, 2017. "Risk preference, option pricing and portfolio hedging with proportional transaction costs," Chaos, Solitons & Fractals, Elsevier, vol. 95(C), pages 111-130.
    6. Ning Cai & Yingda Song & Steven Kou, 2015. "A General Framework for Pricing Asian Options Under Markov Processes," Operations Research, INFORMS, vol. 63(3), pages 540-554, June.
    7. Cajueiro, Daniel O. & Tabak, Benjamin M., 2007. "Long-range dependence and multifractality in the term structure of LIBOR interest rates," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 373(C), pages 603-614.
    8. Alziary, Benedicte & Decamps, Jean-Paul & Koehl, Pierre-Francois, 1997. "A P.D.E. approach to Asian options: analytical and numerical evidence," Journal of Banking & Finance, Elsevier, vol. 21(5), pages 613-640, May.
    9. Cen, Zhongdi & Xu, Aimin & Le, Anbo, 2015. "A hybrid finite difference scheme for pricing Asian options," Applied Mathematics and Computation, Elsevier, vol. 252(C), pages 229-239.
    10. L. C. G. Rogers, 1997. "Arbitrage with Fractional Brownian Motion," Mathematical Finance, Wiley Blackwell, vol. 7(1), pages 95-105, January.
    11. Yong-Jun Liu & Wei-Guo Zhang & Jun-Bo Wang, 2016. "Multi-period cardinality constrained portfolio selection models with interval coefficients," Annals of Operations Research, Springer, vol. 244(2), pages 545-569, September.
    12. Prakasa Rao, B.L.S., 2016. "Pricing geometric Asian power options under mixed fractional Brownian motion environment," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 446(C), pages 92-99.
    13. C. Brown & J. C. Handley & C.-T. Lin & K. J. Palmer, 2016. "Partial differential equations for Asian option prices," Quantitative Finance, Taylor & Francis Journals, vol. 16(3), pages 447-460, March.
    14. Lo, Andrew W, 1991. "Long-Term Memory in Stock Market Prices," Econometrica, Econometric Society, vol. 59(5), pages 1279-1313, September.
    15. Huang, Teng-Ching & Tu, Yu-Chen & Chou, Heng-Chih, 2015. "Long memory and the relation between options and stock prices," Finance Research Letters, Elsevier, vol. 12(C), pages 77-91.
    16. Zhang, Xili & Xiao, Weilin, 2017. "Arbitrage with fractional Gaussian processes," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 471(C), pages 620-628.
    17. Yue-Kuen Kwok & Hoi-Ying Wong, 2000. "Currency-Translated Foreign Equity Options With Path Dependent Features And Their Multi-Asset Extensions," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 3(02), pages 257-278.
    18. Yoshida, Yuji, 2003. "The valuation of European options in uncertain environment," European Journal of Operational Research, Elsevier, vol. 145(1), pages 221-229, February.
    19. Ballestra, Luca Vincenzo & Pacelli, Graziella & Radi, Davide, 2016. "A very efficient approach for pricing barrier options on an underlying described by the mixed fractional Brownian motion," Chaos, Solitons & Fractals, Elsevier, vol. 87(C), pages 240-248.
    20. Chung, Shing Fung & Wong, Hoi Ying, 2014. "Analytical pricing of discrete arithmetic Asian options with mean reversion and jumps," Journal of Banking & Finance, Elsevier, vol. 44(C), pages 130-140.
    21. Ballestra, Luca Vincenzo & Pacelli, Graziella & Radi, Davide, 2016. "A very efficient approach to compute the first-passage probability density function in a time-changed Brownian model: Applications in finance," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 463(C), pages 330-344.
    22. Vadim Linetsky, 2004. "Spectral Expansions for Asian (Average Price) Options," Operations Research, INFORMS, vol. 52(6), pages 856-867, December.
    23. Lixin Wu & Yue Kuen Kwok & Hong Yu, 1999. "Asian Options With The American Early Exercise Feature," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 2(01), pages 101-111.
    24. Boris Podobnik & Duan Wang & H. Eugene Stanley, 2012. "High-frequency trading model for a complex trading hierarchy," Quantitative Finance, Taylor & Francis Journals, vol. 12(4), pages 559-566, October.
    25. R. Cont, 2001. "Empirical properties of asset returns: stylized facts and statistical issues," Quantitative Finance, Taylor & Francis Journals, vol. 1(2), pages 223-236.
    26. Wang, Xingchun, 2016. "Pricing power exchange options with correlated jump risk," Finance Research Letters, Elsevier, vol. 19(C), pages 90-97.
    27. Hoi Ying Wong & Ying Lok Cheung, 2004. "Geometric Asian options: valuation and calibration with stochastic volatility," Quantitative Finance, Taylor & Francis Journals, vol. 4(3), pages 301-314.
    28. Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, vol. 81(3), pages 637-654, May-June.
    29. Ewald, Christian-Oliver & Menkens, Olaf & Hung Marten Ting, Sai, 2013. "Asian and Australian options: A common perspective," Journal of Economic Dynamics and Control, Elsevier, vol. 37(5), pages 1001-1018.
    30. El-Nouty, Charles, 2003. "The fractional mixed fractional Brownian motion," Statistics & Probability Letters, Elsevier, vol. 65(2), pages 111-120, November.
    31. Nowak, Piotr & Romaniuk, Maciej, 2010. "Computing option price for Levy process with fuzzy parameters," European Journal of Operational Research, Elsevier, vol. 201(1), pages 206-210, February.
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    Cited by:

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    3. Ahmadian, D. & Ballestra, L.V., 2020. "Pricing geometric Asian rainbow options under the mixed fractional Brownian motion," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 555(C).
    4. Foad Shokrollahi & Davood Ahmadian & Luca Vincenzo Ballestra, 2021. "Actuarial strategy for pricing Asian options under a mixed fractional Brownian motion with jumps," Papers 2105.06999, arXiv.org.
    5. Panhong Cheng & Zhihong Xu & Zexing Dai, 2023. "Valuation of vulnerable options with stochastic corporate liabilities in a mixed fractional Brownian motion environment," Mathematics and Financial Economics, Springer, volume 17, number 3, June.
    6. Dastranj, Elham & Sahebi Fard, Hossein & Abdolbaghi, Abdolmajid & Reza Hejazi, S., 2020. "Power option pricing under the unstable conditions (Evidence of power option pricing under fractional Heston model in the Iran gold market)," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 537(C).
    7. Wei-Guo Zhang & Zhe Li & Yong-Jun Liu & Yue Zhang, 2021. "Pricing European Option Under Fuzzy Mixed Fractional Brownian Motion Model with Jumps," Computational Economics, Springer;Society for Computational Economics, vol. 58(2), pages 483-515, August.
    8. Lu, Ziqiang & Zhu, Yuanguo & Li, Bo, 2019. "Critical value-based Asian option pricing model for uncertain financial markets," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 525(C), pages 694-703.
    9. Li, Zhe & Zhang, Wei-Guo & Liu, Yong-Jun, 2018. "Analytical valuation for geometric Asian options in illiquid markets," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 507(C), pages 175-191.

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