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Analytical valuation for geometric Asian options in illiquid markets

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

Abstract

Classical option pricing theories are usually built on the paradigm of competitive and frictionless markets, while ignoring the impact of market liquidity on asset prices. In this paper, we study the pricing problems of the continuously monitored fixed and floating strike Asian options in imperfectly liquid markets. First, we assume that the dynamics of the underlying asset price follow a liquidity-adjusted Black–Scholes model. And then we derive the partial differential equation for the continuous fixed and floating strike geometric Asian options based on the delta-hedging strategy. Meanwhile, we also present the PDE governing the price of the continuous fixed and floating strike arithmetic Asian options. Second, the analytical pricing formulas for the continuous fixed and floating strike geometric Asian call options are derived by using the PDE method. Besides, we demonstrate the put–call parity relations for the continuous fixed and floating strike geometric Asian options. Finally, numerical experiments are performed to illustrate the accuracy and efficiency of the proposed liquidity-adjusted option pricing model through comparing the analytical solution with Monte Carlo simulation. Furthermore, we investigate the sensitivity of the continuous arithmetic and geometric Asian option prices to the liquidity factors. The numerical results support our idea of introducing market liquidity effect into option pricing framework.

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  • 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.
    • Handle: RePEc:eee:phsmap:v:507:y:2018:i:c:p:175-191
    DOI: 10.1016/j.physa.2018.05.069
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    as
    1. Chi-Hsiou D. Hung & Qiuliang Chen & Victor Fang, 2015. "Non-Tradable Share Reform, Liquidity, and Stock Returns in China," International Review of Finance, International Review of Finance Ltd., vol. 15(1), pages 27-54, March.
    2. Menachem Brenner & Rafi Eldor & Shmuel Hauser, 2001. "The Price of Options Illiquidity," Journal of Finance, American Finance Association, vol. 56(2), pages 789-805, April.
    3. Francis A. Longstaff & Sanjay Mithal & Eric Neis, 2005. "Corporate Yield Spreads: Default Risk or Liquidity? New Evidence from the Credit Default Swap Market," Journal of Finance, American Finance Association, vol. 60(5), pages 2213-2253, October.
    4. Acharya, Viral V. & Pedersen, Lasse Heje, 2005. "Asset pricing with liquidity risk," Journal of Financial Economics, Elsevier, vol. 77(2), pages 375-410, August.
    5. Akiko Watanabe & Masahiro Watanabe, 2008. "Time-Varying Liquidity Risk and the Cross Section of Stock Returns," The Review of Financial Studies, Society for Financial Studies, vol. 21(6), pages 2449-2486, November.
    6. 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.
    7. U. Çetin & R. Jarrow & P. Protter & M. Warachka, 2008. "Pricing Options in an Extended Black Scholes Economy with Illiquidity: Theory and Empirical Evidence," World Scientific Book Chapters, in: Financial Derivatives Pricing Selected Works of Robert Jarrow, chapter 9, pages 185-221, World Scientific Publishing Co. Pte. Ltd..
    8. Peter Christoffersen & Ruslan Goyenko & Kris Jacobs & Mehdi Karoui, 2018. "Illiquidity Premia in the Equity Options Market," The Review of Financial Studies, Society for Financial Studies, vol. 31(3), pages 811-851.
    9. Pastor, Lubos & Stambaugh, Robert F., 2003. "Liquidity Risk and Expected Stock Returns," Journal of Political Economy, University of Chicago Press, vol. 111(3), pages 642-685, June.
    10. Friedrich Hubalek & Martin Keller-Ressel & Carlo Sgarra, 2017. "Geometric Asian option pricing in general affine stochastic volatility models with jumps," Quantitative Finance, Taylor & Francis Journals, vol. 17(6), pages 873-888, June.
    11. 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.
    12. Umut Çetin & Robert A. Jarrow & Philip Protter, 2008. "Liquidity risk and arbitrage pricing theory," World Scientific Book Chapters, in: Financial Derivatives Pricing Selected Works of Robert Jarrow, chapter 8, pages 153-183, World Scientific Publishing Co. Pte. Ltd..
    13. Feng, Shih-Ping & Hung, Mao-Wei & Wang, Yaw-Huei, 2014. "Option pricing with stochastic liquidity risk: Theory and evidence," Journal of Financial Markets, Elsevier, vol. 18(C), pages 77-95.
    14. repec:bla:jfinan:v:44:y:1989:i:2:p:479-86 is not listed on IDEAS
    15. Leippold, Markus & Schärer, Steven, 2017. "Discrete-time option pricing with stochastic liquidity," Journal of Banking & Finance, Elsevier, vol. 75(C), pages 1-16.
    16. Kemna, A. G. Z. & Vorst, A. C. F., 1990. "A pricing method for options based on average asset values," Journal of Banking & Finance, Elsevier, vol. 14(1), pages 113-129, March.
    17. Amihud, Yakov, 2002. "Illiquidity and stock returns: cross-section and time-series effects," Journal of Financial Markets, Elsevier, vol. 5(1), pages 31-56, January.
    18. Soner, H. Mete & Cetin, Umut & Touzi, Nizar, 2010. "Option hedging for small investors under liquidity costs," LSE Research Online Documents on Economics 28992, London School of Economics and Political Science, LSE Library.
    19. 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.
    20. Robin K. Chou & San‐Lin Chung & Yu‐Jen Hsiao & Yaw‐Huei Wang, 2011. "The impact of liquidity on option prices," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 31(12), pages 1116-1141, December.
    21. Albrecher, Hansjoerg & Guillaume, Florence & Schoutens, Wim, 2013. "Implied liquidity: Model sensitivity," Journal of Empirical Finance, Elsevier, vol. 23(C), pages 48-67.
    22. Cui, Zhenyu & Lee, Chihoon & Liu, Yanchu, 2018. "Single-transform formulas for pricing Asian options in a general approximation framework under Markov processes," European Journal of Operational Research, Elsevier, vol. 266(3), pages 1134-1139.
    23. Vorst, Ton, 1992. "Prices and hedge ratios of average exchange rate options," International Review of Financial Analysis, Elsevier, vol. 1(3), pages 179-193.
    24. Feng, Shih-Ping & Hung, Mao-Wei & Wang, Yaw-Huei, 2016. "The importance of stock liquidity on option pricing," International Review of Economics & Finance, Elsevier, vol. 43(C), pages 457-467.
    25. Turan G. Bali & Lin Peng & Yannan Shen & Yi Tang, 2014. "Liquidity Shocks and Stock Market Reactions," The Review of Financial Studies, Society for Financial Studies, vol. 27(5), pages 1434-1485.
    26. Lee, Min-Ku, 2016. "Asymptotic approach to the pricing of geometric asian options under the CEV model," Chaos, Solitons & Fractals, Elsevier, vol. 91(C), pages 544-548.
    27. Brennan, Michael J. & Subrahmanyam, Avanidhar, 1996. "Market microstructure and asset pricing: On the compensation for illiquidity in stock returns," Journal of Financial Economics, Elsevier, vol. 41(3), pages 441-464, July.
    28. Alessio Caldarera & Celso Brunetti, 2005. "Asset Prices and Asset Correlations in Illiquid Markets," 2005 Meeting Papers 288, Society for Economic Dynamics.
    29. Lee, Kuan-Hui, 2011. "The world price of liquidity risk," Journal of Financial Economics, Elsevier, vol. 99(1), pages 136-161, January.
    30. Cao, Melanie & Wei, Jason, 2010. "Option market liquidity: Commonality and other characteristics," Journal of Financial Markets, Elsevier, vol. 13(1), pages 20-48, February.
    31. John E. Angus, 1999. "A note on pricing Asian derivatives with continuous geometric averaging," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 19(7), pages 845-858, October.
    32. Lars Nordén & Caihong Xu, 2012. "Option happiness and liquidity: Is the dynamics of the volatility smirk affected by relative option liquidity?," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 32(1), pages 47-74, January.
    33. Li, Zhe & Zhang, Wei-Guo & Liu, Yong-Jun, 2018. "European quanto option pricing in presence of liquidity risk," The North American Journal of Economics and Finance, Elsevier, vol. 45(C), pages 230-244.
    34. Bara Kim & In-Suk Wee, 2014. "Pricing of geometric Asian options under Heston's stochastic volatility model," Quantitative Finance, Taylor & Francis Journals, vol. 14(10), pages 1795-1809, October.
    35. Eleswarapu, Venkat R. & Reinganum, Marc R., 1993. "The seasonal behavior of the liquidity premium in asset pricing," Journal of Financial Economics, Elsevier, vol. 34(3), pages 373-386, December.
    36. 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.
    37. Umut Çetin & H. Soner & Nizar Touzi, 2010. "Option hedging for small investors under liquidity costs," Finance and Stochastics, Springer, vol. 14(3), pages 317-341, September.
    38. Ajay Subramanian & Robert A. Jarrow, 2001. "The Liquidity Discount," Mathematical Finance, Wiley Blackwell, vol. 11(4), pages 447-474, October.
    39. 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.
    40. 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.
    41. Liu, Hong & Yong, Jiongmin, 2005. "Option pricing with an illiquid underlying asset market," Journal of Economic Dynamics and Control, Elsevier, vol. 29(12), pages 2125-2156, December.
    42. Michael Ludkovski & Qunying Shen, 2013. "European Option Pricing With Liquidity Shocks," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 16(07), pages 1-30.
    43. Datar, Vinay T. & Y. Naik, Narayan & Radcliffe, Robert, 1998. "Liquidity and stock returns: An alternative test," Journal of Financial Markets, Elsevier, vol. 1(2), pages 203-219, August.
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