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Pricing continuously sampled Asian options with perturbation method

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  • Jin E. Zhang

Abstract

This article explores the price of continuously sampled Asian options. For geometric Asian options, we present pricing formulas for both backward‐starting and forward‐starting cases. For arithmetic Asian options, we demonstrate that the governing partial differential equation (PDE) cannot be transformed into a heat equation with constant coefficients; therefore, these options do not have a closed‐form solution of the Black–Scholes type, that is, the solution is not given in terms of the cumulative normal distribution function. We then solve the PDE with a perturbation method and obtain an analytical solution in a series form. Numerical results show that as compared with Zhang's ( 2001 ) highly accurate numerical results, the series converges very quickly and gives a good approximate value that is more accurate than any other approximate method in the literature, at least for the options tested in this article. Graphical results determine that the solution converges globally very quickly especially near the origin, which is the area in which most of the traded Asian options fall. © 2003 Wiley Periodicals, Inc. Jrl Fut Mark 23:535–560, 2003

Suggested Citation

  • Jin E. Zhang, 2003. "Pricing continuously sampled Asian options with perturbation method," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 23(6), pages 535-560, June.
    • Handle: RePEc:wly:jfutmk:v:23:y:2003:i:6:p:535-560
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    Cited by:

    1. Ning Cai & Steven Kou, 2012. "Pricing Asian Options Under a Hyper-Exponential Jump Diffusion Model," Operations Research, INFORMS, vol. 60(1), pages 64-77, February.
    2. Yishen Li & Jin Zhang, 2004. "Option pricing with Weyl-Titchmarsh theory," Quantitative Finance, Taylor & Francis Journals, vol. 4(4), pages 457-464.
    3. Yulian Fan & Huadong Zhang, 2017. "The pricing of average options with jump diffusion processes in the uncertain volatility model," International Journal of Financial Engineering (IJFE), World Scientific Publishing Co. Pte. Ltd., vol. 4(01), pages 1-31, March.
    4. Juraj Hruška, 2015. "Delta-gamma-theta Hedging of Crude Oil Asian Options," Acta Universitatis Agriculturae et Silviculturae Mendelianae Brunensis, Mendel University Press, vol. 63(6), pages 1897-1903.
    5. 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.
    6. Chuang-Chang Chang & Chueh-Yung Tsao, 2011. "Efficient and accurate quadratic approximation methods for pricing Asian strike options," Quantitative Finance, Taylor & Francis Journals, vol. 11(5), pages 729-748.
    7. J. Lars Kirkby & Duy Nguyen, 2020. "Efficient Asian option pricing under regime switching jump diffusions and stochastic volatility models," Annals of Finance, Springer, vol. 16(3), pages 307-351, September.
    8. Gifty Malhotra & R. Srivastava & H. C. Taneja, 2019. "Pricing of the Geometric Asian Options Under a Multifactor Stochastic Volatility Model," Papers 1912.10640, arXiv.org.
    9. Lu, King-Jeng & Liang, Chiung-Ju & Hsieh, Ming-Hua & Lee, Yi-Hsi, 2020. "An effective hybrid variance reduction method for pricing the Asian options and its variants," The North American Journal of Economics and Finance, Elsevier, vol. 51(C).
    10. Chiu, Chun-Yuan & Dai, Tian-Shyr & Lyuu, Yuh-Dauh, 2015. "Pricing Asian option by the FFT with higher-order error convergence rate under Lévy processes," Applied Mathematics and Computation, Elsevier, vol. 252(C), pages 418-437.
    11. William T. Shaw & Marcus Schofield, 2015. "A model of returns for the post-credit-crunch reality: hybrid Brownian motion with price feedback," Quantitative Finance, Taylor & Francis Journals, vol. 15(6), pages 975-998, June.

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