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Pricing of Arithmetic Average Asian Option by Combining Variance Reduction and Quasi-Monte Carlo Method

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  • Lingling Xu

    (School of Science and Technology, Hong Kong Metropolitan University, Hong Kong 999077, China)

  • Hongjie Zhang

    (Zhongtai Securities Institute for Financial Studies, Shandong University, Jinan 250100, China)

  • Fu Lee Wang

    (School of Science and Technology, Hong Kong Metropolitan University, Hong Kong 999077, China)

Abstract

Financial derivatives have developed rapidly over the past few decades due to their risk-averse nature, with options being the preferred financial derivatives due to their flexible contractual mechanisms, particularly Asian options. The Black–Scholes stock option pricing model is often used in conjunction with Monte Carlo simulations for option pricing. However, the Black–Scholes model assumes that the volatility of asset returns is constant, which does not square with practical financial markets. Additionally, Monte Carlo simulation suffers from slow error convergence. To address these issues, we first correct the asset volatility in the Black–Scholes model using a GARCH model. Then, the low error convergence rate of the Monte Carlo method is improved using variance reduction techniques. Meanwhile, the quasi-Monte Carlo approach based on low discrepancy sequences is used to refine the error convergence rate. We also provide a simulation experiment and result analysis to validate the effectiveness of our proposed method.

Suggested Citation

  • Lingling Xu & Hongjie Zhang & Fu Lee Wang, 2023. "Pricing of Arithmetic Average Asian Option by Combining Variance Reduction and Quasi-Monte Carlo Method," Mathematics, MDPI, vol. 11(3), pages 1-14, January.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:3:p:594-:d:1044657
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    References listed on IDEAS

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    1. Nelson, Daniel B, 1991. "Conditional Heteroskedasticity in Asset Returns: A New Approach," Econometrica, Econometric Society, vol. 59(2), pages 347-370, March.
    2. Ren Raw Chen & Cheng Few Lee & Han-Hsing Lee, 2020. "Empirical Performance of the Constant Elasticity Variance Option Pricing Model," World Scientific Book Chapters, in: Cheng Few Lee & John C Lee (ed.), HANDBOOK OF FINANCIAL ECONOMETRICS, MATHEMATICS, STATISTICS, AND MACHINE LEARNING, chapter 51, pages 1903-1942, World Scientific Publishing Co. Pte. Ltd..
    3. T. Harikumar & Maria E. de Boyrie & Simon J. Pak, 2004. "Evaluation of Black-Scholes and GARCH Models Using Currency Call Options Data," Review of Quantitative Finance and Accounting, Springer, vol. 23(4), pages 299-312, December.
    4. Robert C. Merton, 2005. "Theory of rational option pricing," World Scientific Book Chapters, in: Sudipto Bhattacharya & George M Constantinides (ed.), Theory Of Valuation, chapter 8, pages 229-288, World Scientific Publishing Co. Pte. Ltd..
    5. J. D. Beasley & S. G. Springer, 1977. "The Percentage Points of the Normal Distribution," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 26(1), pages 118-121, March.
    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. Steven P. Clark & Mike Dickson, 2019. "Performance expectations of basic options strategies may be different than you think," Journal of Asset Management, Palgrave Macmillan, vol. 20(2), pages 91-102, March.
    8. Jan Kallsen & Murad S. Taqqu, 1998. "Option Pricing in ARCH‐type Models," Mathematical Finance, Wiley Blackwell, vol. 8(1), pages 13-26, January.
    9. Heston, Steven L, 1993. "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options," The Review of Financial Studies, Society for Financial Studies, vol. 6(2), pages 327-343.
    10. 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.
    11. Cox, John C. & Ross, Stephen A., 1976. "The valuation of options for alternative stochastic processes," Journal of Financial Economics, Elsevier, vol. 3(1-2), pages 145-166.
    12. Corwin Joy & Phelim P. Boyle & Ken Seng Tan, 1996. "Quasi-Monte Carlo Methods in Numerical Finance," Management Science, INFORMS, vol. 42(6), pages 926-938, June.
    13. Glosten, Lawrence R & Jagannathan, Ravi & Runkle, David E, 1993. "On the Relation between the Expected Value and the Volatility of the Nominal Excess Return on Stocks," Journal of Finance, American Finance Association, vol. 48(5), pages 1779-1801, December.
    14. 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.
    15. Bernard Lapeyre & Emmanuel Temam, 2001. "Competitive Monte Carlo methods for the pricing of Asian options," Post-Print hal-01667057, HAL.
    16. Bollerslev, Tim, 1986. "Generalized autoregressive conditional heteroskedasticity," Journal of Econometrics, Elsevier, vol. 31(3), pages 307-327, April.
    17. Cox, John C & Ross, Stephen A, 1976. "A Survey of Some New Results in Financial Option Pricing Theory," Journal of Finance, American Finance Association, vol. 31(2), pages 383-402, May.
    18. 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.
    19. Kleijnen, Jack P.C. & Ridder, A.A.N. & Rubinstein, R.Y., 2010. "Variance Reduction Techniques in Monte Carlo Methods," Other publications TiSEM 87680d1a-53c1-4107-ada4-7, Tilburg University, School of Economics and Management.
    20. 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.
    21. Jin‐Chuan Duan, 1995. "The Garch Option Pricing Model," Mathematical Finance, Wiley Blackwell, vol. 5(1), pages 13-32, January.
    22. Boyle, Phelim P., 1977. "Options: A Monte Carlo approach," Journal of Financial Economics, Elsevier, vol. 4(3), pages 323-338, May.
    23. Engle, Robert F, 1982. "Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation," Econometrica, Econometric Society, vol. 50(4), pages 987-1007, July.
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