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Analysis of Markov chain approximation for Asian options and occupation-time derivatives: Greeks and convergence rates

Author

Listed:
  • Wensheng Yang

    (Southwestern University of Finance and Economics)

  • Jingtang Ma

    (Southwestern University of Finance and Economics)

  • Zhenyu Cui

    (Stevens Institute of Technology)

Abstract

The continuous-time Markov chain (CTMC) approximation method is a powerful tool that has recently been utilized in the valuation of derivative securities, and it has the advantage of yielding closed-form matrix expressions suitable for efficient implementation. For two types of popular path-dependent derivatives, the arithmetic Asian option and the occupation-time derivative, this paper obtains explicit closed-form matrix expressions for the Laplace transforms of their prices and the Greeks of Asian options, through the novel use of pathwise method and Malliavin calculus techniques. We for the first time establish the exact second-order convergence rates of the CTMC methods when applied to the prices and Greeks of Asian options. We propose a new set of error analysis methods for the CTMC methods applied to these path-dependent derivatives, whose payoffs depend on the average of asset prices. A detailed error and convergence analysis of the algorithms and numerical experiments substantiate the theoretical findings.

Suggested Citation

  • Wensheng Yang & Jingtang Ma & Zhenyu Cui, 2021. "Analysis of Markov chain approximation for Asian options and occupation-time derivatives: Greeks and convergence rates," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 93(2), pages 359-412, April.
    • Handle: RePEc:spr:mathme:v:93:y:2021:i:2:d:10.1007_s00186-020-00735-5
    DOI: 10.1007/s00186-020-00735-5
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    1. Gongqiu Zhang & Lingfei Li, 2019. "Analysis of Markov Chain Approximation for Option Pricing and Hedging: Grid Design and Convergence Behavior," Operations Research, INFORMS, vol. 67(2), pages 407-427, March.
    2. Tian, Yisong S., 2013. "Ironing out the kinks in executive compensation: Linking incentive pay to average stock prices," Journal of Banking & Finance, Elsevier, vol. 37(2), pages 415-432.
    3. 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.
    4. Yingda Song & Ning Cai & Steven Kou, 2018. "Computable Error Bounds of Laplace Inversion for Pricing Asian Options," INFORMS Journal on Computing, INFORMS, vol. 30(4), pages 634-645, January.
    5. Eric Fournié & Jean-Michel Lasry & Pierre-Louis Lions & Jérôme Lebuchoux & Nizar Touzi, 1999. "Applications of Malliavin calculus to Monte Carlo methods in finance," Finance and Stochastics, Springer, vol. 3(4), pages 391-412.
    6. Runhuan Feng & Hans W. Volkmer, 2015. "Conditional Asian Options," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 18(06), pages 1-24.
    7. Hélyette Geman & Marc Yor, 1993. "Bessel Processes, Asian Options, And Perpetuities," Mathematical Finance, Wiley Blackwell, vol. 3(4), pages 349-375, October.
    8. Mark Broadie & Paul Glasserman, 1996. "Estimating Security Price Derivatives Using Simulation," Management Science, INFORMS, vol. 42(2), pages 269-285, February.
    9. Borgonovo, E., 2010. "Sensitivity analysis with finite changes: An application to modified EOQ models," European Journal of Operational Research, Elsevier, vol. 200(1), pages 127-138, January.
    10. Søren Asmussen & Jens Ledet Jensen & Leonardo Rojas-Nandayapa, 2016. "Exponential Family Techniques for the Lognormal Left Tail," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 43(3), pages 774-787, September.
    11. Lingfei Li & Gongqiu Zhang, 2018. "Error analysis of finite difference and Markov chain approximations for option pricing," Mathematical Finance, Wiley Blackwell, vol. 28(3), pages 877-919, July.
    12. Plischke, Elmar & Borgonovo, Emanuele, 2019. "Copula theory and probabilistic sensitivity analysis: Is there a connection?," European Journal of Operational Research, Elsevier, vol. 277(3), pages 1046-1059.
    13. Phelan, Carolyn E. & Marazzina, Daniele & Fusai, Gianluca & Germano, Guido, 2018. "Fluctuation identities with continuous monitoring and their application to the pricing of barrier options," European Journal of Operational Research, Elsevier, vol. 271(1), pages 210-223.
    14. Gianluca Fusai & Ioannis Kyriakou, 2016. "General Optimized Lower and Upper Bounds for Discrete and Continuous Arithmetic Asian Options," Mathematics of Operations Research, INFORMS, vol. 41(2), pages 531-559, May.
    15. Borgonovo, Emanuele & Plischke, Elmar, 2016. "Sensitivity analysis: A review of recent advances," European Journal of Operational Research, Elsevier, vol. 248(3), pages 869-887.
    16. Kirkby, J. Lars & Nguyen, Duy & Cui, Zhenyu, 2017. "A unified approach to Bermudan and barrier options under stochastic volatility models with jumps," Journal of Economic Dynamics and Control, Elsevier, vol. 80(C), pages 75-100.
    17. Runhuan Feng & Hans W. Volkmer, 2015. "Conditional Asian Options," Papers 1505.06946, arXiv.org.
    18. Ning Cai & Nan Chen & Xiangwei Wan, 2010. "Occupation Times of Jump-Diffusion Processes with Double Exponential Jumps and the Pricing of Options," Mathematics of Operations Research, INFORMS, vol. 35(2), pages 412-437, May.
    19. Chia Chun Lo & Konstantinos Skindilias, 2014. "An Improved Markov Chain Approximation Methodology: Derivatives Pricing And Model Calibration," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 17(07), pages 1-22.
    20. Plischke, Elmar & Borgonovo, Emanuele & Smith, Curtis L., 2013. "Global sensitivity measures from given data," European Journal of Operational Research, Elsevier, vol. 226(3), pages 536-550.
    21. Boyle, Phelim & Potapchik, Alexander, 2008. "Prices and sensitivities of Asian options: A survey," Insurance: Mathematics and Economics, Elsevier, vol. 42(1), pages 189-211, February.
    22. Borgonovo, Emanuele & Buzzard, Gregery T. & Wendell, Richard E., 2018. "A global tolerance approach to sensitivity analysis in linear programming," European Journal of Operational Research, Elsevier, vol. 267(1), pages 321-337.
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    Cited by:

    1. Kirkby, J. Lars, 2023. "Hybrid equity swap, cap, and floor pricing under stochastic interest by Markov chain approximation," European Journal of Operational Research, Elsevier, vol. 305(2), pages 961-978.

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    More about this item

    Keywords

    Option pricing; Sensitivity analysis; Continuous-time Markov chains; Non-uniform grids; Convergence rates; Path-dependent options; Greeks; Laplace inversion;
    All these keywords.

    JEL classification:

    • C63 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Computational Techniques
    • G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing

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