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A new well-posed algorithm to recover implied local volatility

Author

Listed:
  • Lishang Jiang
  • Qihong Chen
  • Lijun Wang
  • Jin Zhang

Abstract

This paper presents a new algorithm to calibrate the option pricing model, i.e. the algorithm that recovers the implied local volatility function from market option prices in the optimal control framework. A unique optimal control is shown to exist. Our algorithm is well-posed. Our numerical experiments show that, with the help of the techniques developed in the field of optimal control, the local volatility function is recovered very well.

Suggested Citation

  • Lishang Jiang & Qihong Chen & Lijun Wang & Jin Zhang, 2003. "A new well-posed algorithm to recover implied local volatility," Quantitative Finance, Taylor & Francis Journals, vol. 3(6), pages 451-457.
    • Handle: RePEc:taf:quantf:v:3:y:2003:i:6:p:451-457
    DOI: 10.1088/1469-7688/3/6/304
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    References listed on IDEAS

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    1. 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..
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    3. Marco Avellaneda & Craig Friedman & Richard Holmes & Dominick Samperi, 1997. "Calibrating volatility surfaces via relative-entropy minimization," Applied Mathematical Finance, Taylor & Francis Journals, vol. 4(1), pages 37-64.
    4. Mark Rubinstein., 1994. "Implied Binomial Trees," Research Program in Finance Working Papers RPF-232, University of California at Berkeley.
    5. Rubinstein, Mark, 1985. "Nonparametric Tests of Alternative Option Pricing Models Using All Reported Trades and Quotes on the 30 Most Active CBOE Option Classes from August 23, 1976 through August 31, 1978," Journal of Finance, American Finance Association, vol. 40(2), pages 455-480, June.
    6. 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.
    7. 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.
    8. Rubinstein, Mark, 1994. "Implied Binomial Trees," Journal of Finance, American Finance Association, vol. 49(3), pages 771-818, July.
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    Cited by:

    1. He, Xin-Jiang & Zhu, Song-Ping, 2017. "How should a local regime-switching model be calibrated?," Journal of Economic Dynamics and Control, Elsevier, vol. 78(C), pages 149-163.
    2. Liang, J. & Gao, Y., 2012. "Calibration of implied volatility for the exchange rate for the Chinese Yuan from its derivatives," Economic Modelling, Elsevier, vol. 29(4), pages 1278-1285.
    3. S. Gnanavel & N. Barani Balan & K. Balachandran, 2014. "Simultaneous Identification of Two Time Independent Coefficients in a Nonlinear Phase Field System," Journal of Optimization Theory and Applications, Springer, vol. 160(3), pages 992-1008, March.
    4. Deng, Zui-Cha & Yu, Jian-Ning & Yang, Liu, 2008. "Identifying the coefficient of first-order in parabolic equation from final measurement data," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 77(4), pages 421-435.
    5. Yishen Li & Jin Zhang, 2004. "Option pricing with Weyl-Titchmarsh theory," Quantitative Finance, Taylor & Francis Journals, vol. 4(4), pages 457-464.
    6. Gabriel TURINICI, 2008. "Local Volatility Calibration Using An Adjoint Proxy," Review of Economic and Business Studies, Alexandru Ioan Cuza University, Faculty of Economics and Business Administration, issue 2, pages 93-105, November.
    7. Judith Glaser & Pascal Heider, 2012. "Arbitrage-free approximation of call price surfaces and input data risk," Quantitative Finance, Taylor & Francis Journals, vol. 12(1), pages 61-73, August.
    8. Gabriel Turinici, 2009. "Calibration of local volatility using the local and implied instantaneous variance," Post-Print hal-00338114, HAL.
    9. Jin Zhang & Yi Xiang, 2008. "The implied volatility smirk," Quantitative Finance, Taylor & Francis Journals, vol. 8(3), pages 263-284.
    10. David Heath & Eckhard Platen, 2006. "Local volatility function models under a benchmark approach," Quantitative Finance, Taylor & Francis Journals, vol. 6(3), pages 197-206.
    11. Jing Zhao & Hoi Ying Wong, 2012. "A closed-form solution to American options under general diffusion processes," Quantitative Finance, Taylor & Francis Journals, vol. 12(5), pages 725-737, July.
    12. Dai, Min & Tang, Ling & Yue, Xingye, 2016. "Calibration of stochastic volatility models: A Tikhonov regularization approach," Journal of Economic Dynamics and Control, Elsevier, vol. 64(C), pages 66-81.
    13. Min Dai & Hanqing Jin & Xi Yang, 2024. "Data-driven Option Pricing," Papers 2401.11158, arXiv.org.
    14. Zui-Cha Deng & Y.-C. Hon & Liu Yang, 2014. "An Optimal Control Method for Nonlinear Inverse Diffusion Coefficient Problem," Journal of Optimization Theory and Applications, Springer, vol. 160(3), pages 890-910, March.

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