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Convex Regularization Of Local Volatility Estimation

Author

Listed:
  • VINICIUS ALBANI

    (Department of Mathematics, Federal University of Santa Catarina, Trindade, Florianopolis, Santa Catarina 88.040-900, Brazil)

  • ADRIANO DE CEZARO

    (Institute of Mathematics, Statistics and Physics, Federal University of Rio Grande, Av. Italia Km 8, Rio Grande, Rio Grande do Sul 96201-900, Brazil)

  • JORGE P. ZUBELLI

    (Instituto of Pure and Applied Mathematics, Estrada Dona Castorina, 110, Jardim Botnico, Rio de Janeiro, Rio de Janeiro 22460-320, Brazil)

Abstract

We apply convex regularization techniques to the problem of calibrating Dupire’s local volatility surface model taking into account the practical requirement of discrete grids and noisy data. Such requirements are the consequence of bid and ask spreads, quantization of the quoted prices and lack of liquidity of option prices for strikes far away from the at-the-money level. We obtain convergence rates and results comparable to those obtained in the idealized continuous setting. Our results allow us to take into account separately the uncertainties due to the price noise and those due to discretization errors, thus, allowing estimating better discretization levels both in the domain and in the image of the parameter to solution operator by a Morozov’s discrepancy principle. We illustrate the results with simulated as well as real market data. We also validate the results by comparing the implied volatility prices of market data with the computed prices of the calibrated model.

Suggested Citation

  • Vinicius Albani & Adriano De Cezaro & Jorge P. Zubelli, 2017. "Convex Regularization Of Local Volatility Estimation," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 20(01), pages 1-37, February.
    • Handle: RePEc:wsi:ijtafx:v:20:y:2017:i:01:n:s0219024917500066
    DOI: 10.1142/S0219024917500066
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    References listed on IDEAS

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    1. Marco Avellaneda, 1998. "Minimum-Relative-Entropy Calibration of Asset-Pricing Models," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 1(04), pages 447-472.
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    3. Roger W. Lee, 2001. "Implied And Local Volatilities Under Stochastic Volatility," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 4(01), pages 45-89.
    4. Marco Avellaneda & Robert Buff & Craig Friedman & Nicolas Grandechamp & Lukasz Kruk & Joshua Newman, 2001. "Weighted Monte Carlo: A New Technique For Calibrating Asset-Pricing Models," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 4(01), pages 91-119.
    5. 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.
    6. Marco Avellaneda & Robert Buff & Craig Friedman & Nicolas Grandechamp & Lukasz Kruk & Joshua Newman, 2001. "Weighted Monte Carlo: A New Technique For Calibrating Asset-Pricing Models," World Scientific Book Chapters, in: Marco Avellaneda (ed.), Quantitative Analysis In Financial Markets Collected Papers of the New York University Mathematical Finance Seminar(Volume II), chapter 9, pages 239-265, World Scientific Publishing Co. Pte. Ltd..
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    Cited by:

    1. Soobin Kwak & Youngjin Hwang & Yongho Choi & Jian Wang & Sangkwon Kim & Junseok Kim, 2022. "Reconstructing the Local Volatility Surface from Market Option Prices," Mathematics, MDPI, vol. 10(14), pages 1-12, July.
    2. Yuri F. Saporito & Xu Yang & Jorge P. Zubelli, 2017. "The Calibration of Stochastic-Local Volatility Models - An Inverse Problem Perspective," Papers 1711.03023, arXiv.org.
    3. Vinicius V. L. Albani & Jorge P. Zubelli, 2020. "A splitting strategy for the calibration of jump-diffusion models," Finance and Stochastics, Springer, vol. 24(3), pages 677-722, July.

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