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Calibrating volatility surfaces via relative-entropy minimization

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Author Info

  • Marco Avellaneda
  • Craig Friedman
  • Richard Holmes
  • Dominick Samperi
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    Abstract

    A framework for calibrating a pricing model to a prescribed set of options prices quoted in the market is presented. Our algorithm yields an arbitrage-free diffusion process that minimizes the Kullback-Leibler relative entropy distance to a prior diffusion. It consists in solving a constrained (minimax) optimal control problem using a finite-difference scheme for a Bellman parabolic equation combined with a gradient-based optimization routine. The number of unknowns to be solved for in the optimization step is equal to the number of option prices that need to be calibrated, and is independent of the mesh-size used for the scheme. This results in an efficient, non-parametric calibration method that can match an arbitrary number of option prices to any desired degree of accuracy. The algorithm can be used to interpolate, both in strike and expiration date, between implied volatilities of traded options and to price exotics. The stability and qualitative properties of the computed volatility surface are discussed, including the effect of the Bayesian prior on the shape of the surface and on the implied volatility smile/skew. The method is illustrated by calibrating to market prices of Dollar-Deutschmark over-the-counter options and computing interpolated implied-volatility curves.

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    File URL: http://www.tandfonline.com/doi/abs/10.1080/135048697334827
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    Bibliographic Info

    Article provided by Taylor & Francis Journals in its journal Applied Mathematical Finance.

    Volume (Year): 4 (1997)
    Issue (Month): 1 ()
    Pages: 37-64

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    Handle: RePEc:taf:apmtfi:v:4:y:1997:i:1:p:37-64

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    Web page: http://www.tandfonline.com/RAMF20

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    Related research

    Keywords: Option Pricing; Implied Volatility Surface; Calibration; Relative Entropy; Stochastic; Control; Volatility; Smile; Skew;

    References

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    Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.:
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    1. Jens Carsten Jackwerth., 1996. "Implied Binomial Trees: Generalizations and Empirical Tests," Research Program in Finance Working Papers RPF-262, University of California at Berkeley.
    2. Stutzer, Michael, 1995. "A Bayesian approach to diagnosis of asset pricing models," Journal of Econometrics, Elsevier, vol. 68(2), pages 367-397, August.
    3. Jens Carsten Jackwerth, 1998. "Generalized Binomial Trees," Finance 9803004, EconWPA.
    4. Banz, Rolf W & Miller, Merton H, 1978. "Prices for State-contingent Claims: Some Estimates and Applications," The Journal of Business, University of Chicago Press, vol. 51(4), pages 653-72, October.
    5. Mark Rubinstein., 1994. "Implied Binomial Trees," Research Program in Finance Working Papers RPF-232, University of California at Berkeley.
    6. Breeden, Douglas T & Litzenberger, Robert H, 1978. "Prices of State-contingent Claims Implicit in Option Prices," The Journal of Business, University of Chicago Press, vol. 51(4), pages 621-51, October.
    7. M. Avellaneda & A. Levy & A. ParAS, 1995. "Pricing and hedging derivative securities in markets with uncertain volatilities," Applied Mathematical Finance, Taylor & Francis Journals, vol. 2(2), pages 73-88.
    8. Marco Avellaneda & Antonio ParAS, 1996. "Managing the volatility risk of portfolios of derivative securities: the Lagrangian uncertain volatility model," Applied Mathematical Finance, Taylor & Francis Journals, vol. 3(1), pages 21-52.
    9. Rubinstein, Mark, 1994. " Implied Binomial Trees," Journal of Finance, American Finance Association, vol. 49(3), pages 771-818, July.
    10. Jackwerth, Jens Carsten & Rubinstein, Mark, 1996. " Recovering Probability Distributions from Option Prices," Journal of Finance, American Finance Association, vol. 51(5), pages 1611-32, December.
    11. Buchen, Peter W. & Kelly, Michael, 1996. "The Maximum Entropy Distribution of an Asset Inferred from Option Prices," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 31(01), pages 143-159, March.
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    Citations

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    Cited by:
    1. Cristian Homescu, 2011. "Implied Volatility Surface: Construction Methodologies and Characteristics," Papers 1107.1834, arXiv.org.
    2. A. Monteiro & R. Tütüncü & L. Vicente, 2011. "Estimation of risk-neutral density surfaces," Computational Management Science, Springer, vol. 8(4), pages 387-414, November.
    3. Bernd Engelmann & Matthias Fengler & Morten Nalholm & Peter Schwendner, 2006. "Static versus dynamic hedges: an empirical comparison for barrier options," Review of Derivatives Research, Springer, vol. 9(3), pages 239-264, November.
    4. Subbotin, Alexandre, 2009. "Volatility Models: from Conditional Heteroscedasticity to Cascades at Multiple Horizons," Applied Econometrics, Publishing House "SINERGIA PRESS", vol. 15(3), pages 94-138.
    5. Carol Alexander & Leonardo Nogueira, 2004. "Stochastic Local Volatility," ICMA Centre Discussion Papers in Finance icma-dp2008-02, Henley Business School, Reading University, revised Mar 2008.
    6. Bertram Düring & Ansgar Jüngel & S. Volkwein, 2006. "A Sequential Quadratic Programming Method for Volatility Estimation in Option Pricing," CoFE Discussion Paper 06-02, Center of Finance and Econometrics, University of Konstanz.
    7. Vladislav Kargin, 2003. "Consistent Estimation of Pricing Kernels from Noisy Price Data," Papers math/0310223, arXiv.org.
    8. Carol Alexander & Leonardo M. Nogueira, 2004. "Hedging with Stochastic and Local Volatility," ICMA Centre Discussion Papers in Finance icma-dp2004-10, Henley Business School, Reading University, revised Dec 2004.

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