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A linear goal programming method to recover risk neutral probabilities from options prices by maximum entropy

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Listed:
  • José L. Vilar-Zanón

    (Universidad Complutense de Madrid)

  • Olivia Peraita-Ezcurra

    (Banco de Santander)

Abstract

We develop a new methodology to retrieve risk neutral probabilities (equivalent martingale measure) with maximum entropy from quoted option prices. We assume the no arbitrage hypothesis and model the efficient market hypothesis by means of a maximum entropic risk neutral distribution. The method is free of parametric assumption except for the simulation of the distribution support, for which purpose we can choose any stochastic model. Firstly, we innovate in the minimization of a different f-divergence than Kullback–Leibler’s relative entropy, resulting in the total variation distance. We minimize it by means of linear goal programming, thus guaranteeing a fast numerical resolution. The method values non-traded assets finding a RNP minimizing its f-divergence to the maximum entropy distribution over a simulated support—the uniform distribution—calibrated to the benchmarks prices constraints. Our second innovation is that in an incomplete market, we can increase the f-divergence from its minimum to obtain any asset price belonging to the interval satisfying the non-existence of an arbitrage portfolio, without presupposing any utility function for the decision maker. We exemplify our methodology by means of synthetic and real-world cases, showing that our methodology can either price non-traded assets or interpolate and extrapolate a volatility surface.

Suggested Citation

  • José L. Vilar-Zanón & Olivia Peraita-Ezcurra, 2019. "A linear goal programming method to recover risk neutral probabilities from options prices by maximum entropy," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 42(1), pages 259-276, June.
    • Handle: RePEc:spr:decfin:v:42:y:2019:i:1:d:10.1007_s10203-019-00236-z
    DOI: 10.1007/s10203-019-00236-z
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    References listed on IDEAS

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    1. Mr. Kevin C Cheng, 2010. "A New Framework to Estimate the Risk-Neutral Probability Density Functions Embedded in Options Prices," IMF Working Papers 2010/181, International Monetary Fund.
    2. Dupont, Dominique Y., 2001. "Extracting Risk-Neutral Probability Distributions from Option Prices Using Trading Volume as a Filter," Economics Series 104, Institute for Advanced Studies.
    3. 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.
    4. Jim Gatheral & Antoine Jacquier, 2014. "Arbitrage-free SVI volatility surfaces," Quantitative Finance, Taylor & Francis Journals, vol. 14(1), pages 59-71, January.
    5. Dhaene, Jan & Stassen, Ben & Devolder, Pierre & Vellekoop, Michel, 2014. "The Minimal Entropy Martingale Measure in a market of traded financial and actuarial risks," LIDAM Discussion Papers ISBA 2014055, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    6. Jackwerth, Jens Carsten & Rubinstein, Mark, 1996. "Recovering Probability Distributions from Option Prices," Journal of Finance, American Finance Association, vol. 51(5), pages 1611-1632, December.
    7. 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.
    8. Cassio Neri & Lorenz Schneider, 2012. "Maximum entropy distributions inferred from option portfolios on an asset," Finance and Stochastics, Springer, vol. 16(2), pages 293-318, April.
    9. Melick, William R. & Thomas, Charles P., 1997. "Recovering an Asset's Implied PDF from Option Prices: An Application to Crude Oil during the Gulf Crisis," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 32(1), pages 91-115, March.
    10. Marco Frittelli, 2000. "The Minimal Entropy Martingale Measure and the Valuation Problem in Incomplete Markets," Mathematical Finance, Wiley Blackwell, vol. 10(1), pages 39-52, January.
    11. 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..
    12. Robert J. Ritchey, 1990. "Call Option Valuation For Discrete Normal Mixtures," Journal of Financial Research, Southern Finance Association;Southwestern Finance Association, vol. 13(4), pages 285-296, December.
    13. 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(1), pages 143-159, March.
    14. Ritchey, Robert J, 1990. "Call Option Valuation for Discrete Normal Mixtures," Journal of Financial Research, Southern Finance Association;Southwestern Finance Association, vol. 13(4), pages 285-296, Winter.
    15. Monteiro, Ana Margarida & Tutuncu, Reha H. & Vicente, Luis N., 2008. "Recovering risk-neutral probability density functions from options prices using cubic splines and ensuring nonnegativity," European Journal of Operational Research, Elsevier, vol. 187(2), pages 525-542, June.
    16. Rockinger, Michael & Jondeau, Eric, 2002. "Entropy densities with an application to autoregressive conditional skewness and kurtosis," Journal of Econometrics, Elsevier, vol. 106(1), pages 119-142, January.
    17. Christopher Bose & Rua Murray, 2014. "Maximum Entropy Estimates for Risk-Neutral Probability Measures with Non-Strictly-Convex Data," Journal of Optimization Theory and Applications, Springer, vol. 161(1), pages 285-307, April.
    18. Luenberger, David G., 2002. "Arbitrage and universal pricing," Journal of Economic Dynamics and Control, Elsevier, vol. 26(9-10), pages 1613-1628, August.
    19. de Jong, C.M. & Huisman, R., 2000. "From Skews to a Skewed-t," ERIM Report Series Research in Management ERS-2000-12-F&A, Erasmus Research Institute of Management (ERIM), ERIM is the joint research institute of the Rotterdam School of Management, Erasmus University and the Erasmus School of Economics (ESE) at Erasmus University Rotterdam.
    20. Charles J. Corrado, 2001. "Option pricing based on the generalized lambda distribution," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 21(3), pages 213-236, March.
    21. 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.
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