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The Impact of the Prior Density on a Minimum Relative Entropy Density: A Case Study with SPX Option Data

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  • C. Neri

    (Lloyds Banking Group, London, UK)

  • L. Schneider

    (EMLYON Business School, Lyon, France)

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    Abstract

    We study the problem of finding probability densities that match given European call option prices. To allow prior information about such a density to be taken into account, we generalise the algorithm presented in Neri and Schneider (2011) to find the maximum entropy density of an asset price to the relative entropy case. This is applied to study the impact the choice of prior density has in two market scenarios. In the first scenario, call option prices are prescribed at only a small number of strikes, and we see that the choice of prior, or indeed its omission, yields notably different densities. The second scenario is given by CBOE option price data for S&P 500 index options at a large number of strikes. Prior information is now considered to be given by calibrated Heston, Schoebel-Zhu or Variance Gamma models. We find that the resulting digital option prices are essentially the same as those given by the (non-relative) Buchen-Kelly density itself. In other words, in a sufficiently liquid market the influence of the prior density seems to vanish almost completely.

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    File URL: http://arxiv.org/pdf/1201.2616
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    Bibliographic Info

    Paper provided by arXiv.org in its series Papers with number 1201.2616.

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    Date of creation: Jan 2012
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    Handle: RePEc:arx:papers:1201.2616

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    1. Stein, Elias M & Stein, Jeremy C, 1991. "Stock Price Distributions with Stochastic Volatility: An Analytic Approach," Review of Financial Studies, Society for Financial Studies, vol. 4(4), pages 727-52.
    2. Cox, John C & Ingersoll, Jonathan E, Jr & Ross, Stephen A, 1985. "A Theory of the Term Structure of Interest Rates," Econometrica, Econometric Society, vol. 53(2), pages 385-407, March.
    3. Bakshi, Gurdip & Madan, Dilip, 2000. "Spanning and derivative-security valuation," Journal of Financial Economics, Elsevier, vol. 55(2), pages 205-238, February.
    4. 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.
    5. 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.
    6. Madan, Dilip B & Seneta, Eugene, 1990. "The Variance Gamma (V.G.) Model for Share Market Returns," The Journal of Business, University of Chicago Press, vol. 63(4), pages 511-24, October.
    7. Cassio Neri & Lorenz Schneider, 2011. "A Family of Maximum Entropy Densities Matching Call Option Prices," Papers 1102.0224, arXiv.org.
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