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A Family of Maximum Entropy Densities Matching Call Option Prices

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  • Cassio Neri
  • Lorenz Schneider

Abstract

We investigate the position of the Buchen-Kelly density in a family of entropy maximising densities which all match European call option prices for a given maturity observed in the market. Using the Legendre transform which links the entropy function and the cumulant generating function, we show that it is both the unique continuous density in this family and the one with the greatest entropy. We present a fast root-finding algorithm that can be used to calculate the Buchen-Kelly density, and give upper boundaries for three different discrepancies that can be used as convergence criteria. Given the call prices, arbitrage-free digital prices at the same strikes can only move within upper and lower boundaries given by left and right call spreads. As the number of call prices increases, these bounds become tighter, and we give two examples where the densities converge to the Buchen-Kelly density in the sense of relative entropy when we use centered call spreads as proxies for digital prices. As pointed out by Breeden and Litzenberger, in the limit a continuous set of call prices completely determines the density.

Suggested Citation

  • Cassio Neri & Lorenz Schneider, 2011. "A Family of Maximum Entropy Densities Matching Call Option Prices," Papers 1102.0224, arXiv.org.
    • Handle: RePEc:arx:papers:1102.0224
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    References listed on IDEAS

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    1. Michael A. H. Dempster & Elena A. Medova & Seung W. Yang, 2007. "Empirical Copulas For Cdo Tranche Pricing Using Relative Entropy," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 10(04), pages 679-701.
    2. Les Gulko, 2002. "The Entropy Theory Of Bond Option Pricing," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 5(04), pages 355-383.
    3. 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.
    4. Michael A. H. Dempster & Elena A. Medova & Seung W. Yang, 2007. "Erratum: "Empirical Copulas For Cdo Tranche Pricing Using Relative Entropy"," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 10(07), pages 1255-1260.
    5. Les Gulko, 1999. "The Entropic Market Hypothesis," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 2(03), pages 293-329.
    6. Michael A. H. Dempster & Elena A. Medova & Seung W. Yang, 2007. "Empirical Copulas For Cdo Tranche Pricing Using Relative Entropy," World Scientific Book Chapters, in: Alexander Lipton & Andrew Rennie (ed.), Credit Correlation Life After Copulas, chapter 5, pages 87-109, World Scientific Publishing Co. Pte. Ltd..
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    Cited by:

    1. Cassio Neri & Lorenz Schneider, 2012. "A Note on "A Family of Maximum Entropy Densities Matching Call Option Prices"," Papers 1212.4279, arXiv.org.
    2. C. Neri & L. Schneider, 2012. "The Impact of the Prior Density on a Minimum Relative Entropy Density: A Case Study with SPX Option Data," Papers 1201.2616, arXiv.org, revised Sep 2013.

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