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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 (Peter W. Buchen and Michael Kelly. The maximum entropy distribution of an asset inferred from option prices. Journal of Financial and Quantitative Analysis , 31 (1), 143-159, March 1996.) in the family of entropy maximizing densities from Neri and Schneider (Maximum entropy distributions inferred from option portfolios on an asset. Finance and Stochastics , 16 (2), 293-318, April 2012.), 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. The method presented here can also be used to interpolate between call option prices, and we compare it to a method proposed by Kahal� (An arbitrage-free interpolation of volatilities. Risk , 17 (5), 102-106, May 2004). Orozco Rodriguez and Santosa (Estimation of asset distributions from option prices: Analysis and regularization. SIAM Journal on Financial Mathematics , 3 (1), 374-401, 2012.) have produced examples in which the Buchen-Kelly algorithm becomes numerically unstable, and we use these as test cases to show that the algorithm given here remains stable and leads to good results.

Suggested Citation

  • Cassio Neri & Lorenz Schneider, 2013. "A Family of Maximum Entropy Densities Matching Call Option Prices," Applied Mathematical Finance, Taylor & Francis Journals, vol. 20(6), pages 548-577, December.
    • Handle: RePEc:taf:apmtfi:v:20:y:2013:i:6:p:548-577
    DOI: 10.1080/1350486X.2013.780769
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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. Malhotra, Gifty & Srivastava, R. & Taneja, H.C., 2019. "Calibration of the risk-neutral density function by maximization of a two-parameter entropy," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 513(C), pages 45-54.
    3. Gzyl, Henryk & Mayoral, Silvia, 2016. "Determination of zero-coupon and spot rates from treasury data by maximum entropy methods," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 456(C), pages 38-50.
    4. 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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