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Zero bound, option-implied PDFs, and term structure models

  • Don H. Kim
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    This paper points out that several known ways of modeling non-negative nominal interest rates lead to different implications for the risk-neutral distribution of the short rate that can be checked with options data. In particular, Black's boundary models ("interest rates as options") imply a probability density function (pdf) that contains a Dirac delta function and a cumulative distribution function (cdf) that is nonzero at the zero boundary, while models like the CIR and positive-definite quadratic-Gaussian (QG) models have a zero cdf at the boundary. Eurodollar futures options data are found to favor Black's boundary models: the CIR/QG models, even multifactor versions, have difficulty capturing option prices accurately not only in low interest rate environments but also in higher interest rate environments, and data in early 2008 provide an almost tangible signature of the Dirac delta function in Black's boundary pdf models. Options data also contradict the prediction of well-known models whose cdf is zero at the zero boundary, namely that the risk-neutral pdf is always positively skewed.

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    Paper provided by Board of Governors of the Federal Reserve System (U.S.) in its series Finance and Economics Discussion Series with number 2008-31.

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    Date of creation: 2008
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    Handle: RePEc:fip:fedgfe:2008-31
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    1. 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.
    2. Black, Fischer, 1995. " Interest Rates as Options," Journal of Finance, American Finance Association, vol. 50(5), pages 1371-76, December.
    3. Duffie, Darrell & Singleton, Kenneth J, 1997. " An Econometric Model of the Term Structure of Interest-Rate Swap Yields," Journal of Finance, American Finance Association, vol. 52(4), pages 1287-1321, September.
    4. Ait-Sahalia, Yacine, 1996. "Testing Continuous-Time Models of the Spot Interest Rate," Review of Financial Studies, Society for Financial Studies, vol. 9(2), pages 385-426.
    5. Barone-Adesi, Giovanni & Whaley, Robert E, 1987. " Efficient Analytic Approximation of American Option Values," Journal of Finance, American Finance Association, vol. 42(2), pages 301-20, June.
    6. Markus Leippold & Liuren Wu, 2002. "Design and Estimation of Quadratic Term Structure Models," Finance 0207014, EconWPA.
    7. Ahn, Dong-Hyun & Gao, Bin, 1999. "A Parametric Nonlinear Model of Term Structure Dynamics," Review of Financial Studies, Society for Financial Studies, vol. 12(4), pages 721-62.
    8. Longstaff, Francis A., 1992. "Multiple equilibria and term structure models," Journal of Financial Economics, Elsevier, vol. 32(3), pages 333-344, December.
    9. Cecchetti, Stephen G, 1988. "The Case of the Negative Nominal Interest Rates: New Estimates of the Term Structure of Interest Rates during the Great Depression," Journal of Political Economy, University of Chicago Press, vol. 96(6), pages 1111-41, December.
    10. Dong-Hyun Ahn & Robert F. Dittmar, 2002. "Quadratic Term Structure Models: Theory and Evidence," Review of Financial Studies, Society for Financial Studies, vol. 15(1), pages 243-288, March.
    11. Black, Fischer, 1976. "The pricing of commodity contracts," Journal of Financial Economics, Elsevier, vol. 3(1-2), pages 167-179.
    12. 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.
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