Abstract
This paper develops an equilibrium model in which interest rates follow a discontinuous (generalized) gamma process. The gamma process has finite variation, takes an infinite number of “small” jumps in every interval, and includes the Wiener process as a limiting case. The gamma interest rate model produces yield curves that closely resemble those of diffusion models. But in contrast to diffusion models, the curvature of the yield curve does not directly depend on the true volatility of the interest rate process, but instead depends on a different risk-neutral volatility. The gamma model appears to fit the distribution of interest rates changes and the jump characteristics of interest rate paths. Empirical tests reject a diffusion model of interest rates in favor of the more general gamma model because daily interest rate innovations are highly leptokurtic.
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References
Ahn C.M. and Thompson H.E. (1988). Jump-diffusion processes and term structure of interest rates. Journal of Finance 43: 155–174
Ait-Sahalia, Y. (1992). Nonparametric Pricing of Interest Rate Derivative Securities. Working paper, MIT.
Barndorff-Nielsen O. and Sorensen M. (1998). Some stationary processes in discrete and continuouus time. Advanceds in Applied Probability 30: 989–1007
Bertoin, J. (1998). Levy processes (Cambridge tracts in mathematics). Cambridge University Press.
Brown S.J. and Dybvig P.H. (1986). The empirical implications of the Cox, Ingersoll, Ross theory of the term structure of interest rates. Journal of Finance 41: 617–630
Cox J.C., Ingersoll J.E. and Ross S.A. (1985). A theory of the term structure of interest rates. Econometrica 53: 385–408
Daley D.J. and Vere-Jones D. (1988). An introduction to the theory of point processes. Springer-Verlag, New York
Das, S. (1993). Jump-hunting interest rates. Working paper, New York University.
Duffie D., Filipovic D. and Schachermayer W. (2003). Affine processes and applications in finance. Annals of Applied Probability 13: 984–1053
Eberlein E. and Raible S. (1999). Term structure models driven by general levy processes. Mathematical Finance 9: 31–53
Feller W. (1966). An introduction to probability theory and its applications (Vol 2). Wiley and Sons, New York
Filipovic D. (2001). A general characterization of one factor affine term structure models. Finance and Stochastics 5: 389–412
Garman M.B. (1985). Towards a semigroup pricing theory. Journal of Finance 40: 847–861
Hansen L.P. (1982). Large sample properties of the generalized method of moments estimators. Econometrica 50: 1029–1054
Heidari, M., & Wu, L. (2003). Market anticipation of federal reserve policy changes and the term structure of interest rates. Working paper, Baruch College.
Ho T. and Lee S. (1986). Term structure movements and pricing interest rate contingent claims. Journal of Finance 41: 1011–1028
Ingersoll J.E. (1989). Theory of financial decision making. Rowman and Littlefield, Totowa, New Jersey
Ingersoll J.E., Skelton J. and Weil R.L. (1978). Duration forty years later. Journal of Financial and Quantitative Analysis 13: 627–650
Jamshidian F. (1989). An exact bond option formula. Journal of Finance 44: 205–209
Johannes, M. (2003). The statistical and economic role of jumps in continuous-time interest rate models. Journal of Finance (forthcoming).
Litterman, R., Scheinkman, J., & Weiss, L. (1988). Volatility and the yield curve. New YorkL: Goldman Sachs and Co. Financial Strategies Group.
Madan D.B. and Milne F. (1991). Option pricing with V.G. martingale components. Mathematical Finance 1: 39–55
Madan D.B. and Seneta E. (1990). The variance gamma model for share market returns. Journal of Business 63: 511–524
Merton R.C. (1982). On the mathematics and economics assumptions of continuous-time models. In: Sharpe, W.F. and Cootner, C.M. (eds) Financial economics: Essays in honor of Paul Cootner, pp 19–51. Prentice-Hall, Englewood Cliffs, N.J.
Norberg, R. (1998). Vasicek beyond the normal. Working paper, Laboratory of Actuarial Mathematics, University of Copenhagen.
Ray C.I. (1993). The bond market: Trading and risk management. Business One Irwin, Homewood, Illinois
Rishel, R. (1990) Controlled continuous time Markov processes. In: D. P. Heyman, M. J. Sobel (Eds.), Stochastic models, handbooks in operations research (Vol. 2). New York, North-Holland.
Samuelson P. (1970). The fundamental approximation theorem of portfolio analysis in terms of means, variances and higher moments. Review of Economic Studies 37: 537–542
Sorenson, M. (1997). Estimating functions for discretely observed diffusions: A review. In I. V. Basawa, V. P. Godambe, & R. L. Taylor (Eds.), Selected Proceedings of the Symposium on Estimating Functions (pp. 305–325). IMS Lecture Notes – Monograph Series, 32.
Vasicek O.A. (1977). An equilibrium characterization of the term structure. Journal of Financial Economics 5: 177–188
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The author appreciates comments from George Constantinides, Jon Ingersoll, Herbert Johnson, Ray Rishel, and an anonymous referee, computational assistance from Kerry Back and Saikat Nandi, and support from Atlantic Asset Management. Any errors are the responsibility of the author.
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Heston, S.L. A model of discontinuous interest rate behavior, yield curves, and volatility. Rev Deriv Res 10, 205–225 (2007). https://doi.org/10.1007/s11147-008-9020-3
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DOI: https://doi.org/10.1007/s11147-008-9020-3