Black's Consol Rate Conjecture

This paper con rms a version of a conjecture by Fischer Black regarding consol rate models for the term structure of interest rates. A consol rate model is one in which the stochastic behavior of the short rate is in uenced by the consol rate. Since the consol rate is itself determined, via the usual discounted present value formula, by the short rate, such models have an inherent xed point aspect. Under an equivalent martingale measure, purely technical regularity conditions are given for the stochastic di erential equation de ning the short rate and the consol rate to be consistent with the de nition of the consol rate as the yield on a perpetual annuity. The results are based on an extension of the theory for the forward-backward stochastic di erential equations to in nite-horizon settings. Under additional compatibility conditions, we also show that the consol rate is uniquely determined, and given as a function of the short rate. We are grateful for many conversations with Fischer Black, who should not be held responsible for our interpretation of his conjecture. We are also grateful to the IMA for supporting our stay at a workshop at which this research was conceived. Some helpful discussions with P. Protter of Purdue University, A. Conze (formerly of Goldman Sachs), B. Hu of Notre Dame University and M. V. Safonov of The University of Minnesota also deserve acknowledgement. 1 The Graduate School of Business, Stanford University, Stanford, CA 94305-5015, partially supported under a grant from the U.S. National Science Foundation, NSF SES 90-10062. 2 Department of Mathematics, Purdue University, West Lafayette, IN 47907-1395, partially supported by NSF grant # DMS-9301516. 3 IMA, University of Minnesota, Minneapolis, MN 55455; on leave from Department of Mathematics, Fudan University, Shanghai 200433, China. This author is partially supported by NSF of China and Fok Ying Tung Education Foundation.