Consistency conditions for affine term structure models

Affine term structure models are widely applied for pricing of bonds and interest rate derivatives but the consistency of affine term structure models (ATSM) in cases when the short rate may be unbounded from below remains essentially an open question. The main stress in the classification paper Dai and Singleton (2000) is on the overdeterminacy of many ATSM models; however, for wide regions in the parameter's space, standard ATSM models may be inconsistent, and the following issues must be addressed. First, the standard approach to ATSM is based on the reduction to the Riccati equations. The reduction uses the Feynman-Kac formula but the general Feynman-Kac theorem is easily applicable only when the short rate is bounded from below, which excludes many classes used in applications. Second, the solution to the bond pricing problem must be a decreasing function of any state variable for which the corresponding coefficients in the formula for the short rate is positive; the solution must also decrease as the time to maturity increases, if the tuple of state variables belongs to the region where the short rate is positive. In the paper, sufficient conditions for the application of the Feynman-Kac formula, and monotonicity of the bond price are derived, for wide classes of affine term structure models in the pure diffusion case. Necessary conditions for the monotonicity are derived as well. The results can be generalized for jump-diffusion processes. We consider a simple two-factor A_1(2) family, next more general A_1(n) family, and then the family A_2(3) (other families A_m(n) can be studied similarly), and derive, in terms of parameters of the model, I. simple necessary conditions for the decay of the bond price as a function of the time to maturity, in the region where the short rate is positive; II. sufficient conditions for the decay of the bond price; we do not know how wide is the gap between these conditions and the (unknown to us) necessary and sufficient conditions; III. sufficient conditions under which the reduction to the system of the Riccati equations can be justified. For A_1(2) family, and in many other cases, these condition are weaker than the necessary condition in (I). Remarks. a) Necessary and sufficient conditions for the decay of the bond price at infinity, and in a vicinity of 0, are easier to derive, and under these conditions, a ``numerical proof" of the monotonicity of the bond price on a large finite interval can be used to show that for given parameters' values, the model is consistent. b) As our study shows, for the family A_1(n), the monotonicity of the bond price in time to maturity is the main consistency problem for ATSM. On the other hand, should one use the model for a fixed (and sufficiently small) time to maturity then the model can be consistent on this time interval; and it is possible to derive sufficient condition for the decay of the bond price on a small interval near maturity, which depends on parameters of the model. c) When it is necessary to consider more general contingent claims, a sufficient condition for (III), in terms of the rate of growth of the pay-off at infinity, can be derived relatively easily, and the same is true of a necessary condition for the decay of the price. The sufficient conditions for the monotonicity will be more difficult to derive. d) It is plausible that in some empirical studies, the fitted ATSM is inconsistent in the sense that the monotonicity condition fails. It might be possible to construct an arbitrage strategy against a counterparty who uses an inconsistent model