Stochastic Jacobians in affine term-structure models: a local property

Affine diffusions are popular models for risk-factors in mathematical finance and, in particular, form the basis of several term-structure models. We consider the stochastic Jacobian of the stochastic flow associated with an affine diffusion and prove that the conditional expectation under the forward measure of the stochastic Jacobian is deterministic in a neighborhood of the maturity time of a zero-coupon bond. This local result corrects a lemma of Elliott and van der Hoek [7] to the extent possible using the method of proof originally proposed. The local result corresponds to the fact that Riccati equations, which must normally be solved numerically to implement financial models based on affine processes, have solutions in a neighborhood of the boundary condition but not necessarily over a fixed interval.