On the relation between the Stochastic Jacobian and Riccati ODE in Affine Term Structure Models Martino Grasselli Universit`a di Padova * , ESILV † and SAFE ‡ Claudio Tebaldi Universit`adiVerona § and SAFE December 3, 2005 Abstract In Affine Term Structure Models (ATSM) the stochastic Jacobian under the forward measure plays a crucial role for pricing, as discussed in Elliott and van der Hoek (2001). Their approach leads to a deterministic integro-differential equation which, apparently, has the advantage of by-passing the solution to the Riccati ODE obtained by the standard Feynman-Kaˇ c argument. In the generic multi-dimensional case, we find a procedure to reduce such integro- differential equation to a non linear matrix ODE. We prove that its solution does necessarily require the solution of the vector Riccati ODE. This result is obtained proving an extension of the celebrated Radon Lemma, which allows us to highlight a deep relation between the geometry of the Riccati flow and the stochastic calculus of variations for an ATSM. Keywords: Affine Terms Structure Models, Riccati ODE, Radon Lemma, Stochastic Flow, Stochastic Jacobian. * Dipartimento di Matematica Pura ed Applicata, via Belzoni 7, Padova. Email: grassell@math.unipd.it † Ecole Sup´ erieure d’Ing´ enieur L´ eonard de Vinci, D´ epartement Math´ ematiques et Ing´ enierie Financi` ere, 92916 Paris La D´ efense ‡ Center for Studies in Actuarial and Financial Engineering - Economics, Via Giardino Giusti 2, 37129 Verona, Italy. § Department of Economics, Via Giardino Giusti 2, 37129 Verona, Italy. Fax +39 0458054935. E-mail: claudio.tebaldi@univr.it 1