Natural finite dimensional HJM models are NON-affine.
We consider so called HJM-models which model all forward rates, which are future interest rates which can be secured now, dynamically through time. This is in principal a curve valued process and it is known that a given HJM-model is free of arbitrage if the so-called HJM-drift condition is met. We are interested in finite dimensional HJM-models which stay on one fixed given finite dimensional manifold, or otherwise put, where we use a parametrised family of curves with finitely many parameters. It is well known, that a curve valued process stays on a prescribed manifold if the Stratonovich drift is zero at all time, or more simply, if we can instead find a parameter process which selects the curve seen at a given time. From a statistical point of view it would be desirable to leave the diffusion coefficient of the parameter process open for estimation, or in the language of manifolds that means that any tangential diffusion coefficient is possible. In this presentation, we find those finite dimensional manifolds where the diffusion coefficient is open for estimation while still allowing for the HJM-drift condition to be met. It turns out that the resulting manifolds are nowhere locally affine. More so, they are nowhere affinely foliated as has been suggested by earlier work (however under different assumptions).
Area: CS58 - Recent advances in Heath-Jarrow-Morton modelling in finance (Claudio Fontana and Alessandro Gnoatto)
Keywords: HJM-models, Finite dimensional, flexible diffusion
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