Closed Form Solutions for Quadratic and Inverse Quadratic Term Structure Models

We find fundamental solutions in closed form for a family of parabolic equations with two spatial variables, whose symmetry groups had been determined in an earlier paper by Ferdinando Finkel [12]. We show how these results can be applied in finance to yield closed form solutions for special affine and quadratic two factor term structure models as well as a new class of models with inverse square behavior. The latter can be considered a partial extension to two factors of pricing models related to the Bessel process devised by Albanese and Campolieti [3] and Carr Albanese Lipton [2]. A by-product of our results is that Lie’s reduction method in this setting leads only to fundamental solutions that can be factorized as products of functions that depend jointly on time and on one spatial coordinate. Thus all the results in this paper extend immediately to n factor models. The determination of the symmetry group of Fokker-Plank equations has a long tradition in mathematical physics in areas as diverse as fluid mechanics and quantum mechanics. Once the full symmetry group has been determined it is possible in many cases to describe the more restricted group that leaves invariant the solution of its fundamental solution, ie. the probability transition function G : <n×[0, T ] → <, which satisfies the following backward equation 1 Gt(x, t, ξ, t ′) + 1 2 aij(x, t)Gxixj (x, t, ξ, t ′) + bi(x, t)Gxi(x, t, ξ, t ′) = 0 (0.1) G(x, t′, ξ, t′) = δ(x− ξ) (0.2) It is a notable fact that recently, due to its important applications in finance, this problem, for a variable coefficient linear partial differential equation, has, to our knowledge attracted more attention recently in the finance literature than in the mathematics or physics literature. This is due to the importance, in the physics context, of moving beyond linear equations and handling diffusions exhibiting nonlinear effects, see [27]. Although non-linear equations also arise in the finance literature, the linear equations still play a predominant role there. It is well known (see A. Friedman [13], page 41) that an exchange of G’s forward and backward arguments then also yields a solution of the forward Fokker-Planck equation Date: May 18, 2005. 2000 Mathematics Subject Classification. 35K85, 35Q99.