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Interest Rates and Information Geometry

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  • Dorje C. Brody
  • Lane P. Hughston
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    Abstract

    The space of probability distributions on a given sample space possesses natural geometric properties. For example, in the case of a smooth parametric family of probability distributions on the real line, the parameter space has a Riemannian structure induced by the embedding of the family into the Hilbert space of square-integrable functions, and is characterised by the Fisher-Rao metric. In the nonparametric case the relevant geometry is determined by the spherical distance function of Bhattacharyya. In the context of term structure modelling, we show that minus the derivative of the discount function with respect to the maturity date gives rise to a probability density. This follows as a consequence of the positivity of interest rates. Therefore, by mapping the density functions associated with a given family of term structures to Hilbert space, the resulting metrical geometry can be used to analyse the relationship of yield curves to one another. We show that the general arbitrage-free yield curve dynamics can be represented as a process taking values in the convex space of smooth density functions on the positive real line. It follows that the theory of interest rate dynamics can be represented by a class of processes in Hilbert space. We also derive the dynamics for the central moments associated with the distribution determined by the yield curve.

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    File URL: http://arxiv.org/pdf/1111.3757
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    Bibliographic Info

    Paper provided by arXiv.org in its series Papers with number 1111.3757.

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    Date of creation: Nov 2011
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    Publication status: Published in Proc. R. Soc. Lond. A (2001) 457, 1343-1363
    Handle: RePEc:arx:papers:1111.3757

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    1. Björk, Tomas & Svensson, Lars, 1999. "On the Existence of Finite Dimensional Realizations for Nonlinear Forward Rate Models," Working Paper Series in Economics and Finance 338, Stockholm School of Economics.
    2. Nielsen, Lars Tyge, 1999. "Pricing and Hedging of Derivative Securities," OUP Catalogue, Oxford University Press, number 9780198776192.
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