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Lévy–Vasicek Models And The Long-Bond Return Process

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  • DORJE C. BRODY

    (Department of Mathematics, University of Surrey, Guildford, Surrey GU2 7XH, UK2St. Petersburg National Research University, of Information Technologies, Mechanics and Optics, 49 Kronverksky Avenue, St. Petersburg 197101, Russia)

  • LANE P. HUGHSTON

    (Department of Computing, Goldsmiths, University of London, New Cross, London SE14 6NW, UK)

  • DAVID M. MEIER

    (Department of Mathematics, Brunel University London, Uxbridge, Middlesex UB8 3PH, UK)

Abstract

The classical derivation of the well-known Vasicek model for interest rates is reformulated in terms of the associated pricing kernel. An advantage of the pricing kernel method is that it allows one to generalize the construction to the Lévy–Vasicek case, avoiding issues of market incompleteness. In the Lévy–Vasicek model the short rate is taken in the real-world measure to be a mean-reverting process with a general one-dimensional Lévy driver admitting exponential moments. Expressions are obtained for the Lévy–Vasicek bond prices and interest rates, along with a formula for the return on a unit investment in the long bond, defined by Lt =limT→∞PtT/P0T, where PtT is the price at time t of a T-maturity discount bond. We show that the pricing kernel of a Lévy–Vasicek model is uniformly integrable if and only if the long rate of interest is strictly positive.

Suggested Citation

  • Dorje C. Brody & Lane P. Hughston & David M. Meier, 2018. "Lévy–Vasicek Models And The Long-Bond Return Process," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 21(03), pages 1-26, May.
    • Handle: RePEc:wsi:ijtafx:v:21:y:2018:i:03:n:s0219024918500267
    DOI: 10.1142/S0219024918500267
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    References listed on IDEAS

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