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Vasiček Beyond The Normal

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  • Ragnar Norberg

Abstract

A general Ornstein‐Uhlenbeck (OU) process is obtained upon replacing the Brownian motion appearing in the defining stochastic differential equation with a general Lévy process. Certain properties of the Brownian ancestor are distribution‐free and carry over to the general OU process. Explicit expressions are obtainable for expected values of a number of functionals of interest also in the general case. Special attention is paid here to gamma‐ and Poisson‐driven OU processes. The Brownian, Poisson, and gamma versions of the OU process are compared in various respects; in particular, their aptitude to describe stochastic interest rates is discussed in view of some standard issues in financial and actuarial mathematics: prices of zero‐coupon bonds, moments of present values, and probability distributions of present values of perpetuities. The problem of possible negative interest rates finds its resolution in the general setup by taking the driving Lévy process to be nondecreasing.

Suggested Citation

  • Ragnar Norberg, 2004. "Vasiček Beyond The Normal," Mathematical Finance, Wiley Blackwell, vol. 14(4), pages 585-604, October.
    • Handle: RePEc:bla:mathfi:v:14:y:2004:i:4:p:585-604
    DOI: 10.1111/j.0960-1627.2004.00206.x
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    Cited by:

    1. George Bouzianis & Lane P. Hughston, 2019. "Determination Of The Lévy Exponent In Asset Pricing Models," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 22(01), pages 1-18, February.
    2. Dorje C. Brody & Lane P. Hughston & David M. Meier, 2016. "L\'evy-Vasicek Models and the Long-Bond Return Process," Papers 1608.06376, arXiv.org, revised Sep 2016.
    3. Sadoghi, Amirhossein & Vecer, Jan, 2022. "Optimal liquidation problem in illiquid markets," European Journal of Operational Research, Elsevier, vol. 296(3), pages 1050-1066.
    4. Amirhossein Sadoghi & Jan Vecer, 2022. "Optimal liquidation problem in illiquid markets," Post-Print hal-03696768, HAL.

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