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On the Consistency of the Lucas Pricing Formula

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  • Aase, Knut K.

    ()
    (Dept. of Finance and Management Science, Norwegian School of Economics and Business Administration)

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    Abstract

    In order to find the real market value of an asset in an exchange economy, one would typically apply the formula appearing in Lucas (1978), developed in a discrete time framework. This theory has also been extended to continuous time models, in which case the same pricing formula has been universally applied. While the discrete time theory is rather transparent, there has been some confusion regarding the continuous time analogue. In particular, the continuous time pricing formula must contain a certain type of a square covariance term that does not readily follow from the discrete time formulation. As a result, this term has sometimes been missing in situations where it should have been included. In this paper we reformulate the discrete time theory in such a way that this covariance term does not come as a mystery in the continuous time version. It is shown, e.g., that this term is of importance also in the equivalent martingale measure approach to pricing. In most real life situations dividends are paid out in lump sums, not in rates. This leads to a discontinuous model, and adding a continuous time framework, it appears that our framework is a most natural one in finance.

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    Bibliographic Info

    Paper provided by Department of Business and Management Science, Norwegian School of Economics in its series Discussion Papers with number 2005/9.

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    Length: 19 pages
    Date of creation: 30 Nov 2005
    Date of revision:
    Handle: RePEc:hhs:nhhfms:2005_009

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    Postal: NHH, Department of Business and Management Science, Helleveien 30, N-5045 Bergen, Norway
    Phone: +47 55 95 92 93
    Fax: +47 55 95 96 50
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    Web page: http://www.nhh.no/en/research-faculty/department-of-business-and-management-science.aspx
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    Related research

    Keywords: Exchange economy; state price deflator; discrete time; continuous time; equivalent martingale measure; the Gordon growth model;

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    References

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    1. Duffie, Darrell & Zame, William, 1989. "The Consumption-Based Capital Asset Pricing Model," Econometrica, Econometric Society, vol. 57(6), pages 1279-97, November.
    2. Jamshidian, Farshid, 1989. " An Exact Bond Option Formula," Journal of Finance, American Finance Association, vol. 44(1), pages 205-09, March.
    3. Ross, Stephen A, 1978. "A Simple Approach to the Valuation of Risky Streams," The Journal of Business, University of Chicago Press, vol. 51(3), pages 453-75, July.
    4. Lars Nielsen, 2007. "Dividends in the theory of derivative securities pricing," Economic Theory, Springer, vol. 31(3), pages 447-471, June.
    5. Knut K. Aase, 2002. "Equilibrium Pricing in the Presence of Cumulative Dividends Following a Diffusion," Mathematical Finance, Wiley Blackwell, vol. 12(3), pages 173-198.
    6. Lucas, Robert E, Jr, 1978. "Asset Prices in an Exchange Economy," Econometrica, Econometric Society, vol. 46(6), pages 1429-45, November.
    7. Kaushik I. Amin & Robert A. Jarrow, 1992. "Pricing Options On Risky Assets In A Stochastic Interest Rate Economy," Mathematical Finance, Wiley Blackwell, vol. 2(4), pages 217-237.
    8. Duffie, Darrell & Shafer, Wayne, 1985. "Equilibrium in incomplete markets: I : A basic model of generic existence," Journal of Mathematical Economics, Elsevier, vol. 14(3), pages 285-300, June.
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