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Using Option Pricing Theory to Infer About Historical Equity Premiums

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  • Aase, Knut K
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    Abstract

    In this paper we make use of option pricing theory to infer about historical equity premiums. This we do by comparing the prices of an American perpetual put option computed using two different models: One is the standard model with continuous, zero expectation, Gaussian noise, the other is a very similar model, except that the zero expectation noise is of Poissonian type. Since a Poisson random variable is infinitely divisible, by the central limit theorem it is approximately normal. The interesting fact that makes this comparison worthwhile, is that the probability distribution under the risk adjusted measure turns out to depend on the equity premium in the Poisson model, while this is not so for the standard, Brownian motion version. This difference is utilized to find the intertemporal, equilibrium equity premium. We apply this technique to the US equity data of the last century, and find an indication that the risk premium on equity was about two and a half per cent if the risk free short rate was around one per cent. On the other hand, if the latter rate was about four per cent, we similarly find that this corresponds to an equity premium of around four and a half per cent. The advantage with our approach is that we only need equity data and option pricing theory, no consumption data was necessary to arrive at these conclusions. We round off the paper by investigating if the procedure also works for incomplete models.

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    Bibliographic Info

    Paper provided by Anderson Graduate School of Management, UCLA in its series University of California at Los Angeles, Anderson Graduate School of Management with number qt3dd602j5.

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    Date of creation: 10 Mar 2005
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    Handle: RePEc:cdl:anderf:qt3dd602j5

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    Related research

    Keywords: historical equity premiums; perpetual American put option; equity premium puzzle; risk free rate puzzle; geometric Brownian motion; geometric Poisson process; CCAPM;

    References

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    1. Ellen R. McGrattan & Edward C. Prescott, 2003. "Average Debt and Equity Returns: Puzzling?," American Economic Review, American Economic Association, vol. 93(2), pages 392-397, May.
    2. Constantinides, George M, 1990. "Habit Formation: A Resolution of the Equity Premium Puzzle," Journal of Political Economy, University of Chicago Press, vol. 98(3), pages 519-43, June.
    3. Philippe Weil, 1989. "The Equity Premium Puzzle and the Riskfree Rate Puzzle," NBER Working Papers 2829, National Bureau of Economic Research, Inc.
    4. John Y. Campbell, 1993. "Understanding Risk and Return," NBER Working Papers 4554, National Bureau of Economic Research, Inc.
    5. Bick, Avi, 1987. "On the Consistency of the Black-Scholes Model with a General Equilibrium Framework," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 22(03), pages 259-275, September.
    6. Lucas, Robert E, Jr, 1978. "Asset Prices in an Exchange Economy," Econometrica, Econometric Society, vol. 46(6), pages 1429-45, November.
    7. Merton, Robert C., 1976. "Option pricing when underlying stock returns are discontinuous," Journal of Financial Economics, Elsevier, vol. 3(1-2), pages 125-144.
    8. Mehra, Rajnish & Prescott, Edward C., 1985. "The equity premium: A puzzle," Journal of Monetary Economics, Elsevier, vol. 15(2), pages 145-161, March.
    9. S. G. Kou, 2002. "A Jump-Diffusion Model for Option Pricing," Management Science, INFORMS, vol. 48(8), pages 1086-1101, August.
    10. Campbell, John, 1993. "Intertemporal Asset Pricing Without Consumption Data," Scholarly Articles 3221491, Harvard University Department of Economics.
    11. Aase, Knut K., 2004. "Jump Dynamics: The Equity Premium and the Risk-Free Rate Puzzles," Discussion Papers 2004/12, Department of Business and Management Science, Norwegian School of Economics.
    12. Jan Ubøe & Bernt Øksendal & Knut Aase & Nicolas Privault, 2000. "White noise generalizations of the Clark-Haussmann-Ocone theorem with application to mathematical finance," Finance and Stochastics, Springer, vol. 4(4), pages 465-496.
    13. Knut Aase, 1999. "An Equilibrium Model of Catastrophe Insurance Futures and Spreads," The Geneva Risk and Insurance Review, Palgrave Macmillan, vol. 24(1), pages 69-96, June.
    14. Rietz, Thomas A., 1988. "The equity risk premium a solution," Journal of Monetary Economics, Elsevier, vol. 22(1), pages 117-131, July.
    15. Merton, Robert C, 1973. "An Intertemporal Capital Asset Pricing Model," Econometrica, Econometric Society, vol. 41(5), pages 867-87, September.
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    Cited by:
    1. Aase, Knut K, 2005. "The perpetual American put option for jump-diffusions with applications," University of California at Los Angeles, Anderson Graduate School of Management qt31g898nz, Anderson Graduate School of Management, UCLA.
    2. Aase, Knut K., 2005. "The perpetual American put option for jump-diffusions with applications," Discussion Papers 2005/12, Department of Business and Management Science, Norwegian School of Economics.

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