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The perpetual American put option for jump-diffusions: Implications for equity premiums

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  • Aase, Knut K.

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

Abstract

In this paper we solve an optimal stopping problem with an infinite time horizon, when the state variable follows a jump-diffusion. Under certain conditions our solution can be interpreted as the price of an American perpetual put option, when the underlying asset follows this type of process. The probability distribution under the risk adjusted measure turns out to depend on the equity premium, which is not the case for the standard, continuous version. This difference is utilized to find intertemporal, equilibrium equity premiums. 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 need only equity data and option pricing theory, no consumption data was necessary to arrive at these conclusions. Various market models are studied at an increasing level of complexity, ending with the incomplete model in the last part of the paper.

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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 2004/19.

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Length: 36 pages
Date of creation: 17 Dec 2004
Date of revision:
Handle: RePEc:hhs:nhhfms:2004_019

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Postal: NHH, Department of Business and Management Science, Helleveien 30, N-5045 Bergen, Norway
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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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Keywords: Optimal exercise policy; American put option; perpetual option; optimal stopping; incomplete markets; equity premiums; CCAPM;

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References

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  1. Ellen R. McGrattan & Edward C. Prescott, 2003. "Average Debt and Equity Returns: Puzzling?," Levine's Working Paper Archive 506439000000000367, David K. Levine.
  2. Rietz, Thomas A., 1988. "The equity risk premium a solution," Journal of Monetary Economics, Elsevier, vol. 22(1), pages 117-131, July.
  3. Weil, Philippe, 1989. "The equity premium puzzle and the risk-free rate puzzle," Journal of Monetary Economics, Elsevier, vol. 24(3), pages 401-421, November.
  4. R. Mehra & E. Prescott, 2010. "The equity premium: a puzzle," Levine's Working Paper Archive 1401, David K. Levine.
  5. John Y. Campbell, 1995. "Understanding Risk and Return," Harvard Institute of Economic Research Working Papers 1711, Harvard - Institute of Economic Research.
  6. 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.
  7. Lucas, Robert E, Jr, 1978. "Asset Prices in an Exchange Economy," Econometrica, Econometric Society, vol. 46(6), pages 1429-45, November.
  8. 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.
  9. 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.
  10. 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.
  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. Robert C. Merton, 1973. "Theory of Rational Option Pricing," Bell Journal of Economics, The RAND Corporation, vol. 4(1), pages 141-183, Spring.
  13. Merton, Robert C, 1973. "An Intertemporal Capital Asset Pricing Model," Econometrica, Econometric Society, vol. 41(5), pages 867-87, September.
  14. Mehra, Rajnish & Prescott, Edward C., 1988. "The equity risk premium: A solution?," Journal of Monetary Economics, Elsevier, vol. 22(1), pages 133-136, July.
  15. Merton, Robert C., 1976. "Option pricing when underlying stock returns are discontinuous," Journal of Financial Economics, Elsevier, vol. 3(1-2), pages 125-144.
  16. John Y. Campbell, 1992. "Intertemporal Asset Pricing Without Consumption Data," NBER Working Papers 3989, National Bureau of Economic Research, Inc.
  17. McDonald, Robert & Siegel, Daniel, 1986. "The Value of Waiting to Invest," The Quarterly Journal of Economics, MIT Press, vol. 101(4), pages 707-27, November.
  18. 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.
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