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The perpetual American put option for jump-diffusions with applications

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  • Aase, Knut K

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. We present several examples demonstrating when the solution can be interpreted as a perpetual put price. This takes us into a study of how to risk adjust jump-diffusions. One key observation is that the probabililty distribution under the risk adjusted measure depends on the equity premium, which is not the case for the standard, continuous version. This difference may be utilized to find intertemporal, equilibrium equity premiums, for example Our basic solution is exact only when jump sizes can not be negative. We investigate when our solution is an approximation also for negative jumps. 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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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 qt31g898nz.

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Date of creation: 09 Jun 2005
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Handle: RePEc:cdl:anderf:qt31g898nz

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Keywords: Optimal exercise policy; American put option; perpetual option; optimal stopping; incomplete markets; equity premiums; CCAPM;

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  1. 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.
  2. Aase, Knut K, 2005. "Using Option Pricing Theory to Infer About Historical Equity Premiums," University of California at Los Angeles, Anderson Graduate School of Management qt3dd602j5, Anderson Graduate School of Management, UCLA.
  3. 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.
  4. 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.
  5. Campbell, John, 1996. "Understanding Risk and Return," Scholarly Articles 3153293, Harvard University Department of Economics.
  6. Campbell, John, 1993. "Intertemporal Asset Pricing Without Consumption Data," Scholarly Articles 3221491, Harvard University Department of Economics.
  7. Merton, Robert C, 1973. "An Intertemporal Capital Asset Pricing Model," Econometrica, Econometric Society, vol. 41(5), pages 867-87, September.
  8. 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.
  9. Ellen R. McGrattan & Edward C. Prescott, 2003. "Average debt and equity returns: puzzling?," Staff Report 313, Federal Reserve Bank of Minneapolis.
  10. Mehra, Rajnish & Prescott, Edward C., 1985. "The equity premium: A puzzle," Journal of Monetary Economics, Elsevier, vol. 15(2), pages 145-161, March.
  11. Phillippe Weil, 1997. "The Equity Premium Puzzle and the Risk-Free Rate Puzzle," Levine's Working Paper Archive 1833, David K. Levine.
  12. 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.
  13. Robert C. Merton, 1973. "Theory of Rational Option Pricing," Bell Journal of Economics, The RAND Corporation, vol. 4(1), pages 141-183, Spring.
  14. Merton, Robert C., 1975. "Option pricing when underlying stock returns are discontinuous," Working papers 787-75., Massachusetts Institute of Technology (MIT), Sloan School of Management.
  15. Mehra, Rajnish & Prescott, Edward C., 1988. "The equity risk premium: A solution?," Journal of Monetary Economics, Elsevier, vol. 22(1), pages 133-136, July.
  16. 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.
  17. Ernesto Mordecki, 2002. "Optimal stopping and perpetual options for Lévy processes," Finance and Stochastics, Springer, vol. 6(4), pages 473-493.
  18. S. G. Kou, 2002. "A Jump-Diffusion Model for Option Pricing," Management Science, INFORMS, vol. 48(8), pages 1086-1101, August.
  19. Rietz, Thomas A., 1988. "The equity risk premium a solution," Journal of Monetary Economics, Elsevier, vol. 22(1), pages 117-131, July.
  20. Lucas, Robert E, Jr, 1978. "Asset Prices in an Exchange Economy," Econometrica, Econometric Society, vol. 46(6), pages 1429-45, November.
  21. 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.
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Cited by:
  1. Aase, Knut K., 2005. "Using Option Pricing Theory to Infer About Equity Premiums," Discussion Papers 2005/11, Department of Business and Management Science, Norwegian School of Economics.
  2. Moon, Yongma & Yao, Tao & Park, Sungsoon, 2011. "Price negotiation under uncertainty," International Journal of Production Economics, Elsevier, vol. 134(2), pages 413-423, December.

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