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Pricing Perpetual Options for Jump Processes

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  • Hans Gerber
  • Elias Shiu

Abstract

We consider two models in which the logarithm of the price of an asset is a shifted compound Poisson process. Explicit results are obtained for prices and optimal exercise strategies of certain perpetual American options on the asset, in particular for the perpetual put option. In the first model in which the jumps of the asset price are upwards, the results are obtained by the martingale approach and the smooth junction condition. In the second model in which the jumps are downwards, we show that the value of the strategy corresponding to a constant option-exercise boundary satisfies a certain renewal equation. Then the optimal exercise strategy is obtained from the continuous junction condition. Furthermore, the same model can be used to price certain reset options. Finally, we show how the classical model of geometric Brownian motion can be obtained as a limit and also how it can be integrated in the two models.

Suggested Citation

  • Hans Gerber & Elias Shiu, 1998. "Pricing Perpetual Options for Jump Processes," North American Actuarial Journal, Taylor & Francis Journals, vol. 2(3), pages 101-107.
    • Handle: RePEc:taf:uaajxx:v:2:y:1998:i:3:p:101-107
    DOI: 10.1080/10920277.1998.10595736
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    Cited by:

    1. Luis Alvarez & Teppo Rakkolainen, 2009. "Optimal payout policy in presence of downside risk," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 69(1), pages 27-58, March.
    2. Biffis, Enrico & Morales, Manuel, 2010. "On a generalization of the Gerber-Shiu function to path-dependent penalties," Insurance: Mathematics and Economics, Elsevier, vol. 46(1), pages 92-97, February.
    3. Luis H. R. Alvarez & Teppo A. Rakkolainen, 2007. "Optimal Dividend Control in Presence of Downside Risk," Discussion Papers 14, Aboa Centre for Economics.
    4. Ben Salah, Zied & Garrido, José, 2018. "On fair reinsurance premiums; Capital injections in a perturbed risk model," Insurance: Mathematics and Economics, Elsevier, vol. 82(C), pages 11-20.
    5. Marzia De Donno & Zbigniew Palmowski & Joanna Tumilewicz, 2020. "Double continuation regions for American and Swing options with negative discount rate in Lévy models," Mathematical Finance, Wiley Blackwell, vol. 30(1), pages 196-227, January.
    6. Linyi Qian & Zhuo Jin & Wei Wang & Lyu Chen, 2018. "Pricing dynamic fund protections for a hyperexponential jump diffusion process," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 47(1), pages 210-221, January.
    7. Luis H. R. Alvarez & Teppo A. Rakkolainen, 2006. "A Class of Solvable Optimal Stopping Problems of Spectrally Negative Jump Diffusions," Discussion Papers 9, Aboa Centre for Economics.
    8. Morales, Manuel, 2007. "On the expected discounted penalty function for a perturbed risk process driven by a subordinator," Insurance: Mathematics and Economics, Elsevier, vol. 40(2), pages 293-301, March.
    9. S. G. Kou & Hui Wang, 2004. "Option Pricing Under a Double Exponential Jump Diffusion Model," Management Science, INFORMS, vol. 50(9), pages 1178-1192, September.
    10. Avram, Florin & Chan, Terence & Usabel, Miguel, 0. "On the valuation of constant barrier options under spectrally one-sided exponential Lévy models and Carr's approximation for American puts," Stochastic Processes and their Applications, Elsevier, vol. 100(1-2), pages 75-107, July.
    11. Barrieu, Pauline & Bellamy, N., 2007. "Optimal hitting time and perpetual option in a non-Lévy model: application to real options," LSE Research Online Documents on Economics 5099, London School of Economics and Political Science, LSE Library.
    12. Ko, Bangwon & Shiu, Elias S.W. & Wei, Li, 2010. "Pricing maturity guarantee with dynamic withdrawal benefit," Insurance: Mathematics and Economics, Elsevier, vol. 47(2), pages 216-223, October.
    13. Gerber, Hans U. & Landry, Bruno, 1998. "On the discounted penalty at ruin in a jump-diffusion and the perpetual put option," Insurance: Mathematics and Economics, Elsevier, vol. 22(3), pages 263-276, July.

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