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Russian and American put options under exponential phase-type Lévy models


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  • Asmussen, Søren
  • Avram, Florin
  • Pistorius, Martijn R.
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    Consider the American put and Russian option (Ann. Appl. Probab. 3 (1993) 603; Theory Probab. Appl. 39 (1994) 103; Ann. Appl. Probab. 3 (1993) 641) with the stock price modeled as an exponential Lévy process. We find an explicit expression for the price in the dense class of Lévy processes with phase-type jumps in both directions. The solution rests on the reduction to the first passage time problem for (reflected) Lévy processes and on an explicit solution of the latter in the phase-type case via martingale stopping and Wiener-Hopf factorization. The same type of approach is also applied to the more general class of regime switching Lévy processes with phase-type jumps.

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    Bibliographic Info

    Article provided by Elsevier in its journal Stochastic Processes and their Applications.

    Volume (Year): 109 (2004)
    Issue (Month): 1 (January)
    Pages: 79-111

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    Handle: RePEc:eee:spapps:v:109:y:2004:i:1:p:79-111

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    Keywords: Lévy process Markov additive process First passage time Wald martingale Wiener-Hopf factorization Russian option Optimal stopping;


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    1. Benoit Mandelbrot, 1963. "The Variation of Certain Speculative Prices," The Journal of Business, University of Chicago Press, vol. 36, pages 394.
    2. Ernesto Mordecki, 2002. "Optimal stopping and perpetual options for Lévy processes," Finance and Stochastics, Springer, vol. 6(4), pages 473-493.
    3. Rama Cont & Marc Potters & Jean-Philippe Bouchaud, 1997. "Scaling in stock market data: stable laws and beyond," Papers cond-mat/9705087,
    4. Benoit Mandelbrot, 1963. "New Methods in Statistical Economics," Journal of Political Economy, University of Chicago Press, vol. 71, pages 421.
    5. Søren Asmussen, 2000. "Matrix-analytic Models and their Analysis," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics & Finnish Statistical Society & Norwegian Statistical Association & Swedish Statistical Association, vol. 27(2), pages 193-226.
    6. Carr, Peter, 1998. "Randomization and the American Put," Review of Financial Studies, Society for Financial Studies, vol. 11(3), pages 597-626.
    7. Eugene F. Fama, 1965. "Portfolio Analysis in a Stable Paretian Market," Management Science, INFORMS, vol. 11(3), pages 404-419, January.
    8. Eugene F. Fama, 1963. "Mandelbrot and the Stable Paretian Hypothesis," The Journal of Business, University of Chicago Press, vol. 36, pages 420.
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    Cited by:
    1. Doney, Ron & Maller, Ross & Savov, Mladen, 2009. "Renewal theorems and stability for the reflected process," Stochastic Processes and their Applications, Elsevier, vol. 119(4), pages 1270-1297, April.
    2. Chi, Yichun, 2010. "Analysis of the expected discounted penalty function for a general jump-diffusion risk model and applications in finance," Insurance: Mathematics and Economics, Elsevier, vol. 46(2), pages 385-396, April.
    3. Ivanovs, Jevgenijs, 2013. "A note on killing with applications in risk theory," Insurance: Mathematics and Economics, Elsevier, vol. 52(1), pages 29-34.
    4. Bakshi, Gurdip & Panayotov, George, 2010. "First-passage probability, jump models, and intra-horizon risk," Journal of Financial Economics, Elsevier, vol. 95(1), pages 20-40, January.
    5. Boyarchenko, Svetlana & Levendorskii, Sergei, 2010. "Optimal stopping in Levy models, for non-monotone discontinuous payoffs," MPRA Paper 27999, University Library of Munich, Germany.
    6. Christensen, Sören & Irle, Albrecht, 2009. "A note on pasting conditions for the American perpetual optimal stopping problem," Statistics & Probability Letters, Elsevier, vol. 79(3), pages 349-353, February.
    7. Bo, Lijun & Song, Renming & Tang, Dan & Wang, Yongjin & Yang, Xuewei, 2012. "Lévy risk model with two-sided jumps and a barrier dividend strategy," Insurance: Mathematics and Economics, Elsevier, vol. 50(2), pages 280-291.
    8. Ivanovs, Jevgenijs & Boxma, Onno & Mandjes, Michel, 2010. "Singularities of the matrix exponent of a Markov additive process with one-sided jumps," Stochastic Processes and their Applications, Elsevier, vol. 120(9), pages 1776-1794, August.
    9. Hao, Xuemiao & Li, Xuan & Shimizu, Yasutaka, 2013. "Finite-time survival probability and credit default swaps pricing under geometric Lévy markets," Insurance: Mathematics and Economics, Elsevier, vol. 53(1), pages 14-23.
    10. Xing, Xiaoyu & Zhang, Wei & Jiang, Yiming, 2008. "On the time to ruin and the deficit at ruin in a risk model with double-sided jumps," Statistics & Probability Letters, Elsevier, vol. 78(16), pages 2692-2699, November.
    11. Bo, Lijun & Yang, Xuewei, 2012. "Sequential maximum likelihood estimation for reflected generalized Ornstein–Uhlenbeck processes," Statistics & Probability Letters, Elsevier, vol. 82(7), pages 1374-1382.
    12. Hansjörg Albrecher & Jevgenijs Ivanovs, 2013. "A Risk Model with an Observer in a Markov Environment," Risks, MDPI, Open Access Journal, vol. 1(3), pages 148-161, November.
    13. Bayraktar, Erhan & Kyprianou, Andreas E. & Yamazaki, Kazutoshi, 2014. "Optimal dividends in the dual model under transaction costs," Insurance: Mathematics and Economics, Elsevier, vol. 54(C), pages 133-143.
    14. Kuznetsov, A. & Peng, X., 2012. "On the Wiener–Hopf factorization for Lévy processes with bounded positive jumps," Stochastic Processes and their Applications, Elsevier, vol. 122(7), pages 2610-2638.
    15. Jukka Lempa, 2006. "On Infinite Horizon Optimal Stopping of General Random Walk," Discussion Papers 3, Aboa Centre for Economics.
    16. Fourati, Sonia, 2012. "Explicit solutions of the exit problem for a class of Lévy processes; applications to the pricing of double-barrier options," Stochastic Processes and their Applications, Elsevier, vol. 122(3), pages 1034-1067.
    17. Gerber, Hans U. & Shiu, Elias S.W. & Yang, Hailiang, 2013. "Valuing equity-linked death benefits in jump diffusion models," Insurance: Mathematics and Economics, Elsevier, vol. 53(3), pages 615-623.
    18. Boyarchenko, Svetlana & Levendorskii, Sergei, 2008. "Exit problems in regime-switching models," Journal of Mathematical Economics, Elsevier, vol. 44(2), pages 180-206, January.
    19. Florian Kleinert & Kees van Schaik, 2013. "A variation of the Canadisation algorithm for the pricing of American options driven by L\'evy processes," Papers 1304.4534,


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