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European and American options: The semi-Markov case

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  • D’Amico, Guglielmo
  • Janssen, Jacques
  • Manca, Raimondo
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    Abstract

    In this paper, we assume that the log return of the underlying asset follows a semi-Markov process, then from the knowledge of the kernel we derive an explicit expression for the value of the option and for the bare risk in the case of the European call (put) option and, by means of a recursive system, we derive the value and the bare risk in the case of the American option. The prices and risks we obtained depend explicitly on the waiting-time distributions of the asset and they are duration dependent. The link with models based on Markov Chains and Continuous Time Random Walks is debated.

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    Bibliographic Info

    Article provided by Elsevier in its journal Physica A: Statistical Mechanics and its Applications.

    Volume (Year): 388 (2009)
    Issue (Month): 15 ()
    Pages: 3181-3194

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    Handle: RePEc:eee:phsmap:v:388:y:2009:i:15:p:3181-3194

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    Web page: http://www.journals.elsevier.com/physica-a-statistical-mechpplications/

    Related research

    Keywords: Semi-Markov processes; Backward recurrence time; European and American options; Wealth balance equation; Option value; Bare risk;

    References

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    1. Marco Raberto & Enrico Scalas & Francesco Mainardi, 2004. "Waiting-times and returns in high-frequency financial data: an empirical study," Finance, EconWPA 0411014, EconWPA.
    2. Enrico Scalas & Rudolf Gorenflo & Francesco Mainardi, 2004. "Fractional calculus and continuous-time finance," Finance, EconWPA 0411007, EconWPA.
    3. Mainardi, Francesco & Raberto, Marco & Gorenflo, Rudolf & Scalas, Enrico, 2000. "Fractional calculus and continuous-time finance II: the waiting-time distribution," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 287(3), pages 468-481.
    4. Scalas, Enrico, 2006. "The application of continuous-time random walks in finance and economics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 362(2), pages 225-239.
    5. Repetowicz, Przemysław & Richmond, Peter, 2004. "Modeling share price evolution as a continuous time random walk (CTRW) with non-independent price changes and waiting times," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 344(1), pages 108-111.
    6. Bouchaud, Jean-Philippe, 2002. "An introduction to statistical finance," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 313(1), pages 238-251.
    7. Jean-Philippe Bouchaud, 2002. "An introduction to statistical finance," Science & Finance (CFM) working paper archive 313238, Science & Finance, Capital Fund Management.
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    Cited by:
    1. Bhat, Harish S. & Kumar, Nitesh, 2012. "Option pricing under a normal mixture distribution derived from the Markov tree model," European Journal of Operational Research, Elsevier, Elsevier, vol. 223(3), pages 762-774.
    2. D’Amico, Guglielmo & Petroni, Filippo & Prattico, Flavio, 2013. "First and second order semi-Markov chains for wind speed modeling," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 392(5), pages 1194-1201.
    3. Preda, Vasile & Dedu, Silvia & Sheraz, Muhammad, 2014. "New measure selection for Hunt–Devolder semi-Markov regime switching interest rate models," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 407(C), pages 350-359.
    4. Guglielmo D'Amico & Filippo Petroni, 2011. "A semi-Markov model with memory for price changes," Papers 1109.4259, arXiv.org, revised Dec 2011.
    5. G. D'Amico & F. Petroni & F. Prattico, 2013. "Semi-Markov Models in High Frequency Finance: A Review," Papers 1312.3894, arXiv.org.
    6. Hunt, Julien & Devolder, Pierre, 2011. "Semi-Markov regime switching interest rate models and minimal entropy measure," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 390(21), pages 3767-3781.

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