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European and American options: The semi-Markov case

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

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.

Suggested Citation

  • D’Amico, Guglielmo & Janssen, Jacques & Manca, Raimondo, 2009. "European and American options: The semi-Markov case," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 388(15), pages 3181-3194.
    • Handle: RePEc:eee:phsmap:v:388:y:2009:i:15:p:3181-3194
    DOI: 10.1016/j.physa.2009.04.016
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    References listed on IDEAS

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    1. 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.
    2. Jin-Chuan Duan & Evan Dudley & Geneviève Gauthier & Jean-Guy Simonato, 1999. "Pricing Discretely Monitored Barrier Options by a Markov Chain," CIRANO Working Papers 99s-15, CIRANO.
    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 & Gorenflo, Rudolf & Mainardi, Francesco, 2000. "Fractional calculus and continuous-time finance," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 284(1), pages 376-384.
    5. 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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    7. Raberto, Marco & Scalas, Enrico & Mainardi, Francesco, 2002. "Waiting-times and returns in high-frequency financial data: an empirical study," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 314(1), pages 749-755.
    8. 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.
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    Citations

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    Cited by:

    1. 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.
    2. Guglielmo D'Amico & Filippo Petroni, 2011. "A semi-Markov model with memory for price changes," Papers 1109.4259, arXiv.org, revised Dec 2011.
    3. 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, vol. 223(3), pages 762-774.
    4. G. D'Amico & F. Petroni & F. Prattico, 2013. "Semi-Markov Models in High Frequency Finance: A Review," Papers 1312.3894, arXiv.org.
    5. 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.
    6. Guglielmo D'Amico & Filippo Petroni, 2020. "A micro-to-macro approach to returns, volumes and waiting times," Papers 2007.06262, arXiv.org.
    7. Guglielmo D'Amico & Filippo Petroni, 2013. "Multivariate high-frequency financial data via semi-Markov processes," Papers 1305.0436, arXiv.org.
    8. 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.

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