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A note on the existence of unique equivalent martingale measures in a Markovian setting

Author

Listed:
  • Tina Hviid Rydberg

    (University of Aarhus, Department of Theoretical Statistics, Ny Munkegade Bldg. 530, DK-8000 århus C, Denmark)

Abstract

Simple sufficient conditions for the existence of a unique equivalent martingale measure are provided. Furthermore, these conditions give us a handle on situations where an equivalent martingale measure cannot exist. The existence of a unique equivalent martingale measure is of relevance to problems in mathematical finance. Two examples of models for which the question of existence was unresolved are studied. By means of our results existence of a unique equivalent measure up to an explosion time is proved.

Suggested Citation

  • Tina Hviid Rydberg, 1997. "A note on the existence of unique equivalent martingale measures in a Markovian setting," Finance and Stochastics, Springer, vol. 1(3), pages 251-257.
    • Handle: RePEc:spr:finsto:v:1:y:1997:i:3:p:251-257
    Note: received: May 1996; final version received: March 1997
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    Citations

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    Cited by:

    1. Blei, Stefan & Engelbert, Hans-Jürgen, 2009. "On exponential local martingales associated with strong Markov continuous local martingales," Stochastic Processes and their Applications, Elsevier, vol. 119(9), pages 2859-2880, September.
    2. Ahdida, Abdelkoddousse & Alfonsi, Aurélien, 2013. "A mean-reverting SDE on correlation matrices," Stochastic Processes and their Applications, Elsevier, vol. 123(4), pages 1472-1520.
    3. Ren'e Aid & Giorgia Callegaro & Luciano Campi, 2019. "No-Arbitrage Commodity Option Pricing with Market Manipulation," Papers 1909.07896, arXiv.org, revised Mar 2020.
    4. Takuji Arai & Yuto Imai & Ryo Nakashima, 2018. "Numerical analysis on quadratic hedging strategies for normal inverse Gaussian models," Papers 1801.05597, arXiv.org.
    5. Neil Shephard & Ole E. Barndorff-Nielsen, 2012. "Basics of Levy processes," Economics Series Working Papers 610, University of Oxford, Department of Economics.
    6. Barndorff-Nielsen, Ole E. & Pérez-Abreu, Victor, 1999. "Stationary and self-similar processes driven by Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 84(2), pages 357-369, December.
    7. Chernov, Mikhail & Ghysels, Eric, 2000. "A study towards a unified approach to the joint estimation of objective and risk neutral measures for the purpose of options valuation," Journal of Financial Economics, Elsevier, vol. 56(3), pages 407-458, June.
    8. Damiano Brigo & Fabio Mercurio, 2008. "Discrete Time vs Continuous Time Stock-price Dynamics and implications for Option Pricing," Papers 0812.4010, arXiv.org.
    9. Abdelkoddousse Ahdida & Aur'elien Alfonsi, 2011. "A Mean-Reverting SDE on Correlation matrices," Papers 1108.5264, arXiv.org, revised Feb 2012.
    10. Paul Fearnhead & Omiros Papaspiliopoulos & Gareth O. Roberts & Andrew Stuart, 2010. "Random‐weight particle filtering of continuous time processes," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 72(4), pages 497-512, September.
    11. Giorgos Sermaidis & Omiros Papaspiliopoulos & Gareth O. Roberts & Alexandros Beskos & Paul Fearnhead, 2013. "Markov Chain Monte Carlo for Exact Inference for Diffusions," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 40(2), pages 294-321, June.
    12. Abdelkoddousse Ahdida & Aur'elien Alfonsi & Ernesto Palidda, 2014. "Smile with the Gaussian term structure model," Papers 1412.7412, arXiv.org, revised Nov 2015.
    13. Aïd, René & Callegaro, Giorgia & Campi, Luciano, 2020. "No-arbitrage commodity option pricing with market manipulation," LSE Research Online Documents on Economics 103815, London School of Economics and Political Science, LSE Library.
    14. David R. Ba~nos & Salvador Ortiz-Latorre & Oriol Zamora Font, 2022. "Change of measure in a Heston-Hawkes stochastic volatility model," Papers 2210.15343, arXiv.org.
    15. Alfonsi, Aurélien & Kebaier, Ahmed & Rey, Clément, 2016. "Maximum likelihood estimation for Wishart processes," Stochastic Processes and their Applications, Elsevier, vol. 126(11), pages 3243-3282.
    16. Abdelkoddousse Ahdida & Aurélien Alfonsi, 2013. "A Mean-Reverting SDE on Correlation matrices," Post-Print hal-00617111, HAL.
    17. Jourdain Benjamin & Sbai Mohamed, 2007. "Exact retrospective Monte Carlo computation of arithmetic average Asian options," Monte Carlo Methods and Applications, De Gruyter, vol. 13(2), pages 135-171, July.

    More about this item

    Keywords

    Explosion time; hyperbolic diffusion processes; normal inverse Gaussian diffusion processes; stochastic differential equations; unique solution in law;
    All these keywords.

    JEL classification:

    • G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing

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