IDEAS home Printed from https://ideas.repec.org/a/spr/jotpro/v35y2022i4d10.1007_s10959-021-01154-w.html
   My bibliography  Save this article

Characterisation of Honest Times and Optional Semimartingales of Class- $$(\Sigma )$$ ( Σ )

Author

Listed:
  • Libo Li

    (University of New South Wales)

Abstract

Given a finite honest time, we first show that the associated Azéma optional supermartingale can be expressed as the drawdown and the relative drawdown of some local optional supermartingales with continuous running supremum. The relative drawdown representation then allows us to provide a characterisation of finite honest times using a family of non-negative local optional supermartingales with continuous running supremum which converges to zero at infinity. Then we extend the notion of semimartingales of class- $$(\Sigma )$$ ( Σ ) by allowing for jumps in its finite variation part of the semimartingale decomposition. This enables one to establish the Madan–Roynette–Yor option pricing formula for a larger class of processes, and finally, we apply the extended formula to the construction of finite honest times.

Suggested Citation

  • Libo Li, 2022. "Characterisation of Honest Times and Optional Semimartingales of Class- $$(\Sigma )$$ ( Σ )," Journal of Theoretical Probability, Springer, vol. 35(4), pages 2145-2175, December.
    • Handle: RePEc:spr:jotpro:v:35:y:2022:i:4:d:10.1007_s10959-021-01154-w
    DOI: 10.1007/s10959-021-01154-w
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10959-021-01154-w
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s10959-021-01154-w?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Kardaras, Constantinos, 2014. "On the characterisation of honest times that avoid all stopping times," Stochastic Processes and their Applications, Elsevier, vol. 124(1), pages 373-384.
    2. Kardaras, Constantinos, 2015. "On the stochastic behaviour of optional processes up to random times," LSE Research Online Documents on Economics 64965, London School of Economics and Political Science, LSE Library.
    3. Ashkan Nikeghbali & Eckhard Platen, 2013. "A reading guide for last passage times with financial applications in view," Finance and Stochastics, Springer, vol. 17(3), pages 615-640, July.
    4. Nikeghbali, Ashkan, 2006. "A class of remarkable submartingales," Stochastic Processes and their Applications, Elsevier, vol. 116(6), pages 917-938, June.
    5. Claudio Fontana & Monique Jeanblanc & Shiqi Song, 2014. "On arbitrages arising with honest times," Finance and Stochastics, Springer, vol. 18(3), pages 515-543, July.
    6. Grigorova, Miryana & Imkeller, Peter & Ouknine, Youssef & Quenez, Marie-Claire, 2020. "Optimal stopping with f-expectations: The irregular case," Stochastic Processes and their Applications, Elsevier, vol. 130(3), pages 1258-1288.
    7. Madan, D. & Roynette, B. & Yor, Marc, 2008. "Option prices as probabilities," Finance Research Letters, Elsevier, vol. 5(2), pages 79-87, June.
    8. Patrick Cheridito & Ashkan Nikeghbali & Eckhard Platen, 2012. "Processes of Class Sigma, Last Passage Times, and Drawdowns," Published Paper Series 2012-4, Finance Discipline Group, UTS Business School, University of Technology, Sydney.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Libo Li, 2018. "Characterisation of honest times and optional semimartingales of class-($\Sigma$)," Papers 1801.03873, arXiv.org, revised Dec 2021.
    2. Hongzhong Zhang, 2018. "Stochastic Drawdowns," World Scientific Books, World Scientific Publishing Co. Pte. Ltd., number 10078, December.
    3. Claudio Fontana & Monique Jeanblanc & Shiqi Song, 2014. "On arbitrages arising with honest times," Finance and Stochastics, Springer, vol. 18(3), pages 515-543, July.
    4. Ashkan Nikeghbali & Eckhard Platen, 2013. "A reading guide for last passage times with financial applications in view," Finance and Stochastics, Springer, vol. 17(3), pages 615-640, July.
    5. Anna Aksamit & Tahir Choulli & Jun Deng & Monique Jeanblanc, 2013. "Non-Arbitrage up to Random Horizon for Semimartingale Models," Papers 1310.1142, arXiv.org, revised Feb 2014.
    6. Claudio Fontana & Monique Jeanblanc & Shiqi Song, 2012. "On arbitrages arising from honest times," Papers 1207.1759, arXiv.org, revised Jul 2013.
    7. Acciaio, Beatrice & Fontana, Claudio & Kardaras, Constantinos, 2016. "Arbitrage of the first kind and filtration enlargements in semimartingale financial models," LSE Research Online Documents on Economics 65150, London School of Economics and Political Science, LSE Library.
    8. Fulgence Eyi-Obiang & Youssef Ouknine & Octave Moutsinga, 2017. "On the Study of Processes of $$\sum (H)$$ ∑ ( H ) and $$\sum _\mathrm{s}(H)$$ ∑ s ( H ) Classes," Journal of Theoretical Probability, Springer, vol. 30(1), pages 117-142, March.
    9. Libo Li & Ruyi Liu & Marek Rutkowski, 2022. "Vulnerable European and American Options in a Market Model with Optional Hazard Process," Papers 2212.12860, arXiv.org.
    10. Acciaio, Beatrice & Fontana, Claudio & Kardaras, Constantinos, 2016. "Arbitrage of the first kind and filtration enlargements in semimartingale financial models," Stochastic Processes and their Applications, Elsevier, vol. 126(6), pages 1761-1784.
    11. Kreher, Dörte, 2017. "Change of measure up to a random time: Details," Stochastic Processes and their Applications, Elsevier, vol. 127(5), pages 1565-1598.
    12. Ernst, Philip A. & Rogers, L.C.G. & Zhou, Quan, 2017. "The value of foresight," Stochastic Processes and their Applications, Elsevier, vol. 127(12), pages 3913-3927.
    13. Amel Bentata & Marc Yor, 2008. "From Black-Scholes and Dupire formulae to last passage times of local martingales. Part A : The infinite time horizon," Papers 0806.0239, arXiv.org.
    14. Vladimir Cherny & Jan Obłój, 2013. "Portfolio optimisation under non-linear drawdown constraints in a semimartingale financial model," Finance and Stochastics, Springer, vol. 17(4), pages 771-800, October.
    15. Hongzhong Zhang & Olympia Hadjiliadis, 2012. "Drawdowns and the Speed of Market Crash," Methodology and Computing in Applied Probability, Springer, vol. 14(3), pages 739-752, September.
    16. Fontana, Claudio & Grbac, Zorana & Jeanblanc, Monique & Li, Qinghua, 2014. "Information, no-arbitrage and completeness for asset price models with a change point," Stochastic Processes and their Applications, Elsevier, vol. 124(9), pages 3009-3030.
    17. Cui, Zhenyu & Nguyen, Duy, 2016. "Omega diffusion risk model with surplus-dependent tax and capital injections," Insurance: Mathematics and Economics, Elsevier, vol. 68(C), pages 150-161.
    18. Najnudel, Joseph & Nikeghbali, Ashkan, 2012. "On some universal σ-finite measures related to a remarkable class of submartingales," Stochastic Processes and their Applications, Elsevier, vol. 122(4), pages 1582-1600.
    19. Lingjiong Zhu, 2015. "Short maturity options for Azéma–Yor martingales," International Journal of Financial Engineering (IJFE), World Scientific Publishing Co. Pte. Ltd., vol. 2(04), pages 1-32, December.
    20. Konstantinos Konstantaras & Vasilios Sogiakas, 2019. "Is stock liquidity transferred and upgraded in acquisitions? Evidence from liquidity synergies in US freeze-outs," Annals of Operations Research, Springer, vol. 282(1), pages 179-216, November.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:jotpro:v:35:y:2022:i:4:d:10.1007_s10959-021-01154-w. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.