IDEAS home Printed from https://ideas.repec.org/p/ajf/louvlr/2020006.html
   My bibliography  Save this paper

SDES with uniform distributions: Peacocks, conic martingales and mean reverting uniform diffusions

Author

Listed:
  • Brigo, Damiano
  • Jeanblanc, Monique
  • Vrins, Frédéric

Abstract

Peacocks are increasing processes for the convex order. To any peacock, one can associate martingales with the same marginal laws. We are interested in finding the diffusion associated to the uniform peacock, i.e., the peacock with uniform law at all times on a time-varying support [a(t),b(t)]. Following an idea from Dupire (1994), Madan and Yor (2002) propose a construction to find a diffusion martingale associated to a Peacock, under the assumption of existence of a solution to a particular stochastic differential equation (SDE). In this paper we study the SDE associated to the uniform Peacock and give sufficient conditions on the (conic) boundary to have a unique strong or weak solution and analyze the local time at the boundary. Eventually, we focus on the constant support case. Given that the only uniform martingale with time-independent support seems to be a constant, we consider more general (mean-reverting) diffusions. We prove existence of a solution to the related SDE and derive the moments of transition densities. Limit-laws and ergodic results show that the transition law tends to a uniform distribution.
(This abstract was borrowed from another version of this item.)

Suggested Citation

  • Brigo, Damiano & Jeanblanc, Monique & Vrins, Frédéric, 2019. "SDES with uniform distributions: Peacocks, conic martingales and mean reverting uniform diffusions," LIDAM Reprints LFIN 2020006, Université catholique de Louvain, Louvain Finance (LFIN).
    • Handle: RePEc:ajf:louvlr:2020006
    Note: In : Stochastic Processes and Their Applications, Vol. 130, no. 7, p. 3895-3919 (2020)
    as

    Download full text from publisher

    To our knowledge, this item is not available for download. To find whether it is available, there are three options:
    1. Check below whether another version of this item is available online.
    2. Check on the provider's web page whether it is in fact available.
    3. Perform a search for a similarly titled item that would be available.

    Other versions of this item:

    References listed on IDEAS

    as
    1. Brigo, Damiano & Jeanblanc, Monique & Vrins, Frédéric, 2020. "SDEs with uniform distributions: Peacocks, conic martingales and mean reverting uniform diffusions," Stochastic Processes and their Applications, Elsevier, vol. 130(7), pages 3895-3919.
    2. Monique Jeanblanc & Frédéric Vrins, 2018. "Conic martingales from stochastic integrals," Mathematical Finance, Wiley Blackwell, vol. 28(2), pages 516-535, April.
    3. Peter Carr, 2017. "Bounded Brownian Motion," Risks, MDPI, vol. 5(4), pages 1-11, November.
    4. Brigo, Damiano, 2000. "On SDEs with marginal laws evolving in finite-dimensional exponential families," Statistics & Probability Letters, Elsevier, vol. 49(2), pages 127-134, August.
    5. Damiano Brigo & Fabio Mercurio, 2000. "Option pricing impact of alternative continuous-time dynamics for discretely-observed stock prices," Finance and Stochastics, Springer, vol. 4(2), pages 147-159.
    6. Christophe Profeta & Frédéric Vrins, 2019. "Piecewise constant martingales and lazy clocks," LIDAM Reprints CORE 2990, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Brigo, Damiano & Jeanblanc, Monique & Vrins, Frédéric, 2020. "SDEs with uniform distributions: Peacocks, conic martingales and mean reverting uniform diffusions," Stochastic Processes and their Applications, Elsevier, vol. 130(7), pages 3895-3919.

    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. D. Brigo, 2023. "Probability-Free Models in Option Pricing: Statistically Indistinguishable Dynamics and Historical vs Implied Volatility," World Scientific Book Chapters, in: David Gershon & Alexander Lipton & Mathieu Rosenbaum & Zvi Wiener (ed.), Options — 45 years since the Publication of the Black–Scholes–Merton Model The Gershon Fintech Center Conference, chapter 4, pages 47-61, World Scientific Publishing Co. Pte. Ltd..
    2. Cheikh Mbaye & Fr'ed'eric Vrins, 2019. "An arbitrage-free conic martingale model with application to credit risk," Papers 1909.02474, arXiv.org.
    3. Damiano Brigo & Fabio Mercurio, 2008. "Discrete Time vs Continuous Time Stock-price Dynamics and implications for Option Pricing," Papers 0812.4010, arXiv.org.
    4. Damiano Brigo, 2008. "The general mixture-diffusion SDE and its relationship with an uncertain-volatility option model with volatility-asset decorrelation," Papers 0812.4052, arXiv.org.
    5. Brigo, Damiano, 2000. "On SDEs with marginal laws evolving in finite-dimensional exponential families," Statistics & Probability Letters, Elsevier, vol. 49(2), pages 127-134, August.
    6. Cheikh Mbaye & Frédéric Vrins, 2018. "A Subordinated Cir Intensity Model With Application To Wrong-Way Risk Cva," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 21(07), pages 1-22, November.
    7. Albert Cohen & Nick Costanzino, 2017. "A General Framework for Incorporating Stochastic Recovery in Structural Models of Credit Risk," Risks, MDPI, vol. 5(4), pages 1-19, December.
    8. Tianyao Chen & Xue Cheng & Jingping Yang, 2019. "Common Decomposition of Correlated Brownian Motions and its Financial Applications," Papers 1907.03295, arXiv.org, revised Nov 2020.
    9. Frédéric Vrins, 2017. "Wrong-Way Risk Cva Models With Analytical Epe Profiles Under Gaussian Exposure Dynamics," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 20(07), pages 1-35, November.
    10. Zura Kakushadze, 2020. "Option Pricing: Channels, Target Zones and Sideways Markets," Bulletin of Applied Economics, Risk Market Journals, vol. 7(2), pages 25-33.
    11. John Armstrong & Claudio Bellani & Damiano Brigo & Thomas Cass, 2021. "Option pricing models without probability: a rough paths approach," Mathematical Finance, Wiley Blackwell, vol. 31(4), pages 1494-1521, October.
    12. László Márkus & Ashish Kumar, 2021. "Modelling Joint Behaviour of Asset Prices Using Stochastic Correlation," Methodology and Computing in Applied Probability, Springer, vol. 23(1), pages 341-354, March.
    13. Cheikh Mbaye & Frédéric Vrins, 2022. "Affine term structure models: A time‐change approach with perfect fit to market curves," Mathematical Finance, Wiley Blackwell, vol. 32(2), pages 678-724, April.
    14. BRIGO, Damiano & VRINS, Frédéric, 2018. "Disentangling wrong-way risk: pricing credit valuation adjustment via change of measures," European Journal of Operational Research, Elsevier, vol. 269(3), pages 1154-1164.
    15. Nassim Nicholas Taleb, 2017. "Election Predictions as Martingales: An Arbitrage Approach," Papers 1703.06351, arXiv.org, revised Jul 2019.
    16. Luciano Campi, 2004. "Arbitrage and completeness in financial markets with given N-dimensional distributions," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 27(1), pages 57-80, August.
    17. Zura Kakushadze, 2020. "Option Pricing: Channels, Target Zones and Sideways Markets," Papers 2006.14121, arXiv.org.
    18. Damiano Brigo & Fr'ed'eric Vrins, 2016. "Disentangling wrong-way risk: pricing CVA via change of measures and drift adjustment," Papers 1611.02877, arXiv.org.
    19. Samuel Drapeau & Yunbo Zhang, 2019. "Pricing and Hedging Performance on Pegged FX Markets Based on a Regime Switching Model," Papers 1910.08344, arXiv.org, revised May 2020.
    20. Albert Cohen, 2018. "Editorial: A Celebration of the Ties That Bind Us: Connections between Actuarial Science and Mathematical Finance," Risks, MDPI, vol. 6(1), pages 1-3, January.

    More about this item

    Statistics

    Access and download statistics

    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:ajf:louvlr:2020006. 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: Séverine De Visscher (email available below). General contact details of provider: https://edirc.repec.org/data/lfuclbe.html .

    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.