IDEAS home Printed from https://ideas.repec.org/a/eee/spapps/v145y2022icp117-142.html
   My bibliography  Save this article

Density estimates and short-time asymptotics for a hypoelliptic diffusion process

Author

Listed:
  • Pigato, Paolo

Abstract

We study a system of n differential equations, each in dimension d. Only the first equation is forced by a Brownian motion and the dependence structure is such that, under a local weak Hörmander condition, the noise propagates to the whole system. We prove upper bounds for the transition density (heat kernel) and its derivatives of any order. Then we give precise short-time asymptotics of the density at a suitable central limit time scale. Both these results account for the different non-diffusive scales of propagation in the various components. Finally, we provide a valuation formula for short-maturity at-the-money Asian basket options under correlated local volatility dynamics.

Suggested Citation

  • Pigato, Paolo, 2022. "Density estimates and short-time asymptotics for a hypoelliptic diffusion process," Stochastic Processes and their Applications, Elsevier, vol. 145(C), pages 117-142.
  • Handle: RePEc:eee:spapps:v:145:y:2022:i:c:p:117-142
    DOI: 10.1016/j.spa.2021.11.012
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0304414921002015
    Download Restriction: Full text for ScienceDirect subscribers only

    File URL: https://libkey.io/10.1016/j.spa.2021.11.012?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. Alziary, Benedicte & Decamps, Jean-Paul & Koehl, Pierre-Francois, 1997. "A P.D.E. approach to Asian options: analytical and numerical evidence," Journal of Banking & Finance, Elsevier, vol. 21(5), pages 613-640, May.
    2. Archil Gulisashvili, 2018. "Gaussian stochastic volatility models: Scaling regimes, large deviations, and moment explosions," Papers 1808.00421, arXiv.org, revised Jun 2019.
    3. Dan Pirjol & Lingjiong Zhu, 2016. "Short Maturity Asian Options in Local Volatility Models," Papers 1609.07559, arXiv.org.
    4. Ditlevsen, Susanne & Löcherbach, Eva, 2017. "Multi-class oscillating systems of interacting neurons," Stochastic Processes and their Applications, Elsevier, vol. 127(6), pages 1840-1869.
    5. Deelstra, G. & Liinev, J. & Vanmaele, M., 2004. "Pricing of arithmetic basket options by conditioning," Insurance: Mathematics and Economics, Elsevier, vol. 34(1), pages 55-77, February.
    6. Griselda Deelstra & Jan Liinev & Michèle Vanmaele, 2004. "Pricing of arithmetic basket options by conditioning," ULB Institutional Repository 2013/7600, ULB -- Universite Libre de Bruxelles.
    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. Kenichiro Shiraya & Akihiko Takahashi, 2014. "Pricing Basket Options under Local Stochastic Volatility with Jumps," CIRJE F-Series CIRJE-F-913, CIRJE, Faculty of Economics, University of Tokyo.
    2. Griselda Deelstra & Michèle Vanmaele & David Vyncke, 2010. "Minimizing the Risk of a Financial Product Using a Put Option," Journal of Risk & Insurance, The American Risk and Insurance Association, vol. 77(4), pages 767-800, December.
    3. Kenichiro Shiraya & Akihiko Takahashi, 2015. "An approximation formula for basket option prices under local stochastic volatility with jumps: an application to commodity markets," CARF F-Series CARF-F-361, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo, revised Aug 2015.
    4. Griselda Deelstra & Alexandre Petkovic & Michèle Vanmaele, 2008. "Pricing and Hedging Asian Basket Spread Options," Working Papers ECARES 2008_004, ULB -- Universite Libre de Bruxelles.
    5. Brückner, Karsten, 2008. "Quantifying the error of convex order bounds for truncated first moments," Insurance: Mathematics and Economics, Elsevier, vol. 42(1), pages 261-270, February.
    6. Kenichiro Shiraya & Akihiko Takahashi, 2015. "An Approximation Formula for Basket Option Prices under Local Stochastic Volatility with Jumps: an Application to Commodity Markets," CIRJE F-Series CIRJE-F-973, CIRJE, Faculty of Economics, University of Tokyo.
    7. Kenichiro Shiraya & Akihiko Takahashi, 2013. "Pricing Basket Options under Local Stochastic Volatility with Jumps," CARF F-Series CARF-F-336, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo, revised May 2014.
    8. Ng, Andrew C.Y. & Li, Johnny Siu-Hang & Chan, Wai-Sum, 2013. "Pricing options on stocks denominated in different currencies: Theory and illustrations," The North American Journal of Economics and Finance, Elsevier, vol. 26(C), pages 339-354.
    9. Leccadito, Arturo & Paletta, Tommaso & Tunaru, Radu, 2016. "Pricing and hedging basket options with exact moment matching," Insurance: Mathematics and Economics, Elsevier, vol. 69(C), pages 59-69.
    10. Xu, Guoping & Zheng, Harry, 2009. "Approximate basket options valuation for a jump-diffusion model," Insurance: Mathematics and Economics, Elsevier, vol. 45(2), pages 188-194, October.
    11. Dan Pirjol & Lingjiong Zhu, 2017. "Asymptotics for the Discrete-Time Average of the Geometric Brownian Motion and Asian Options," Papers 1706.09659, arXiv.org.
    12. Dingeç, Kemal Dinçer & Hörmann, Wolfgang, 2013. "Control variates and conditional Monte Carlo for basket and Asian options," Insurance: Mathematics and Economics, Elsevier, vol. 52(3), pages 421-434.
    13. Karel in 't Hout & Jacob Snoeijer, 2021. "Numerical valuation of American basket options via partial differential complementarity problems," Papers 2106.01200, arXiv.org.
    14. Hobson, David & Laurence, Peter & Wang, Tai-Ho, 2005. "Static-arbitrage optimal subreplicating strategies for basket options," Insurance: Mathematics and Economics, Elsevier, vol. 37(3), pages 553-572, December.
    15. Yijuan Liang & Xiuchuan Xu, 2019. "Variance and Dimension Reduction Monte Carlo Method for Pricing European Multi-Asset Options with Stochastic Volatilities," Sustainability, MDPI, vol. 11(3), pages 1-21, February.
    16. Guoping Xu & Harry Zheng, 2012. "Lower Bound Approximation to Basket Option Values for Local Volatility Jump-Diffusion Models," Papers 1212.3147, arXiv.org, revised Oct 2013.
    17. Dan Pirjol & Lingjiong Zhu, 2023. "Asymptotics for Short Maturity Asian Options in Jump-Diffusion models with Local Volatility," Papers 2308.15672, arXiv.org, revised Feb 2024.
    18. De Luigi Christophe & Maire Sylvain, 2010. "Adaptive integration and approximation over hyper-rectangular regions with applications to basket option pricing," Monte Carlo Methods and Applications, De Gruyter, vol. 16(3-4), pages 265-282, January.
    19. Shiraya, Kenichiro & Takahashi, Akihiko, 2017. "A general control variate method for multi-dimensional SDEs: An application to multi-asset options under local stochastic volatility with jumps models in finance," European Journal of Operational Research, Elsevier, vol. 258(1), pages 358-371.
    20. David Hobson & Peter Laurence & Tai-Ho Wang, 2005. "Static-arbitrage upper bounds for the prices of basket options," Quantitative Finance, Taylor & Francis Journals, vol. 5(4), pages 329-342.

    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:eee:spapps:v:145:y:2022:i:c:p:117-142. 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: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/wps/find/journaldescription.cws_home/505572/description#description .

    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.