IDEAS home Printed from https://ideas.repec.org/a/spr/metcap/v23y2021i4d10.1007_s11009-020-09827-5.html
   My bibliography  Save this article

Small-t Expansion for the Hartman-Watson Distribution

Author

Listed:
  • Dan Pirjol

    (Stevens Institute of Technology)

Abstract

The Hartman-Watson distribution with density f r ( t ) = 1 I 0 ( r ) θ ( r , t ) $f_{r}(t)=\frac {1}{I_{0}(r)} \theta (r,t)$ with r > 0 is a probability distribution defined on t ∈ ℝ + $t \in \mathbb {R}_{+}$ , which appears in several problems of applied probability. The density of this distribution is given by an integral θ(r, t) which is difficult to evaluate numerically for small t → 0. Using saddle point methods, we obtain the first two terms of the t → 0 expansion of θ(ρ/t, t) at fixed ρ > 0.

Suggested Citation

  • Dan Pirjol, 2021. "Small-t Expansion for the Hartman-Watson Distribution," Methodology and Computing in Applied Probability, Springer, vol. 23(4), pages 1537-1549, December.
    • Handle: RePEc:spr:metcap:v:23:y:2021:i:4:d:10.1007_s11009-020-09827-5
    DOI: 10.1007/s11009-020-09827-5
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s11009-020-09827-5
    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/s11009-020-09827-5?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. Daniel Dufresne, 2000. "Laguerre Series for Asian and Other Options," Mathematical Finance, Wiley Blackwell, vol. 10(4), pages 407-428, October.
    2. Stefan Gerhold, 2010. "The Hartman-Watson Distribution revisited: Asymptotics for Pricing Asian Options," Papers 1011.4830, arXiv.org, revised May 2011.
    3. Ning Cai & Yingda Song & Nan Chen, 2017. "Exact Simulation of the SABR Model," Operations Research, INFORMS, vol. 65(4), pages 931-951, August.
    4. Boyle, Phelim & Potapchik, Alexander, 2008. "Prices and sensitivities of Asian options: A survey," Insurance: Mathematics and Economics, Elsevier, vol. 42(1), pages 189-211, February.
    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. Dan Pirjol, 2020. "Asymptotic expansion for the Hartman-Watson distribution," Papers 2001.09579, arXiv.org, revised Feb 2021.
    2. Runhuan Feng & Hans W. Volkmer, 2015. "Conditional Asian Options," Papers 1505.06946, arXiv.org.
    3. Chueh-Yung Tsao & Chao-Ching Liu, 2012. "Asian Options with Credit Risks: Pricing and Sensitivity Analysis," Emerging Markets Finance and Trade, Taylor & Francis Journals, vol. 48(S3), pages 96-115, September.
    4. Runhuan Feng & Hans W. Volkmer, 2015. "Conditional Asian Options," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 18(06), pages 1-24.
    5. Dan Pirjol & Jing Wang & Lingjiong Zhu, 2017. "Short Maturity Forward Start Asian Options in Local Volatility Models," Papers 1710.03160, arXiv.org.
    6. Brignone, Riccardo & Kyriakou, Ioannis & Fusai, Gianluca, 2021. "Moment-matching approximations for stochastic sums in non-Gaussian Ornstein–Uhlenbeck models," Insurance: Mathematics and Economics, Elsevier, vol. 96(C), pages 232-247.
    7. Jianqiang Sun & Langnan Chen & Shiyin Li, 2013. "A Quasi‐Analytical Pricing Model for Arithmetic Asian Options," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 33(12), pages 1143-1166, December.
    8. Dai, Min & Li, Peifan & Zhang, Jin E., 2010. "A lattice algorithm for pricing moving average barrier options," Journal of Economic Dynamics and Control, Elsevier, vol. 34(3), pages 542-554, March.
    9. Antoine Jacquier & Patrick Roome, 2015. "Black-Scholes in a CEV random environment," Papers 1503.08082, arXiv.org, revised Nov 2017.
    10. Jean-Yves Datey & Genevieve Gauthier & Jean-Guy Simonato, 2003. "The Performance of Analytical Approximations for the Computation of Asian Quanto-Basket Option Prices," Multinational Finance Journal, Multinational Finance Journal, vol. 7(1-2), pages 55-82, March-Jun.
    11. Jurczenko, Emmanuel & Maillet, Bertrand & Negrea, Bogdan, 2002. "Revisited multi-moment approximate option pricing models: a general comparison (Part 1)," LSE Research Online Documents on Economics 24950, London School of Economics and Political Science, LSE Library.
    12. Dan Pirjol, 2024. "Subleading correction to the Asian options volatility in the Black-Scholes model," Papers 2407.05142, arXiv.org, revised Aug 2024.
    13. Ewald, Christian-Oliver & Menkens, Olaf & Hung Marten Ting, Sai, 2013. "Asian and Australian options: A common perspective," Journal of Economic Dynamics and Control, Elsevier, vol. 37(5), pages 1001-1018.
    14. Cen, Zhongdi & Xu, Aimin & Le, Anbo, 2015. "A hybrid finite difference scheme for pricing Asian options," Applied Mathematics and Computation, Elsevier, vol. 252(C), pages 229-239.
    15. Dan Pirjol & Lingjiong Zhu, 2023. "Sensitivities of Asian options in the Black-Scholes model," Papers 2301.06460, arXiv.org.
    16. Vadim Linetsky, 2004. "Spectral Expansions for Asian (Average Price) Options," Operations Research, INFORMS, vol. 52(6), pages 856-867, December.
    17. Zhang, Sumei & Gao, Xiong, 2019. "An asymptotic expansion method for geometric Asian options pricing under the double Heston model," Chaos, Solitons & Fractals, Elsevier, vol. 127(C), pages 1-9.
    18. 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.
    19. Florian Stebegg, 2014. "Model-Independent Pricing of Asian Options via Optimal Martingale Transport," Papers 1412.1429, arXiv.org.
    20. Cui, Zhenyu & Kirkby, J. Lars & Nguyen, Duy, 2021. "Efficient simulation of generalized SABR and stochastic local volatility models based on Markov chain approximations," European Journal of Operational Research, Elsevier, vol. 290(3), pages 1046-1062.

    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:metcap:v:23:y:2021:i:4:d:10.1007_s11009-020-09827-5. 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.