IDEAS home Printed from https://ideas.repec.org/p/arx/papers/1810.02390.html
   My bibliography  Save this paper

On the First Hitting Time Density of an Ornstein-Uhlenbeck Process

Author

Listed:
  • Alexander Lipton
  • Vadim Kaushansky

Abstract

In this paper, we study the classical problem of the first passage hitting density of an Ornstein--Uhlenbeck process. We give two complementary (forward and backward) formulations of this problem and provide semi-analytical solutions for both. The corresponding problems are comparable in complexity. By using the method of heat potentials, we show how to reduce these problems to linear Volterra integral equations of the second kind. For small values of $t$, we solve these equations analytically by using Abel equation approximation; for larger $t$ we solve them numerically. We also provide a comparison with other known methods for finding the hitting density of interest, and argue that our method has considerable advantages and provides additional valuable insights.

Suggested Citation

  • Alexander Lipton & Vadim Kaushansky, 2018. "On the First Hitting Time Density of an Ornstein-Uhlenbeck Process," Papers 1810.02390, arXiv.org, revised Oct 2018.
    • Handle: RePEc:arx:papers:1810.02390
    as

    Download full text from publisher

    File URL: http://arxiv.org/pdf/1810.02390
    File Function: Latest version
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Pierre Collin‐Dufresne & Robert S. Goldstein, 2001. "Do Credit Spreads Reflect Stationary Leverage Ratios?," Journal of Finance, American Finance Association, vol. 56(5), pages 1929-1957, October.
    2. Olivier Scaillet & Boris Leblanc, 1998. "Path dependent options on yields in the affine term structure model," Finance and Stochastics, Springer, vol. 2(4), pages 349-367.
    3. Delia Coculescu & Hélyette Geman & Monique Jeanblanc, 2008. "Valuation of default-sensitive claims under imperfect information," Finance and Stochastics, Springer, vol. 12(2), pages 195-218, April.
    4. Alexander Lipton & Vadim Kaushansky & Christoph Reisinger, 2018. "Semi-analytical solution of a McKean-Vlasov equation with feedback through hitting a boundary," Papers 1808.05311, arXiv.org, revised Aug 2018.
    5. Chuang Yi, 2010. "On the first passage time distribution of an Ornstein-Uhlenbeck process," Quantitative Finance, Taylor & Francis Journals, vol. 10(9), pages 957-960.
    6. Alexander Lipton, 2001. "Mathematical Methods for Foreign Exchange:A Financial Engineer's Approach," World Scientific Books, World Scientific Publishing Co. Pte. Ltd., number 4694, February.
    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. Alexander Lipton, 2020. "Old Problems, Classical Methods, New Solutions," Papers 2003.06903, arXiv.org.
    2. Kyriakos Georgiou & Athanasios N. Yannacopoulos, 2023. "Probability of Default modelling with L\'evy-driven Ornstein-Uhlenbeck processes and applications in credit risk under the IFRS 9," Papers 2309.12384, arXiv.org.
    3. Andrey Itkin & Dmitry Muravey, 2020. "Semi-analytic pricing of double barrier options with time-dependent barriers and rebates at hit," Papers 2009.09342, arXiv.org, revised Oct 2020.
    4. Andrey Itkin & Dmitry Muravey, 2020. "Semi-closed form prices of barrier options in the Hull-White model," Papers 2004.09591, arXiv.org, revised Sep 2020.
    5. Alexander Lipton & Marcos Lopez de Prado, 2020. "A closed-form solution for optimal mean-reverting trading strategies," Papers 2003.10502, arXiv.org.

    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. Lingfei Li & Vadim Linetsky, 2015. "Discretely monitored first passage problems and barrier options: an eigenfunction expansion approach," Finance and Stochastics, Springer, vol. 19(4), pages 941-977, October.
    2. Alexander Lipton, 2020. "Old Problems, Classical Methods, New Solutions," Papers 2003.06903, arXiv.org.
    3. Stefano De Marco & Caroline Hillairet & Antoine Jacquier, 2017. "Shapes of implied volatility with positive mass at zero," Working Papers 2017-77, Center for Research in Economics and Statistics.
    4. Stefano De Marco & Caroline Hillairet & Antoine Jacquier, 2013. "Shapes of implied volatility with positive mass at zero," Papers 1310.1020, arXiv.org, revised May 2017.
    5. Xing, Xiaoyu & Xing, Yongsheng & Yang, Xuewei, 2012. "A note on transition density for the reflected Ornstein–Uhlenbeck process," Statistics & Probability Letters, Elsevier, vol. 82(3), pages 586-591.
    6. Chi Man Leung & Yue Kuen Kwok, 2017. "NUMERICAL PRICING OF CoCo BONDS WITH PARISIAN TRIGGER FEATURE USING THE FORTET METHOD," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 20(07), pages 1-22, November.
    7. Stylianos Perrakis & Rui Zhong, 2017. "Liquidity Risk and Volatility Risk in Credit Spread Models: A Unified Approach," European Financial Management, European Financial Management Association, vol. 23(5), pages 873-901, October.
    8. Chuang Yi, 2010. "On the first passage time distribution of an Ornstein-Uhlenbeck process," Quantitative Finance, Taylor & Francis Journals, vol. 10(9), pages 957-960.
    9. Zhijian (James) Huang & Yuchen Luo, 2016. "Revisiting Structural Modeling of Credit Risk—Evidence from the Credit Default Swap (CDS) Market," JRFM, MDPI, vol. 9(2), pages 1-20, May.
    10. Chenxu Li, 2014. "Closed-Form Expansion, Conditional Expectation, and Option Valuation," Mathematics of Operations Research, INFORMS, vol. 39(2), pages 487-516, May.
    11. Zhao, Hong & Shen, Hao & Wang, Haizhi & Zhu, Yun, 2024. "State corporate tax changes and bond pricing: U.S. evidence," International Review of Financial Analysis, Elsevier, vol. 92(C).
    12. Diaz Weigel, Diana & Gemmill, Gordon, 2006. "What drives credit risk in emerging markets? The roles of country fundamentals and market co-movements," Journal of International Money and Finance, Elsevier, vol. 25(3), pages 476-502, April.
    13. Ye, Zhen & Zhang, Fangzhu & Coffman, D’Maris & Xia, Senmao & Wang, Zhifeng & Zhu, Zhonghua, 2022. "China’s urban construction investment bond: Contextualising a financial tool for local government," Land Use Policy, Elsevier, vol. 112(C).
    14. Sangwon Suh & Inwon Jang & Misun Ahn, 2013. "A Simple Method For Measuring Systemic Risk Using Credit Default Swap Market Data," Journal of Economic Development, Chung-Ang Unviersity, Department of Economics, vol. 38(4), pages 75-100, December.
    15. Nusrat Jahan, 2022. "Macroeconomic Determinants of Corporate Credit Spreads: Evidence from Canada," Carleton Economic Papers 22-07, Carleton University, Department of Economics.
    16. Nan Chen & S. G. Kou, 2009. "Credit Spreads, Optimal Capital Structure, And Implied Volatility With Endogenous Default And Jump Risk," Mathematical Finance, Wiley Blackwell, vol. 19(3), pages 343-378, July.
    17. John Y. Campbell & Glen B. Taksler, 2003. "Equity Volatility and Corporate Bond Yields," Journal of Finance, American Finance Association, vol. 58(6), pages 2321-2350, December.
    18. Ratner, Mitchell & Chiu, Chih-Chieh (Jason), 2013. "Hedging stock sector risk with credit default swaps," International Review of Financial Analysis, Elsevier, vol. 30(C), pages 18-25.
    19. Dionne, Georges & Laajimi, Sadok, 2012. "On the determinants of the implied default barrier," Journal of Empirical Finance, Elsevier, vol. 19(3), pages 395-408.
    20. Alexander Lipton, 2020. "Physics and Derivatives -- Interview Questions and Answers," Papers 2003.11471, arXiv.org.

    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:arx:papers:1810.02390. 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: arXiv administrators (email available below). General contact details of provider: http://arxiv.org/ .

    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.