IDEAS home Printed from https://ideas.repec.org/a/spr/metcap/v21y2019i4d10.1007_s11009-018-9677-5.html
   My bibliography  Save this article

Joint Distribution of First-Passage Time and First-Passage Area of Certain Lévy Processes

Author

Listed:
  • Mario Abundo

    (Università “Tor Vergata”)

  • Sara Furia

    (Università “Tor Vergata”)

Abstract

Let be X(t) = x − μt + σBt − Nt a Lévy process starting from x > 0, where μ ≥ 0, σ ≥ 0, Bt is a standard BM, and Nt is a homogeneous Poisson process with intensity 𝜃 > 0, starting from zero. We study the joint distribution of the first-passage time below zero, τ(x), and the first-passage area, A(x), swept out by X till the time τ(x). In particular, we establish differential-difference equations with outer conditions for the Laplace transforms of τ(x) and A(x), and for their joint moments. In a special case (μ = σ = 0), we show an algorithm to find recursively the moments E[τ(x)mA(x)n], for any integers m and n; moreover, we obtain the expected value of the time average of X till the time τ(x).

Suggested Citation

  • Mario Abundo & Sara Furia, 2019. "Joint Distribution of First-Passage Time and First-Passage Area of Certain Lévy Processes," Methodology and Computing in Applied Probability, Springer, vol. 21(4), pages 1283-1302, December.
  • Handle: RePEc:spr:metcap:v:21:y:2019:i:4:d:10.1007_s11009-018-9677-5
    DOI: 10.1007/s11009-018-9677-5
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s11009-018-9677-5
    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/s11009-018-9677-5?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. Bates, David S, 1996. "Jumps and Stochastic Volatility: Exchange Rate Processes Implicit in Deutsche Mark Options," The Review of Financial Studies, Society for Financial Studies, vol. 9(1), pages 69-107.
    2. Maria Teresa Giraudo & Laura Sacerdote & Cristina Zucca, 2001. "A Monte Carlo Method for the Simulation of First Passage Times of Diffusion Processes," Methodology and Computing in Applied Probability, Springer, vol. 3(2), pages 215-231, June.
    3. Mario Abundo & Danilo Del Vescovo, 2017. "On the Joint Distribution of First-passage Time and First-passage Area of Drifted Brownian Motion," Methodology and Computing in Applied Probability, Springer, vol. 19(3), pages 985-996, September.
    4. Philippe Jorion, 1988. "On Jump Processes in the Foreign Exchange and Stock Markets," The Review of Financial Studies, Society for Financial Studies, vol. 1(4), pages 427-445.
    5. Frank B. Knight, 2000. "The moments of the area under reflected Brownian bridge conditional on its local time at zero," International Journal of Stochastic Analysis, Hindawi, vol. 13, pages 1-26, January.
    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. Chernov, Mikhail & Graveline, Jeremy & Zviadadze, Irina, 2018. "Crash Risk in Currency Returns," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 53(1), pages 137-170, February.
    2. Lim, Terence & Lo, Andrew W. & Merton, Robert C. & Scholes, Myron S., 2006. "The Derivatives Sourcebook," Foundations and Trends(R) in Finance, now publishers, vol. 1(5–6), pages 365-572, April.
    3. Calvet, Laurent E. & Fisher, Adlai J., 2008. "Multifrequency jump-diffusions: An equilibrium approach," Journal of Mathematical Economics, Elsevier, vol. 44(2), pages 207-226, January.
    4. Volk-Makarewicz, Warren & Borovkova, Svetlana & Heidergott, Bernd, 2022. "Assessing the impact of jumps in an option pricing model: A gradient estimation approach," European Journal of Operational Research, Elsevier, vol. 298(2), pages 740-751.
    5. Torben G. Andersen & Luca Benzoni & Jesper Lund, 2002. "An Empirical Investigation of Continuous‐Time Equity Return Models," Journal of Finance, American Finance Association, vol. 57(3), pages 1239-1284, June.
    6. Chunsheng Zhou, 1997. "A jump-diffusion approach to modeling credit risk and valuing defaultable securities," Finance and Economics Discussion Series 1997-15, Board of Governors of the Federal Reserve System (U.S.).
    7. Pezzo, Rosanna & Uberti, Mariacristina, 2006. "Approaches to forecasting volatility: Models and their performances for emerging equity markets," Chaos, Solitons & Fractals, Elsevier, vol. 29(3), pages 556-565.
    8. Kim, Tae-Hwan & White, Halbert, 2004. "On more robust estimation of skewness and kurtosis," Finance Research Letters, Elsevier, vol. 1(1), pages 56-73, March.
    9. Nicola Bruti-Liberati & Eckhard Platen, 2005. "On the Strong Approximation of Jump-Diffusion Processes," Research Paper Series 157, Quantitative Finance Research Centre, University of Technology, Sydney.
    10. Chernov, Mikhail & Graveline, Jeremy & Zviadadze, Irina, 2012. "Sources of Risk in Currency Returns," CEPR Discussion Papers 8745, C.E.P.R. Discussion Papers.
    11. Jose Giancarlo Gasha & Mr. Andre O Santos & Mr. Jorge A Chan-Lau & Mr. Carlos I. Medeiros & Mr. Marcos R Souto & Christian Capuano, 2009. "Recent Advances in Credit Risk Modeling," IMF Working Papers 2009/162, International Monetary Fund.
    12. Carl Chiarella & Andrew Ziogas, 2009. "American Call Options Under Jump-Diffusion Processes - A Fourier Transform Approach," Applied Mathematical Finance, Taylor & Francis Journals, vol. 16(1), pages 37-79.
    13. Hu, May & Park, Jason, 2019. "Valuation of collateralized debt obligations: An equilibrium model," Economic Modelling, Elsevier, vol. 82(C), pages 119-135.
    14. Ignacio Mauleon & Javier Perote, 2000. "Testing densities with financial data: an empirical comparison of the Edgeworth-Sargan density to the Student's t," The European Journal of Finance, Taylor & Francis Journals, vol. 6(2), pages 225-239.
    15. Bruti-Liberati Nicola & Nikitopoulos-Sklibosios Christina & Platen Eckhard, 2006. "First Order Strong Approximations of Jump Diffusions," Monte Carlo Methods and Applications, De Gruyter, vol. 12(3), pages 191-209, October.
    16. Khalaf, Lynda & Saphores, Jean-Daniel & Bilodeau, Jean-François, 2000. "Simulation-Based Exact Tests with Unidentified Nuisance Parameters Under the Null Hypothesis: the Case of Jumps Tests in Models with Conditional Heteroskedasticity," Cahiers de recherche 0004, GREEN.
    17. Chan, Kam Fong & Powell, John G. & Treepongkaruna, Sirimon, 2014. "Currency jumps and crises: Do developed and emerging market currencies jump together?," Pacific-Basin Finance Journal, Elsevier, vol. 30(C), pages 132-157.
    18. Sun, Qi & Xu, Weidong, 2015. "Pricing foreign equity option with stochastic volatility," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 437(C), pages 89-100.
    19. Jean-Thomas Bernard & Lynda Khalaf & Maral Kichian & Sebastien Mcmahon, 2008. "Forecasting commodity prices: GARCH, jumps, and mean reversion," Journal of Forecasting, John Wiley & Sons, Ltd., vol. 27(4), pages 279-291.
    20. Torben G. Andersen & Tim Bollerslev & Francis X. Diebold & Paul Labys, 1999. "The Distribution of Exchange Rate Volatility," New York University, Leonard N. Stern School Finance Department Working Paper Seires 99-059, New York University, Leonard N. Stern School of Business-.

    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:21:y:2019:i:4:d:10.1007_s11009-018-9677-5. 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.