IDEAS home Printed from https://ideas.repec.org/a/spr/metcap/v17y2015i3d10.1007_s11009-013-9380-5.html
   My bibliography  Save this article

Multilevel Simulation of Functionals of Bernoulli Random Variables with Application to Basket Credit Derivatives

Author

Listed:
  • K. Bujok

    (Oxford University)

  • B. M. Hambly

    (Oxford University)

  • C. Reisinger

    (Oxford University)

Abstract

We consider N Bernoulli random variables, which are independent conditional on a common random factor determining their probability distribution. We show that certain expected functionals of the proportion L N of variables in a given state converge at rate 1/N as N → ∞. Based on these results, we propose a multi-level simulation algorithm using a family of sequences with increasing length, to obtain estimators for these expected functionals with a mean-square error of ϵ 2 and computational complexity of order ϵ −2, independent of N. In particular, this optimal complexity order also holds for the infinite-dimensional limit. Numerical examples are presented for tranche spreads of basket credit derivatives.

Suggested Citation

  • K. Bujok & B. M. Hambly & C. Reisinger, 2015. "Multilevel Simulation of Functionals of Bernoulli Random Variables with Application to Basket Credit Derivatives," Methodology and Computing in Applied Probability, Springer, vol. 17(3), pages 579-604, September.
    • Handle: RePEc:spr:metcap:v:17:y:2015:i:3:d:10.1007_s11009-013-9380-5
    DOI: 10.1007/s11009-013-9380-5
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s11009-013-9380-5
    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/s11009-013-9380-5?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. Edward I. Altman & Brooks Brady & Andrea Resti & Andrea Sironi, 2005. "The Link between Default and Recovery Rates: Theory, Empirical Evidence, and Implications," The Journal of Business, University of Chicago Press, vol. 78(6), pages 2203-2228, November.
    2. Denis Belomestny & John Schoenmakers & Fabian Dickmann, 2013. "Multilevel dual approach for pricing American style derivatives," Finance and Stochastics, Springer, vol. 17(4), pages 717-742, October.
    3. Paul Glasserman & Wanmo Kang & Perwez Shahabuddin, 2007. "Large Deviations In Multifactor Portfolio Credit Risk," Mathematical Finance, Wiley Blackwell, vol. 17(3), pages 345-379, July.
    4. Amir Dembo & Jean-Dominique Deuschel & Darrell Duffie, 2004. "Large portfolio losses," Finance and Stochastics, Springer, vol. 8(1), pages 3-16, January.
    5. Michael B. Giles, 2008. "Multilevel Monte Carlo Path Simulation," Operations Research, INFORMS, vol. 56(3), pages 607-617, June.
    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. Bourgey Florian & De Marco Stefano & Gobet Emmanuel & Zhou Alexandre, 2020. "Multilevel Monte Carlo methods and lower–upper bounds in initial margin computations," Monte Carlo Methods and Applications, De Gruyter, vol. 26(2), pages 131-161, June.
    2. Giorgi Daphné & Lemaire Vincent & Pagès Gilles, 2017. "Limit theorems for weighted and regular Multilevel estimators," Monte Carlo Methods and Applications, De Gruyter, vol. 23(1), pages 43-70, March.
    3. Michael B. Giles & Abdul-Lateef Haji-Ali & Jonathan Spence, 2023. "Efficient Risk Estimation for the Credit Valuation Adjustment," Papers 2301.05886, arXiv.org.
    4. Alfonsi, Aurélien & Cherchali, Adel & Infante Acevedo, Jose Arturo, 2021. "Multilevel Monte-Carlo for computing the SCR with the standard formula and other stress tests," Insurance: Mathematics and Economics, Elsevier, vol. 100(C), pages 234-260.
    5. Michael B. Giles & Abdul-Lateef Haji-Ali, 2019. "Sub-sampling and other considerations for efficient risk estimation in large portfolios," Papers 1912.05484, arXiv.org, revised Apr 2022.
    6. Daphné Giorgi & Vincent Lemaire & Gilles Pagès, 2020. "Weak Error for Nested Multilevel Monte Carlo," Methodology and Computing in Applied Probability, Springer, vol. 22(3), pages 1325-1348, September.
    7. Michael B. Giles & Francisco Bernal, 2017. "Multilevel estimation of expected exit times and other functionals of stopped diffusions," Papers 1710.07492, arXiv.org, revised Sep 2018.
    8. Goda, Takashi & Kitade, Wataru, 2023. "Constructing unbiased gradient estimators with finite variance for conditional stochastic optimization," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 204(C), pages 743-763.
    9. Aur'elien Alfonsi & Adel Cherchali & Jose Arturo Infante Acevedo, 2020. "Multilevel Monte-Carlo for computing the SCR with the standard formula and other stress tests," Papers 2010.12651, arXiv.org, revised Apr 2021.
    10. F Bourgey & S de Marco & Emmanuel Gobet & Alexandre Zhou, 2020. "Multilevel Monte-Carlo methods and lower-upper bounds in Initial Margin computations," Post-Print hal-02430430, HAL.
    11. F Bourgey & S de Marco & Emmanuel Gobet & Alexandre Zhou, 2020. "Multilevel Monte-Carlo methods and lower-upper bounds in Initial Margin computations," Working Papers hal-02430430, HAL.

    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. Spiliopoulos, Konstantinos & Sowers, Richard B., 2011. "Recovery rates in investment-grade pools of credit assets: A large deviations analysis," Stochastic Processes and their Applications, Elsevier, vol. 121(12), pages 2861-2898.
    2. Karolina Bujok & Ben Hambly & Christoph Reisinger, 2012. "Multilevel simulation of functionals of Bernoulli random variables with application to basket credit derivatives," Papers 1211.0707, arXiv.org, revised Feb 2018.
    3. Tang, Qihe & Tong, Zhiwei & Yang, Yang, 2021. "Large portfolio losses in a turbulent market," European Journal of Operational Research, Elsevier, vol. 292(2), pages 755-769.
    4. Kay Giesecke & Konstantinos Spiliopoulos & Richard B. Sowers, 2011. "Default clustering in large portfolios: Typical events," Papers 1104.1773, arXiv.org, revised Feb 2013.
    5. Mike Giles & Lukasz Szpruch, 2012. "Multilevel Monte Carlo methods for applications in finance," Papers 1212.1377, arXiv.org.
    6. Dirk Becherer & Plamen Turkedjiev, 2014. "Multilevel approximation of backward stochastic differential equations," Papers 1412.3140, arXiv.org.
    7. Konstantinos Spiliopoulos & Richard B. Sowers, 2013. "Default Clustering in Large Pools: Large Deviations," Papers 1311.0498, arXiv.org, revised Feb 2015.
    8. Devang Sinha & Siddhartha P. Chakrabarty, 2022. "Multilevel Monte Carlo and its Applications in Financial Engineering," Papers 2209.14549, arXiv.org.
    9. Denis Belomestny & Fabian Dickmann & Tigran Nagapetyan, 2013. "Pricing American options via multi-level approximation methods," Papers 1303.1334, arXiv.org, revised Dec 2013.
    10. Tang, Qihe & Tang, Zhaofeng & Yang, Yang, 2019. "Sharp asymptotics for large portfolio losses under extreme risks," European Journal of Operational Research, Elsevier, vol. 276(2), pages 710-722.
    11. Amogh Deshpande, 2014. "Comparing the Value at Risk Performance of the CreditRisk + and its Enhancement: A Large Deviations Approach," Methodology and Computing in Applied Probability, Springer, vol. 16(4), pages 1009-1023, December.
    12. Konstantinos Spiliopoulos, 2014. "Systemic Risk and Default Clustering for Large Financial Systems," Papers 1402.5352, arXiv.org, revised Feb 2015.
    13. Yi Chen & Jing Dong & Hao Ni, 2021. "ɛ-Strong Simulation of Fractional Brownian Motion and Related Stochastic Differential Equations," Mathematics of Operations Research, INFORMS, vol. 46(2), pages 559-594, May.
    14. Patrick Gagliardini & Christian Gouriéroux, 2011. "Approximate Derivative Pricing for Large Classes of Homogeneous Assets with Systematic Risk," Journal of Financial Econometrics, Oxford University Press, vol. 9(2), pages 237-280, Spring.
    15. Parrini, Alessandro, 2013. "Importance Sampling for Portfolio Credit Risk in Factor Copula Models," MPRA Paper 103745, University Library of Munich, Germany.
    16. Andrea Cipollini & Giuseppe Missaglia, 2008. "Measuring bank capital requirements through Dynamic Factor analysis," Center for Economic Research (RECent) 010, University of Modena and Reggio E., Dept. of Economics "Marco Biagi".
    17. Bank for International Settlements, 2005. "The role of ratings in structured finance: issues and implications," CGFS Papers, Bank for International Settlements, number 23, december.
    18. Jian Wang & Xiang Gao & Zhili Sun, 2021. "A Multilevel Simulation Method for Time-Variant Reliability Analysis," Sustainability, MDPI, vol. 13(7), pages 1-16, March.
    19. Ahmed Kebaier & J'er^ome Lelong, 2015. "Coupling Importance Sampling and Multilevel Monte Carlo using Sample Average Approximation," Papers 1510.03590, arXiv.org, revised Jul 2017.
    20. Stéphane Crépey & Noufel Frikha & Azar Louzi & Gilles Pagès, 2023. "Asymptotic Error Analysis of Multilevel Stochastic Approximations for the Value-at-Risk and Expected Shortfall," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) hal-04304985, HAL.

    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:17:y:2015:i:3:d:10.1007_s11009-013-9380-5. 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.