IDEAS home Printed from https://ideas.repec.org/a/bpj/mcmeap/v20y2014i2p77-100n1.html
   My bibliography  Save this article

Rare event simulation for diffusion processes via two-stage importance sampling

Author

Listed:
  • Metzler Adam

    (Wilfrid Laurier University, 75 University Avenue West, Waterloo, Ontario, Canada)

  • Scott Alexandre

    (Western University, 1151 Richmond Street North, London, Ontario, Canada)

Abstract

We consider the problem of estimating expected values of functionals of real-valued diffusions over regions in path space that have very small probability. We propose a two-stage importance sampling procedure that first converts the problem into one involving standard Brownian motion and then addresses the rare event problem in this simpler setting. In order to identify an effective yet practical importance measure we propose using a time-dependent deterministic drift that minimizes the relative entropy between the corresponding importance measure and the conditional law of the standard Brownian motion, given that its trajectory lies in the region of interest. We provide numerical evidence that (i) our entropy-based criteria performs favourably with an alternative, but less general and less practical, criteria based on large deviations and (ii) our two-stage procedure performs admirably in cases where the region of interest is so rare that crude estimators fail completely.

Suggested Citation

  • Metzler Adam & Scott Alexandre, 2014. "Rare event simulation for diffusion processes via two-stage importance sampling," Monte Carlo Methods and Applications, De Gruyter, vol. 20(2), pages 77-100, June.
    • Handle: RePEc:bpj:mcmeap:v:20:y:2014:i:2:p:77-100:n:1
    DOI: 10.1515/mcma-2013-0019
    as

    Download full text from publisher

    File URL: https://doi.org/10.1515/mcma-2013-0019
    Download Restriction: For access to full text, subscription to the journal or payment for the individual article is required.
    File URL: https://libkey.io/10.1515/mcma-2013-0019?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. Kay Giesecke & Dmitry Smelov, 2013. "Exact Sampling of Jump Diffusions," Operations Research, INFORMS, vol. 61(4), pages 894-907, August.
    2. DiCesare, Joe & Mcleish, Don, 2008. "Simulation of jump diffusions and the pricing of options," Insurance: Mathematics and Economics, Elsevier, vol. 43(3), pages 316-326, December.
    3. Kurbanmuradov O. & Rannik U. & Sabelfeld K. & Vesala T., 1999. "Direct and Adjoint Monte Carlo Algorithms for the Footprint Problem," Monte Carlo Methods and Applications, De Gruyter, vol. 5(2), pages 85-112, December.
    4. Paolo Guasoni & Scott Robertson, 2008. "Optimal importance sampling with explicit formulas in continuous time," Finance and Stochastics, Springer, vol. 12(1), pages 1-19, 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. Giesecke, K. & Schwenkler, G., 2019. "Simulated likelihood estimators for discretely observed jump–diffusions," Journal of Econometrics, Elsevier, vol. 213(2), pages 297-320.
    2. Li, Chenxu & Chen, Dachuan, 2016. "Estimating jump–diffusions using closed-form likelihood expansions," Journal of Econometrics, Elsevier, vol. 195(1), pages 51-70.
    3. Aleksandar Arandjelović & Thorsten Rheinländer & Pavel V. Shevchenko, 2025. "Importance sampling for option pricing with feedforward neural networks," Finance and Stochastics, Springer, vol. 29(1), pages 97-141, January.
    4. Genin, Adrien & Tankov, Peter, 2020. "Optimal importance sampling for Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 130(1), pages 20-46.
    5. Madalina Deaconu & Samuel Herrmann, 2023. "Strong Approximation of Bessel Processes," Methodology and Computing in Applied Probability, Springer, vol. 25(1), pages 1-24, March.
    6. 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.
    7. Dan Pirjol & Lingjiong Zhu, 2017. "Asymptotics for the Discrete-Time Average of the Geometric Brownian Motion and Asian Options," Papers 1706.09659, arXiv.org.
    8. 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.
    9. Erdinc Akyildirim & Ahmet Goncu & Alper Hekimoglu & Duc Khuong Nguyen & Ahmet Sensoy, 2023. "Statistical arbitrage: factor investing approach," OR Spectrum: Quantitative Approaches in Management, Springer;Gesellschaft für Operations Research e.V., vol. 45(4), pages 1295-1331, December.
    10. Adrien Genin & Peter Tankov, 2016. "Optimal importance sampling for L\'evy Processes," Papers 1608.04621, arXiv.org.
    11. Hermann, Simone & Ickstadt, Katja & Müller, Christine H., 2018. "Bayesian prediction for a jump diffusion process – With application to crack growth in fatigue experiments," Reliability Engineering and System Safety, Elsevier, vol. 179(C), pages 83-96.
    12. Ning Cai & Yingda Song & Nan Chen, 2017. "Exact Simulation of the SABR Model," Operations Research, INFORMS, vol. 65(4), pages 931-951, August.
    13. Dan Pirjol & Lingjiong Zhu, 2016. "Short Maturity Asian Options in Local Volatility Models," Papers 1609.07559, arXiv.org.
    14. Aur'elien Alfonsi & David Krief & Peter Tankov, 2018. "Long-time large deviations for the multi-asset Wishart stochastic volatility model and option pricing," Papers 1806.06883, arXiv.org.
    15. Sabelfeld K. & Shalimova I., 2001. "Forward and Backward Stochastic Lagrangian Models for turbulent transport and the well-mixed condition," Monte Carlo Methods and Applications, De Gruyter, vol. 7(3-4), pages 369-382, December.
    16. Kahalé, Nabil, 2020. "General multilevel Monte Carlo methods for pricing discretely monitored Asian options," European Journal of Operational Research, Elsevier, vol. 287(2), pages 739-748.
    17. Wanmo Kang & Jong Mun Lee, 2019. "Unbiased Sensitivity Estimation of One-Dimensional Diffusion Processes," Mathematics of Operations Research, INFORMS, vol. 44(1), pages 334-353, February.
    18. Herrmann, Samuel & Massin, Nicolas, 2023. "Exact simulation of the first passage time through a given level of jump diffusions," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 203(C), pages 553-576.
    19. Zorana Grbac & David Krief & Peter Tankov, 2021. "Long-Time Trajectorial Large Deviations and Importance Sampling for Affine Stochastic Volatility Models," Post-Print hal-03899237, HAL.
    20. Fernández Lexuri & Hieber Peter & Scherer Matthias, 2013. "Double-barrier first-passage times of jump-diffusion processes," Monte Carlo Methods and Applications, De Gruyter, vol. 19(2), pages 107-141, July.

    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:bpj:mcmeap:v:20:y:2014:i:2:p:77-100:n:1. 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: Peter Golla (email available below). General contact details of provider: https://www.degruyter.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.