IDEAS home Printed from
   My bibliography  Save this article

Analyzing model robustness via a distortion of the stochastic root: A Dirichlet prior approach


  • Mai Jan-Frederik

    () (XAIA Investment GmbH, Sonnenstr. 19, 80331 München, Germany)

  • Schenk Steffen


  • Scherer Matthias

    () (Lehrstuhl für Finanzmathematik, Technische Universität München, Parkring 11, 85748 Garching-Hochbrück, Germany)


It is standard in quantitative risk management to model a random vector 𝐗:={Xtk}k=1,...,d${\mathbf {X}:=\lbrace X_{t_k}\rbrace _{k=1,\ldots ,d}}$ of consecutive log-returns to ultimately analyze the probability law of the accumulated return Xt1+⋯+Xtd${X_{t_1}+\cdots +X_{t_d}}$. By the Markov regression representation (see [25]), any stochastic model for 𝐗${\mathbf {X}}$ can be represented as Xtk=fk(Xt1,...,Xtk-1,Uk)${X_{t_k}=f_k(X_{t_1},\ldots ,X_{t_{k-1}},U_k)}$, k=1,...,d${k=1,\ldots ,d}$, yielding a decomposition into a vector 𝐔:={Uk}k=1,...,d${\mathbf {U}:=\lbrace U_{k}\rbrace _{k=1,\ldots ,d}}$ of i.i.d. random variables accounting for the randomness in the model, and a function f:={fk}k=1,...,d${f:=\lbrace f_k\rbrace _{k=1,\ldots ,d}}$ representing the economic reasoning behind. For most models, f is known explicitly and Uk may be interpreted as an exogenous risk factor affecting the return Xtk in time step k. While existing literature addresses model uncertainty by manipulating the function f, we introduce a new philosophy by distorting the source of randomness 𝐔${\mathbf {U}}$ and interpret this as an analysis of the model's robustness. We impose consistency conditions for a reasonable distortion and present a suitable probability law and a stochastic representation for 𝐔${\mathbf {U}}$ based on a Dirichlet prior. The resulting framework has one parameter c∈[0,∞]${c\in [0,\infty ]}$ tuning the severity of the imposed distortion. The universal nature of the methodology is illustrated by means of a case study comparing the effect of the distortion to different models for 𝐗${\mathbf {X}}$. As a mathematical byproduct, the consistency conditions of the suggested distortion function reveal interesting insights into the dependence structure between samples from a Dirichlet prior.

Suggested Citation

  • Mai Jan-Frederik & Schenk Steffen & Scherer Matthias, 2015. "Analyzing model robustness via a distortion of the stochastic root: A Dirichlet prior approach," Statistics & Risk Modeling, De Gruyter, vol. 32(3-4), pages 177-195, December.
  • Handle: RePEc:bpj:strimo:v:32:y:2015:i:3-4:p:177-195:n:2

    Download full text from publisher

    File URL:
    Download Restriction: For access to full text, subscription to the journal or payment for the individual article is required.

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    1. Barrieu, Pauline & Scandolo, Giacomo, 2015. "Assessing financial model risk," European Journal of Operational Research, Elsevier, vol. 242(2), pages 546-556.
    2. Rama Cont & Romain Deguest & Giacomo Scandolo, 2010. "Robustness and sensitivity analysis of risk measurement procedures," Post-Print hal-00413729, HAL.
    3. Pérignon, Christophe & Smith, Daniel R., 2010. "The level and quality of Value-at-Risk disclosure by commercial banks," Journal of Banking & Finance, Elsevier, vol. 34(2), pages 362-377, February.
    4. Rama Cont, 2006. "Model Uncertainty And Its Impact On The Pricing Of Derivative Instruments," Mathematical Finance, Wiley Blackwell, vol. 16(3), pages 519-547, July.
    5. Bollerslev, Tim, 1986. "Generalized autoregressive conditional heteroskedasticity," Journal of Econometrics, Elsevier, vol. 31(3), pages 307-327, April.
    6. Rama Cont & Romain Deguest & Giacomo Scandolo, 2010. "Robustness and sensitivity analysis of risk measurement procedures," Quantitative Finance, Taylor & Francis Journals, vol. 10(6), pages 593-606.
    7. Rama Cont, 2006. "Model uncertainty and its impact on the pricing of derivative instruments," Post-Print halshs-00002695, HAL.
    8. Philippe Artzner & Freddy Delbaen & Jean‐Marc Eber & David Heath, 1999. "Coherent Measures of Risk," Mathematical Finance, Wiley Blackwell, vol. 9(3), pages 203-228, July.
    9. Rüschendorf, Ludger & de Valk, Vincent, 1993. "On regression representations of stochastic processes," Stochastic Processes and their Applications, Elsevier, vol. 46(2), pages 183-198, June.
    10. M. Avellaneda & A. Levy & A. ParAS, 1995. "Pricing and hedging derivative securities in markets with uncertain volatilities," Applied Mathematical Finance, Taylor & Francis Journals, vol. 2(2), pages 73-88.
    11. Embrechts, Paul & Puccetti, Giovanni & Rüschendorf, Ludger, 2013. "Model uncertainty and VaR aggregation," Journal of Banking & Finance, Elsevier, vol. 37(8), pages 2750-2764.
    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. Farkas, Walter & Fringuellotti, Fulvia & Tunaru, Radu, 2020. "A cost-benefit analysis of capital requirements adjusted for model risk," Journal of Corporate Finance, Elsevier, vol. 65(C).
    2. Marcelo Brutti Righi & Fernanda Maria Muller & Marlon Ruoso Moresco, 2017. "On a robust risk measurement approach for capital determination errors minimization," Papers 1707.09829,, revised Oct 2020.
    3. Marcelo Brutti Righi, 2018. "A theory for combinations of risk measures," Papers 1807.01977,, revised Aug 2020.
    4. Claußen, Arndt & Rösch, Daniel & Schmelzle, Martin, 2019. "Hedging parameter risk," Journal of Banking & Finance, Elsevier, vol. 100(C), pages 111-121.
    5. Barrieu, Pauline & Scandolo, Giacomo, 2014. "Assessing financial model risk," LSE Research Online Documents on Economics 60084, London School of Economics and Political Science, LSE Library.
    6. Radu Tunaru, 2015. "Model Risk in Financial Markets:From Financial Engineering to Risk Management," World Scientific Books, World Scientific Publishing Co. Pte. Ltd., number 9524, February.
    7. Asimit, Alexandru V. & Bignozzi, Valeria & Cheung, Ka Chun & Hu, Junlei & Kim, Eun-Seok, 2017. "Robust and Pareto optimality of insurance contracts," European Journal of Operational Research, Elsevier, vol. 262(2), pages 720-732.
    8. Barrieu, Pauline & Scandolo, Giacomo, 2015. "Assessing financial model risk," European Journal of Operational Research, Elsevier, vol. 242(2), pages 546-556.
    9. Lazar, Emese & Zhang, Ning, 2019. "Model risk of expected shortfall," Journal of Banking & Finance, Elsevier, vol. 105(C), pages 74-93.
    10. repec:hal:journl:hal-00921283 is not listed on IDEAS
    11. Alexander, Gordon J. & Baptista, Alexandre M. & Yan, Shu, 2012. "When more is less: Using multiple constraints to reduce tail risk," Journal of Banking & Finance, Elsevier, vol. 36(10), pages 2693-2716.
    12. Mark H. A. Davis, 2014. "Verification of internal risk measure estimates," Papers 1410.4382,, revised Nov 2015.
    13. Carol Alexander & Jose Maria Sarabia, 2010. "Endogenizing Model Risk to Quantile Estimates," ICMA Centre Discussion Papers in Finance icma-dp2010-07, Henley Business School, Reading University.
    14. Shige Peng & Shuzhen Yang, 2020. "Autoregressive models of the time series under volatility uncertainty and application to VaR model," Papers 2011.09226,
    15. Bernard, Carole & Vanduffel, Steven, 2015. "A new approach to assessing model risk in high dimensions," Journal of Banking & Finance, Elsevier, vol. 58(C), pages 166-178.
    16. Liu, Peng & Wang, Ruodu & Wei, Linxiao, 2020. "Is the inf-convolution of law-invariant preferences law-invariant?," Insurance: Mathematics and Economics, Elsevier, vol. 91(C), pages 144-154.
    17. Susanne Emmer & Marie Kratz & Dirk Tasche, 2013. "What is the best risk measure in practice? A comparison of standard measures," Papers 1312.1645,, revised Apr 2015.
    18. Marcelo Brutti Righi & Paulo Sergio Ceretta, 2015. "Shortfall Deviation Risk: An alternative to risk measurement," Papers 1501.02007,, revised May 2016.
    19. Bannör, Karl & Kiesel, Rüdiger & Nazarova, Anna & Scherer, Matthias, 2016. "Parametric model risk and power plant valuation," Energy Economics, Elsevier, vol. 59(C), pages 423-434.
    20. Bernard, Carole & Kazzi, Rodrigue & Vanduffel, Steven, 2020. "Range Value-at-Risk bounds for unimodal distributions under partial information," Insurance: Mathematics and Economics, Elsevier, vol. 94(C), pages 9-24.
    21. Coqueret, Guillaume & Tavin, Bertrand, 2016. "An investigation of model risk in a market with jumps and stochastic volatility," European Journal of Operational Research, Elsevier, vol. 253(3), pages 648-658.

    More about this item


    Access and download statistics


    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:strimo:v:32:y:2015:i:3-4:p:177-195:n:2. See general information about how to correct material in RePEc.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (Peter Golla). General contact details of provider: .

    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 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.

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service hosted by the Research Division of the Federal Reserve Bank of St. Louis . RePEc uses bibliographic data supplied by the respective publishers.