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Computing Tails of Compound Distributions Using Direct Numerical Integration

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  • Xiaolin Luo
  • Pavel V. Shevchenko
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    Abstract

    An efficient adaptive direct numerical integration (DNI) algorithm is developed for computing high quantiles and conditional Value at Risk (CVaR) of compound distributions using characteristic functions. A key innovation of the numerical scheme is an effective tail integration approximation that reduces the truncation errors significantly with little extra effort. High precision results of the 0.999 quantile and CVaR were obtained for compound losses with heavy tails and a very wide range of loss frequencies using the DNI, Fast Fourier Transform (FFT) and Monte Carlo (MC) methods. These results, particularly relevant to operational risk modelling, can serve as benchmarks for comparing different numerical methods. We found that the adaptive DNI can achieve high accuracy with relatively coarse grids. It is much faster than MC and competitive with FFT in computing high quantiles and CVaR of compound distributions in the case of moderate to high frequencies and heavy tails.

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    File URL: http://arxiv.org/pdf/0904.0830
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    Bibliographic Info

    Paper provided by arXiv.org in its series Papers with number 0904.0830.

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    Date of creation: Apr 2009
    Date of revision: Feb 2010
    Publication status: Published in Journal of Computational Finance. 13(2), 73-111. (2009)
    Handle: RePEc:arx:papers:0904.0830

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    Web page: http://arxiv.org/

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    References

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    1. Mark Craddock & David Heath & Eckhard Platen, 1999. "Numerical Inversion of Laplace Transforms: A Survey of Techniques with Applications to Derivative Pricing," Research Paper Series 27, Quantitative Finance Research Centre, University of Technology, Sydney.
    2. Panjer, Harry H. & Willmot, Gordon E., 1986. "Computational aspects of recursive evaluation of compound distributions," Insurance: Mathematics and Economics, Elsevier, vol. 5(1), pages 113-116, January.
    3. S. Illeris & G. Akehurst, 2002. "Introduction," The Service Industries Journal, Taylor & Francis Journals, vol. 22(1), pages 1-3, January.
    4. Shephard, N.G., 1991. "From Characteristic Function to Distribution Function: A Simple Framework for the Theory," Econometric Theory, Cambridge University Press, vol. 7(04), pages 519-529, December.
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    Cited by:
    1. Gareth W. Peters & Pavel V. Shevchenko & Mario V. W\"uthrich, 2009. "Dynamic operational risk: modeling dependence and combining different sources of information," Papers 0904.4074, arXiv.org, revised Jul 2009.
    2. repec:hal:cesptp:halshs-00443846 is not listed on IDEAS
    3. Pavel V. Shevchenko & Grigory Temnov, 2009. "Modeling operational risk data reported above a time-varying threshold," Papers 0904.4075, arXiv.org, revised Jul 2009.
    4. Pavel V. Shevchenko, 2009. "Implementing Loss Distribution Approach for Operational Risk," Papers 0904.1805, arXiv.org, revised Jul 2009.

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