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Tail behavior of sums and differences of log-normal random variables

Author

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  • Archil Gulisashvili
  • Peter Tankov

Abstract

We present sharp tail asymptotics for the density and the distribution function of linear combinations of correlated log-normal random variables, that is, exponentials of components of a correlated Gaussian vector. The asymptotic behavior turns out to depend on the correlation between the components, and the explicit solution is found by solving a tractable quadratic optimization problem. These results can be used either to approximate the probability of tail events directly, or to construct variance reduction procedures to estimate these probabilities by Monte Carlo methods. In particular, we propose an efficient importance sampling estimator for the left tail of the distribution function of the sum of log-normal variables. As a corollary of the tail asymptotics, we compute the asymptotics of the conditional law of a Gaussian random vector given a linear combination of exponentials of its components. In risk management applications, this finding can be used for the systematic construction of stress tests, which the financial institutions are required to conduct by the regulators. We also characterize the asymptotic behavior of the Value at Risk for log-normal portfolios in the case where the confidence level tends to one.

Suggested Citation

  • Archil Gulisashvili & Peter Tankov, 2013. "Tail behavior of sums and differences of log-normal random variables," Papers 1309.3057, arXiv.org, revised Jan 2016.
  • Handle: RePEc:arx:papers:1309.3057
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    References listed on IDEAS

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    1. Embrechts, Paul & Puccetti, Giovanni, 2006. "Bounds for functions of multivariate risks," Journal of Multivariate Analysis, Elsevier, vol. 97(2), pages 526-547, February.
    2. Serguei Foss & Andrew Richards, 2010. "On Sums of Conditionally Independent Subexponential Random Variables," Mathematics of Operations Research, INFORMS, vol. 35(1), pages 102-119, February.
    3. Paul Embrechts & Giovanni Puccetti, 2006. "Aggregating risk capital, with an application to operational risk," The Geneva Papers on Risk and Insurance Theory, Springer;International Association for the Study of Insurance Economics (The Geneva Association), vol. 31(2), pages 71-90, December.
    4. Asmussen, Søren & Rojas-Nandayapa, Leonardo, 2008. "Asymptotics of sums of lognormal random variables with Gaussian copula," Statistics & Probability Letters, Elsevier, vol. 78(16), pages 2709-2714, November.
    5. Søren Asmussen & José Blanchet & Sandeep Juneja & Leonardo Rojas-Nandayapa, 2011. "Efficient simulation of tail probabilities of sums of correlated lognormals," Annals of Operations Research, Springer, vol. 189(1), pages 5-23, September.
    6. Paul Embrechts & Giovanni Puccetti, 2006. "Bounds for Functions of Dependent Risks," Finance and Stochastics, Springer, vol. 10(3), pages 341-352, September.
    7. Dominik Kortschak & Hansjörg Albrecher, 2009. "Asymptotic Results for the Sum of Dependent Non-identically Distributed Random Variables," Methodology and Computing in Applied Probability, Springer, vol. 11(3), pages 279-306, September.
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    Citations

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    Cited by:

    1. Dan Pirjol & Lingjiong Zhu, 2016. "Discrete Sums of Geometric Brownian Motions, Annuities and Asian Options," Papers 1609.07558, arXiv.org.
    2. Ankush Agarwal & Stefano de Marco & Emmanuel Gobet & Gang Liu, 2017. "Rare event simulation related to financial risks: efficient estimation and sensitivity analysis," Working Papers hal-01219616, HAL.
    3. Chakrabarty, Arijit & Samorodnitsky, Gennady, 2018. "Asymptotic behaviour of high Gaussian minima," Stochastic Processes and their Applications, Elsevier, vol. 128(7), pages 2297-2324.
    4. Alouini Mohamed-Slim & Ben Rached Nadhir & Kammoun Abla & Tempone Raul, 2018. "On the efficient simulation of the left-tail of the sum of correlated log-normal variates," Monte Carlo Methods and Applications, De Gruyter, vol. 24(2), pages 101-115, June.
    5. Nakatsu, Tomonori, 2023. "On density functions related to discrete time maximum of some one-dimensional diffusion processes," Applied Mathematics and Computation, Elsevier, vol. 441(C).
    6. Kemal Dinçer Dingeç & Wolfgang Hörmann, 2022. "Efficient Algorithms for Tail Probabilities of Exchangeable Lognormal Sums," Methodology and Computing in Applied Probability, Springer, vol. 24(3), pages 2093-2121, September.
    7. Peter Tankov, 2014. "Tails of weakly dependent random vectors," Papers 1402.4683, arXiv.org, revised Jan 2016.
    8. Archil Gulisashvili & Peter Tankov, 2014. "Implied volatility of basket options at extreme strikes," Papers 1406.0394, arXiv.org.
    9. Pirjol, Dan & Zhu, Lingjiong, 2016. "Discrete sums of geometric Brownian motions, annuities and Asian options," Insurance: Mathematics and Economics, Elsevier, vol. 70(C), pages 19-37.
    10. Zdravko I. Botev & Robert Salomone & Daniel Mackinlay, 2019. "Fast and accurate computation of the distribution of sums of dependent log-normals," Annals of Operations Research, Springer, vol. 280(1), pages 19-46, September.
    11. Furman, Edward & Hackmann, Daniel & Kuznetsov, Alexey, 2020. "On log-normal convolutions: An analytical–numerical method with applications to economic capital determination," Insurance: Mathematics and Economics, Elsevier, vol. 90(C), pages 120-134.
    12. Boyle, Phelim & Jiang, Ruihong, 2023. "A note on portfolios of averages of lognormal variables," Insurance: Mathematics and Economics, Elsevier, vol. 112(C), pages 97-109.
    13. Robert Parham, 2023. "The Difference-of-Log-Normals Distribution: Properties, Estimation, and Growth," Papers 2302.02486, arXiv.org.

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