IDEAS home Printed from https://ideas.repec.org/a/wly/jnljam/v2012y2012i1n838397.html

The Sum and Difference of Two Lognormal Random Variables

Author

Listed:
  • C. F. Lo

Abstract

We have presented a new unified approach to model the dynamics of both the sum and difference of two correlated lognormal stochastic variables. By the Lie‐Trotter operator splitting method, both the sum and difference are shown to follow a shifted lognormal stochastic process, and approximate probability distributions are determined in closed form. Illustrative numerical examples are presented to demonstrate the validity and accuracy of these approximate distributions. In terms of the approximate probability distributions, we have also obtained an analytical series expansion of the exact solutions, which can allow us to improve the approximation in a systematic manner. Moreover, we believe that this new approach can be extended to study both (1) the algebraic sum of N lognormals, and (2) the sum and difference of other correlated stochastic processes, for example, two correlated CEV processes, two correlated CIR processes, and two correlated lognormal processes with mean‐reversion.

Suggested Citation

  • C. F. Lo, 2012. "The Sum and Difference of Two Lognormal Random Variables," Journal of Applied Mathematics, John Wiley & Sons, vol. 2012(1).
    • Handle: RePEc:wly:jnljam:v:2012:y:2012:i:1:n:838397
    DOI: 10.1155/2012/838397
    as

    Download full text from publisher

    File URL: https://doi.org/10.1155/2012/838397
    Download Restriction: no
    File URL: https://libkey.io/10.1155/2012/838397?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
    ---><---

    References listed on IDEAS

    as
    1. Xin Gao & Hong Xu & Dong Ye, 2009. "Asymptotic Behavior of Tail Density for Sum of Correlated Lognormal Variables," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 2009, pages 1-28, August.
    2. Moshe Arye Milevsky & Steven E. Posner, 1999. "Asian Options, The Sum Of Lognormals, And The Reciprocal Gamma Distribution," World Scientific Book Chapters, in: Marco Avellaneda (ed.), Quantitative Analysis In Financial Markets Collected Papers of the New York University Mathematical Finance Seminar, chapter 7, pages 203-218, World Scientific Publishing Co. Pte. Ltd..
    3. Dhaene, J. & Denuit, M. & Goovaerts, M. J. & Kaas, R. & Vyncke, D., 2002. "The concept of comonotonicity in actuarial science and finance: applications," Insurance: Mathematics and Economics, Elsevier, vol. 31(2), pages 133-161, October.
    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. Dagmara Dudek & Anna Kuczmaszewska, 2025. "A note on an approximation and estimation of distribution function of difference of random variables," Statistical Papers, Springer, vol. 66(7), pages 1-31, December.
    2. Yong-Ki Ma, 2021. "Correlated Log‐Normal Random Variables under a Multiscale Volatility Model," Advances in Mathematical Physics, John Wiley & Sons, vol. 2021(1).

    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. Van Weert, Koen & Dhaene, Jan & Goovaerts, Marc, 2010. "Optimal portfolio selection for general provisioning and terminal wealth problems," Insurance: Mathematics and Economics, Elsevier, vol. 47(1), pages 90-97, August.
    2. Xu, Guoping & Zheng, Harry, 2009. "Approximate basket options valuation for a jump-diffusion model," Insurance: Mathematics and Economics, Elsevier, vol. 45(2), pages 188-194, October.
    3. Jianqiang Sun & Langnan Chen & Shiyin Li, 2013. "A Quasi‐Analytical Pricing Model for Arithmetic Asian Options," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 33(12), pages 1143-1166, December.
    4. Kijima, Masaaki & Wong, Tony, 2007. "Pricing of Ratchet equity-indexed annuities under stochastic interest rates," Insurance: Mathematics and Economics, Elsevier, vol. 41(3), pages 317-338, November.
    5. Yong-Ki Ma, 2017. "Stochastic Volatility Effects on Correlated Log‐Normal Random Variables," Advances in Mathematical Physics, John Wiley & Sons, vol. 2017(1).
    6. Geon Ho Choe & Minseok Kim, 2021. "Closed‐form lower bounds for the price of arithmetic average Asian options by multiple conditioning," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 41(12), pages 1916-1932, December.
    7. Boyle, Phelim & Potapchik, Alexander, 2008. "Prices and sensitivities of Asian options: A survey," Insurance: Mathematics and Economics, Elsevier, vol. 42(1), pages 189-211, February.
    8. Dimitrios G. Konstantinides & Georgios C. Zachos, 2019. "Exhibiting Abnormal Returns Under a Risk Averse Strategy," Methodology and Computing in Applied Probability, Springer, vol. 21(2), pages 551-566, June.
    9. Zheng, Yanting & Yang, Jingping & Huang, Jianhua Z., 2011. "Approximation of bivariate copulas by patched bivariate Fréchet copulas," Insurance: Mathematics and Economics, Elsevier, vol. 48(2), pages 246-256, March.
    10. Said Khalil, 2022. "Expectile-based capital allocation," Working Papers hal-03816525, HAL.
    11. Denuit, Michel & Hieber, Peter & Robert, Christian Y., 2021. "Mortality credits within large survivor funds," LIDAM Discussion Papers ISBA 2021038, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    12. Nam, Hee Seok & Tang, Qihe & Yang, Fan, 2011. "Characterization of upper comonotonicity via tail convex order," Insurance: Mathematics and Economics, Elsevier, vol. 48(3), pages 368-373, May.
    13. Denuit, Michel & Dhaene, Jan & Ghossoub, Mario & Robert, Christian Y., 2025. "Comonotonicity and Pareto optimality, with application to collaborative insurance," Insurance: Mathematics and Economics, Elsevier, vol. 120(C), pages 1-16.
    14. Kaas, Rob & Tang, Qihe, 2005. "A large deviation result for aggregate claims with dependent claim occurrences," Insurance: Mathematics and Economics, Elsevier, vol. 36(3), pages 251-259, June.
    15. Keyi Zeng & Zhenfeng Zou & Yuting Su & Taizhong Hu, 2025. "Further Developments on Stochastic Dominance for Convex Combinations of Infinite-Mean Random Variables," Papers 2511.00764, arXiv.org, revised Apr 2026.
    16. Wenjun Jiang & Jiandong Ren & Ričardas Zitikis, 2017. "Optimal Reinsurance Policies under the VaR Risk Measure When the Interests of Both the Cedent and the Reinsurer Are Taken into Account," Risks, MDPI, vol. 5(1), pages 1-22, February.
    17. He, Junnan & Tang, Qihe & Zhang, Huan, 2016. "Risk reducers in convex order," Insurance: Mathematics and Economics, Elsevier, vol. 70(C), pages 80-88.
    18. Asimit, Alexandru V. & Badescu, Alexandru M. & Cheung, Ka Chun, 2013. "Optimal reinsurance in the presence of counterparty default risk," Insurance: Mathematics and Economics, Elsevier, vol. 53(3), pages 690-697.
    19. Xiaoou Li & Jingchen Liu & Gongjun Xu, 2016. "On the Tail Probabilities of Aggregated Lognormal Random Fields with Small Noise," Mathematics of Operations Research, INFORMS, vol. 41(1), pages 236-246, February.
    20. Raj Kumari Bahl & Sotirios Sabanis, 2017. "General Price Bounds for Guaranteed Annuity Options," Papers 1707.00807, arXiv.org.

    More about this item

    Statistics

    Access and download statistics

    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:wly:jnljam:v:2012:y:2012:i:1:n:838397. 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: Wiley Content Delivery (email available below). General contact details of provider: https://onlinelibrary.wiley.com/journal/4185 .

    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.