IDEAS home Printed from https://ideas.repec.org/a/spr/metcap/v27y2025i3d10.1007_s11009-025-10193-3.html
   My bibliography  Save this article

A Large Class of Bilateral Distributions for Financial Applications

Author

Listed:
  • Khouzeima Moutanabbir

    (University of Cape Town)

  • Houenansi Placide Ezin

    (University of Johannesburg)

Abstract

This paper introduces the new class of Bilateral Mixed Erlang (BME) distributions. Capitalizing on the different properties of the Mixed Erlang distribution, we show that the BME class has some helpful closeness properties, among other interesting characteristics. It also exhibits a high degree of tractability since many quantities of interest, such as the probability density function, the characteristic function, the raw moments, and the expected-shortfall risk measure, are given in closed-form expressions. In this paper, it is shown that the class of BME distributions is dense in the class of all distributions on the real line, which allows the use of BME distributions as an approximation. We also give several examples of distributions that belong to the BME class. In addition to presenting the different properties of this new distribution, we illustrate how this distribution could be used in finance through a few important applications. First, we demonstrate how the BME distribution could approximate other bilateral distributions and their related risk measures, VaR and ES. Then, the BME distribution is applied to fit financial returns, and we assess how it outperforms many other well-known distributions. Also, using an FFT algorithm, we show that this distribution can produce option prices that are volatility-smile consistent with a very good fit to the market options data.

Suggested Citation

  • Khouzeima Moutanabbir & Houenansi Placide Ezin, 2025. "A Large Class of Bilateral Distributions for Financial Applications," Methodology and Computing in Applied Probability, Springer, vol. 27(3), pages 1-29, September.
    • Handle: RePEc:spr:metcap:v:27:y:2025:i:3:d:10.1007_s11009-025-10193-3
    DOI: 10.1007/s11009-025-10193-3
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s11009-025-10193-3
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s11009-025-10193-3?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

    for a different version of it.

    References listed on IDEAS

    as
    1. Dilip B. Madan & King Wang, 2020. "Additive Processes with Bilateral Gamma Marginals," Applied Mathematical Finance, Taylor & Francis Journals, vol. 27(3), pages 171-188, May.
    2. Gordon Willmot & Jae-Kyung Woo, 2007. "On the Class of Erlang Mixtures with Risk Theoretic Applications," North American Actuarial Journal, Taylor & Francis Journals, vol. 11(2), pages 99-115.
    3. Madan, Dilip B & Seneta, Eugene, 1990. "The Variance Gamma (V.G.) Model for Share Market Returns," The Journal of Business, University of Chicago Press, vol. 63(4), pages 511-524, October.
    4. Gordon E. Willmot & X. Sheldon Lin, 2011. "Risk modelling with the mixed Erlang distribution," Applied Stochastic Models in Business and Industry, John Wiley & Sons, vol. 27(1), pages 2-16, January.
    5. Merton, Robert C., 1976. "Option pricing when underlying stock returns are discontinuous," Journal of Financial Economics, Elsevier, vol. 3(1-2), pages 125-144.
    6. Fabio Bellini & Lorenzo Mercuri, 2014. "Option pricing in a conditional Bilateral Gamma model," Central European Journal of Operations Research, Springer;Slovak Society for Operations Research;Hungarian Operational Research Society;Czech Society for Operations Research;Österr. Gesellschaft für Operations Research (ÖGOR);Slovenian Society Informatika - Section for Operational Research;Croatian Operational Research Society, vol. 22(2), pages 373-390, June.
    7. S. G. Kou, 2002. "A Jump-Diffusion Model for Option Pricing," Management Science, INFORMS, vol. 48(8), pages 1086-1101, August.
    8. Küchler, Uwe & Tappe, Stefan, 2008. "Bilateral gamma distributions and processes in financial mathematics," Stochastic Processes and their Applications, Elsevier, vol. 118(2), pages 261-283, February.
    9. J. G. Shanthikumar, 1985. "Bilateral phase‐type distributions," Naval Research Logistics Quarterly, John Wiley & Sons, vol. 32(1), pages 119-136, February.
    10. Xie, Haibin & Wang, Shouyang & Lu, Zudi, 2018. "The behavioral implications of the bilateral gamma process," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 500(C), pages 259-264.
    11. Simon Lee & X. Lin, 2010. "Modeling and Evaluating Insurance Losses Via Mixtures of Erlang Distributions," North American Actuarial Journal, Taylor & Francis Journals, vol. 14(1), pages 107-130.
    12. Cossette, Hélène & Côté, Marie-Pier & Marceau, Etienne & Moutanabbir, Khouzeima, 2013. "Multivariate distribution defined with Farlie–Gumbel–Morgenstern copula and mixed Erlang marginals: Aggregation and capital allocation," Insurance: Mathematics and Economics, Elsevier, vol. 52(3), pages 560-572.
    13. Yoshihiro Shirai, 2023. "Acceptable Bilateral Gamma Parameters," Papers 2301.05333, arXiv.org.
    14. Alexander J. McNeil & Rüdiger Frey & Paul Embrechts, 2015. "Quantitative Risk Management: Concepts, Techniques and Tools Revised edition," Economics Books, Princeton University Press, edition 2, number 10496.
    15. Küchler, Uwe & Tappe, Stefan, 2008. "On the shapes of bilateral Gamma densities," Statistics & Probability Letters, Elsevier, vol. 78(15), pages 2478-2484, October.
    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. Xie, Haibin & Wang, Shouyang & Lu, Zudi, 2018. "The behavioral implications of the bilateral gamma process," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 500(C), pages 259-264.
    2. Cossette, Hélène & Marceau, Etienne & Mtalai, Itre, 2019. "Collective risk models with dependence," Insurance: Mathematics and Economics, Elsevier, vol. 87(C), pages 153-168.
    3. Nacira Agram & Bernt Øksendal & Jan Rems, 2024. "Deep learning for quadratic hedging in incomplete jump market," Digital Finance, Springer, vol. 6(3), pages 463-499, September.
    4. Rong Du & Duy-Minh Dang, 2023. "Fourier Neural Network Approximation of Transition Densities in Finance," Papers 2309.03966, arXiv.org, revised Sep 2024.
    5. Zbigniew Palmowski & Jos'e Luis P'erez & Kazutoshi Yamazaki, 2020. "Double continuation regions for American options under Poisson exercise opportunities," Papers 2004.03330, arXiv.org.
    6. Michael C. Fu & Bingqing Li & Guozhen Li & Rongwen Wu, 2017. "Option Pricing for a Jump-Diffusion Model with General Discrete Jump-Size Distributions," Management Science, INFORMS, vol. 63(11), pages 3961-3977, November.
    7. Buckley, Winston & Long, Hongwei & Marshall, Mario, 2016. "Numerical approximations of optimal portfolios in mispriced asymmetric Lévy markets," European Journal of Operational Research, Elsevier, vol. 252(2), pages 676-686.
    8. Markus Leippold & Nikola Vasiljević, 2020. "Option-Implied Intrahorizon Value at Risk," Management Science, INFORMS, vol. 66(1), pages 397-414, January.
    9. Luis Rincón & David J. Santana, 2022. "Ruin Probability for Finite Erlang Mixture Claims Via Recurrence Sequences," Methodology and Computing in Applied Probability, Springer, vol. 24(3), pages 2213-2236, September.
    10. Ascione, Giacomo & Mehrdoust, Farshid & Orlando, Giuseppe & Samimi, Oldouz, 2023. "Foreign Exchange Options on Heston-CIR Model Under Lévy Process Framework," Applied Mathematics and Computation, Elsevier, vol. 446(C).
    11. Reynkens, Tom & Verbelen, Roel & Beirlant, Jan & Antonio, Katrien, 2017. "Modelling censored losses using splicing: A global fit strategy with mixed Erlang and extreme value distributions," Insurance: Mathematics and Economics, Elsevier, vol. 77(C), pages 65-77.
    12. Lee, Hangsuck & Kye, Yisub & Kong, Byungdoo & Song, Seongjoo, 2025. "Multi-step double barrier options under time-varying interest rates," The North American Journal of Economics and Finance, Elsevier, vol. 76(C).
    13. Grechuk, Bogdan & Zabarankin, Michael, 2018. "Direct data-based decision making under uncertainty," European Journal of Operational Research, Elsevier, vol. 267(1), pages 200-211.
    14. Phelan, Carolyn E. & Marazzina, Daniele & Fusai, Gianluca & Germano, Guido, 2018. "Fluctuation identities with continuous monitoring and their application to the pricing of barrier options," European Journal of Operational Research, Elsevier, vol. 271(1), pages 210-223.
    15. Victor Korolev & Alexander Zeifman, 2023. "Quasi-Exponentiated Normal Distributions: Mixture Representations and Asymmetrization," Mathematics, MDPI, vol. 11(17), pages 1-14, September.
    16. Carr, Peter & Wu, Liuren, 2007. "Stochastic skew in currency options," Journal of Financial Economics, Elsevier, vol. 86(1), pages 213-247, October.
    17. Liming Feng & Vadim Linetsky, 2008. "Pricing Options in Jump-Diffusion Models: An Extrapolation Approach," Operations Research, INFORMS, vol. 56(2), pages 304-325, April.
    18. Carolyn E. Phelan & Daniele Marazzina & Gianluca Fusai & Guido Germano, 2019. "Hilbert transform, spectral filters and option pricing," Annals of Operations Research, Springer, vol. 282(1), pages 273-298, November.
    19. M. Gardini & P. Sabino & E. Sasso, 2021. "The Variance Gamma++ Process and Applications to Energy Markets," Papers 2106.15452, arXiv.org.
    20. Liming Feng & Vadim Linetsky, 2008. "Pricing Discretely Monitored Barrier Options And Defaultable Bonds In Lévy Process Models: A Fast Hilbert Transform Approach," Mathematical Finance, Wiley Blackwell, vol. 18(3), pages 337-384, July.

    More about this item

    Keywords

    ;
    ;
    ;
    ;
    ;
    ;

    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:spr:metcap:v:27:y:2025:i:3:d:10.1007_s11009-025-10193-3. 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: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.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.