IDEAS home Printed from https://ideas.repec.org/p/crs/wpaper/2017-75.html
   My bibliography  Save this paper

Pricing formulae for derivatives in insurance using the Malliavin calculus

Author

Listed:
  • Caroline Hillairet

    (ENSAE; Université Paris Saclay)

  • Ying Jiao

    (Université Claude Bernard - Lyon 1; Institut de Science Financière et d’Assurances)

Abstract

In this paper we provide a valuation formula for different classes of actuarial and financial contracts which depend on a general loss process, by using the Malliavin calculus. In analogy with the celebrated Black-Scholes formula, we aim at expressing the expected cash flow in terms of a building block. The former is related to the loss process which is a cumulated sum indexed by a doubly stochastic Poisson process of claims allowed to be dependent on the intensity and the jump times of the counting process. For example, in the context of Stop-Loss contracts the building block is given by the distribution function of the terminal cumulated loss, taken at the Value at Risk when computing the Expected Shortfall risk measure.

Suggested Citation

  • Caroline Hillairet & Ying Jiao, 2017. "Pricing formulae for derivatives in insurance using the Malliavin calculus," Working Papers 2017-75, Center for Research in Economics and Statistics.
  • Handle: RePEc:crs:wpaper:2017-75
    as

    Download full text from publisher

    File URL: http://crest.science/RePEc/wpstorage/2017-75.pdf
    File Function: CREST working paper version
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Albrecher, Hansjörg & Constantinescu, Corina & Loisel, Stephane, 2011. "Explicit ruin formulas for models with dependence among risks," Insurance: Mathematics and Economics, Elsevier, vol. 48(2), pages 265-270, March.
    2. Denuit, Michel & Dhaene, Jan & Ribas, Carmen, 2001. "Does positive dependence between individual risks increase stop-loss premiums?," Insurance: Mathematics and Economics, Elsevier, vol. 28(3), pages 305-308, June.
    3. Gerber, Hans U., 1982. "On the numerical evaluation of the distribution of aggregate claims and its stop-loss premiums," Insurance: Mathematics and Economics, Elsevier, vol. 1(1), pages 13-18, January.
    4. Albrecher, Hansjorg & Boxma, Onno J., 2004. "A ruin model with dependence between claim sizes and claim intervals," Insurance: Mathematics and Economics, Elsevier, vol. 35(2), pages 245-254, October.
    5. Albers, Willem, 1999. "Stop-loss premiums under dependence," Insurance: Mathematics and Economics, Elsevier, vol. 24(3), pages 173-185, May.
    6. Stéphane Loisel, 2011. "Explicit ruin formulas for dependent risks," Post-Print hal-00600093, HAL.
    7. de Lourdes Centeno, Maria, 2005. "Dependent risks and excess of loss reinsurance," Insurance: Mathematics and Economics, Elsevier, vol. 37(2), pages 229-238, October.
    8. Panjer, Harry H., 1981. "Recursive Evaluation of a Family of Compound Distributions," ASTIN Bulletin, Cambridge University Press, vol. 12(1), pages 22-26, June.
    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. Caroline Hillairet & Ying Jiao & Anthony Réveillac, 2018. "Pricing formulae for derivatives in insurance using the Malliavin calculus ," Post-Print hal-01561987, HAL.
    2. Caroline Hillairet & Ying Jiao & Anthony Réveillac, 2017. "Pricing formulae for derivatives in insurance using the Malliavin calculus ," Working Papers hal-01561987, HAL.
    3. Caroline Hillairet & Ying Jiao & Anthony R'eveillac, 2017. "Pricing formulae for derivatives in insurance using the Malliavin calculus," Papers 1707.05061, arXiv.org.
    4. Hillairet, Caroline & Réveillac, Anthony & Rosenbaum, Mathieu, 2023. "An expansion formula for Hawkes processes and application to cyber-insurance derivatives," Stochastic Processes and their Applications, Elsevier, vol. 160(C), pages 89-119.
    5. Paul Embrechts & Marco Frei, 2009. "Panjer recursion versus FFT for compound distributions," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 69(3), pages 497-508, July.
    6. Dutang, C. & Lefèvre, C. & Loisel, S., 2013. "On an asymptotic rule A+B/u for ultimate ruin probabilities under dependence by mixing," Insurance: Mathematics and Economics, Elsevier, vol. 53(3), pages 774-785.
    7. Rafał Wójcik & Charlie Wusuo Liu & Jayanta Guin, 2019. "Direct and Hierarchical Models for Aggregating Spatially Dependent Catastrophe Risks," Risks, MDPI, vol. 7(2), pages 1-22, May.
    8. Su, Jianxi & Furman, Edward, 2017. "Multiple risk factor dependence structures: Distributional properties," Insurance: Mathematics and Economics, Elsevier, vol. 76(C), pages 56-68.
    9. Corina Constantinescu & Suhang Dai & Weihong Ni & Zbigniew Palmowski, 2016. "Ruin Probabilities with Dependence on the Number of Claims within a Fixed Time Window," Risks, MDPI, vol. 4(2), pages 1-23, June.
    10. repec:hal:wpaper:hal-00746251 is not listed on IDEAS
    11. Bae, Taehan & Kim, Changki & Kulperger, Reginald J., 2009. "Securitization of motor insurance loss rate risks," Insurance: Mathematics and Economics, Elsevier, vol. 44(1), pages 48-58, February.
    12. Emilio Gómez-Déniz & Jorge V. Pérez-Rodríguez & Simón Sosvilla-Rivero, 2022. "Analyzing How the Social Security Reserve Fund in Spain Affects the Sustainability of the Pension System," Risks, MDPI, vol. 10(6), pages 1-17, June.
    13. Florin Avram & Romain Biard & Christophe Dutang & Stéphane Loisel & Landy Rabehasaina, 2014. "A survey of some recent results on Risk Theory," Post-Print hal-01616178, HAL.
    14. Zhimin Zhang & Hailiang Yang & Hu Yang, 2012. "On a Sparre Andersen Risk Model with Time-Dependent Claim Sizes and Jump-Diffusion Perturbation," Methodology and Computing in Applied Probability, Springer, vol. 14(4), pages 973-995, December.
    15. Yujuan Huang & Jing Li & Hengyu Liu & Wenguang Yu, 2021. "Estimating Ruin Probability in an Insurance Risk Model with Stochastic Premium Income Based on the CFS Method," Mathematics, MDPI, vol. 9(9), pages 1-17, April.
    16. Muhsin Tamturk & Dominic Cortis & Mark Farrell, 2020. "Examining the Effects of Gradual Catastrophes on Capital Modelling and the Solvency of Insurers: The Case of COVID-19," Risks, MDPI, vol. 8(4), pages 1-13, December.
    17. Jae-Kyung Woo & Haibo Liu, 2018. "Discounted Aggregate Claim Costs Until Ruin in the Discrete-Time Renewal Risk Model," Methodology and Computing in Applied Probability, Springer, vol. 20(4), pages 1285-1318, December.
    18. Sundt, Bjorn, 2002. "Recursive evaluation of aggregate claims distributions," Insurance: Mathematics and Economics, Elsevier, vol. 30(3), pages 297-322, June.
    19. Anh Ninh, 2021. "Robust newsvendor problems with compound Poisson demands," Annals of Operations Research, Springer, vol. 302(1), pages 327-338, July.
    20. Cossette, Hélène & Marceau, Etienne & Mtalai, Itre & Veilleux, Déry, 2018. "Dependent risk models with Archimedean copulas: A computational strategy based on common mixtures and applications," Insurance: Mathematics and Economics, Elsevier, vol. 78(C), pages 53-71.
    21. Janic-Wroblewska, A. & Kallenberg, W. C. M. & Ledwina, T., 2004. "Detecting positive quadrant dependence and positive function dependence," Insurance: Mathematics and Economics, Elsevier, vol. 34(3), pages 467-487, June.

    More about this item

    NEP fields

    This paper has been announced in the following NEP Reports:

    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:crs:wpaper:2017-75. 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: Secretariat General (email available below). General contact details of provider: https://edirc.repec.org/data/crestfr.html .

    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.