Optimal Exercise Strategies for Operational Risk Insurance via Multiple Stopping Times

  • Rodrigo S. TarginoEmail author
  • Gareth W. Peters
  • Georgy Sofronov
  • Pavel V. Shevchenko
Open Access


In this paper we demonstrate how to develop analytic closed form solutions to optimal multiple stopping time problems arising in the setting in which the value function acts on a compound process that is modified by the actions taken at the stopping times. This class of problem is particularly relevant in insurance and risk management settings and we demonstrate this on an important application domain based on insurance strategies in Operational Risk management for financial institutions. In this area of risk management the most prevalent class of loss process models is the Loss Distribution Approach (LDA) framework which involves modelling annual losses via a compound process. Given an LDA model framework, we consider Operational Risk insurance products that mitigate the risk for such loss processes and may reduce capital requirements. In particular, we consider insurance products that grant the policy holder the right to insure k of its annual Operational losses in a horizon of T years. We consider two insurance product structures and two general model settings, the first are families of relevant LDA loss models that we can obtain closed form optimal stopping rules for under each generic insurance mitigation structure and then secondly classes of LDA models for which we can develop closed form approximations of the optimal stopping rules. In particular, for losses following a compound Poisson process with jump size given by an Inverse-Gaussian distribution and two generic types of insurance mitigation, we are able to derive analytic expressions for the loss process modified by the insurance application, as well as closed form solutions for the optimal multiple stopping rules in discrete time (annually). When the combination of insurance mitigation and jump size distribution does not lead to tractable stopping rules we develop a principled class of closed form approximations to the optimal decision rule. These approximations are developed based on a class of orthogonal Askey polynomial series basis expansion representations of the annual loss compound process distribution and functions of this annual loss.


Insurance Multiple stopping rules Operational risk 

Mathematics Subject Classification (2010)

60G40 62P05 91B30 41A58 


  1. Aase KK (1993) Equilibrum in a reinsurance syndicate: existence, uniquiness and characterization. ASTIN Bull 23(2):185–211MathSciNetCrossRefGoogle Scholar
  2. Allen L, Boudoukh J, Saunders A (2009) Understanding market, credit, and operational risk: the value at risk approach. WileyGoogle Scholar
  3. Arrow KJ (1953) Le rôle des valeurs boursières pour la répartition la meilleure des risques. Colloques Internationaux du Centre National de la Recherche Scientifique 11:41–47MathSciNetzbMATHGoogle Scholar
  4. Arrow KJ (1965) Aspects of the theory of risk-bearing. Yrjö Jahnssonin SäätiöGoogle Scholar
  5. Bazzarello D, Crielaard B, Piacenza F, Soprano A (2006) Modeling insurance mitigation on operational risk capital. J Oper Risk 1(1):57–65CrossRefGoogle Scholar
  6. BCBS (2006) International Convergence of Capital Measurement and Capital Standards - A Revised Framework (Comprehensive Version). Technical report, Bank for International SettlementsGoogle Scholar
  7. BCBS (2010) Basel III: a global regulatory framework for more resilient banks and baning systems. Technical report, Bank for International SettlementsGoogle Scholar
  8. Bender C, Schoenmakers J (2006) An iterative method for multiple stopping: convergence and stability. Adv Appl Probab:729–749Google Scholar
  9. Bender C, Schoenmakers J, Zhang J (2013) Dual representations for general multiple stopping problems. Mathematical FinanceGoogle Scholar
  10. Berliner B. (1982) Limits of insurability of risks. Prentice-Hall Englewood Cliff, NJGoogle Scholar
  11. Borch K (1962) Equilibrium in a reinsurance market. Econometrica: J Econ Soc 30(3):424–444CrossRefzbMATHGoogle Scholar
  12. Bowers NL Jr (1966) Expansion of probability density functions as a sum of gamma densities with applications in risk theory. Trans Soc Actuar 18:125–137Google Scholar
  13. Brandts S (2004) Operational risk and insurance: quantitative and qualitative aspects. Working paper, Goethe University, FrankfurtGoogle Scholar
  14. Carmona R, Touzi N (2008) Optimal multiple stopping and valuation of swing options. Math Financ 18(2):239–268MathSciNetCrossRefzbMATHGoogle Scholar
  15. Chernobai AS, Rachev ST, Fabozzi FJ (2008) Operational risk: a guide to Basel II capital requirements, models, and analysis, vol 180. WileyGoogle Scholar
  16. Chhikara R, Folks J (1977) The inverse gaussian distribution as a lifetime model. Technometrics 19(4):461–468CrossRefzbMATHGoogle Scholar
  17. EBA (2015) Final draft on ama assessment for operational risk. Technical reportGoogle Scholar
  18. Embrechts P (1983) A property of the generalized inverse gaussian distribution with some applications. J Appl Probab:537–544Google Scholar
  19. Folks J, Chhikara R (1978) The inverse gaussian distribution and its statistical application–a review. J R Stat Soc Series B 40(3):263–289MathSciNetzbMATHGoogle Scholar
  20. Franzetti C (2011) Operational risk modelling and management. Taylor & Francis, USGoogle Scholar
  21. Ghossoub M (2012) Belief heterogeneity in the arrow-borch-raviv insurance model. Available at SSRN 2028550Google Scholar
  22. Gollier C (2005) Some aspects of the economics of catastrophe risk insurance. Technical report. CESifo Working Paper SeriesGoogle Scholar
  23. Jackson D (1941) Fourier series and orthogonal polynomials. Courier Dover PublicationsGoogle Scholar
  24. Jaillet P, Ronn EI, Tompaidis S (2004) Valuation of commodity-based swing options. Manag Sci 50(7):909–921CrossRefzbMATHGoogle Scholar
  25. Jondeau E, Rockinger M (2001) Gram–charlier densities. J Econ Dyn Control 25(10):1457–1483CrossRefzbMATHGoogle Scholar
  26. Jørgensen B (1982) Statistical properties of the generalized inverse Gaussian distribution, volume 9 of lecture notes in statistics. Springer-Verlag, New YorkCrossRefGoogle Scholar
  27. Longstaff FA, Schwartz ES (2001) Valuing american options by simulation: a simple least-squares approach. Rev Financ Stud 14(1):113–147CrossRefGoogle Scholar
  28. Mehr RI, Cammack E, Rose T (1980) Principles of insurance, vol 8. RD IrwinGoogle Scholar
  29. Nikolaev M, Sofronov G (2007) A multiple optimal stopping rule for sums of independent random variables. Discret Math Appl dma 17(5):463–473zbMATHGoogle Scholar
  30. Peters GW, Byrnes AD, Shevchenko PV (2011) Impact of insurance for operational risk: is it worthwhile to insure or be insured for severe losses? Insur Math Econ 48(2):287–303MathSciNetCrossRefzbMATHGoogle Scholar
  31. Peters GW, Targino RS, Shevchenko PV (2013) Understanding operational risk capital approximations: first and second orders. J Govern Regulat 2(3):58–78CrossRefGoogle Scholar
  32. Raviv A (1979) The design of an optimal insurance policy. Amer Econ Rev 69 (1):84–96Google Scholar
  33. Sofronov G (2013) An optimal sequential procedure for a multiple selling problem with independent observations. Eur J Oper Res 225(2):332–336MathSciNetCrossRefzbMATHGoogle Scholar
  34. Sofronov G, Keith J, Kroese D (2006) An optimal sequential procedure for a buying-selling problem with independent observations. J Appl Probab 43(2):454–462MathSciNetCrossRefzbMATHGoogle Scholar
  35. Tweedie M (1957) Statistical properties of inverse gaussian distributions. Ann Math Stat 28(2):362–377MathSciNetCrossRefzbMATHGoogle Scholar
  36. Van den Brink GJ (2002) Operational risk: the new challenge for banks. Palgrave MacmillanGoogle Scholar
  37. Watson G (1922) A treatise on the theory of Bessel functions. Cambridge University PressGoogle Scholar

Copyright information

© The Author(s) 2016

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Department of Statistical ScienceUniversity College LondonLondonUK
  2. 2.Department of StatisticsMacquarie UniversitySydneyAustralia
  3. 3.CSIROSydneyAustralia

Personalised recommendations