Zeitschrift für Operations Research

, Volume 39, Issue 1, pp 1–34 | Cite as

Modelling of extremal events in insurance and finance

  • Paul Embrechts
  • Hanspeter Schmidli
Articles
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Abstract

Extremal events play an increasingly important role in stochastic modelling in insurance and finance. Over many years, probabilists and statisticians have developed techniques for the description, analysis and prediction of such events. In the present paper, we review the relevant theory which may also be used in the wider context of Operation Research. Various applications from the field of insurance and finance are discussed. Via an extensive list of references, the reader is guided towards further material related to the above problem areas.

Key words

extreme value theory Pareto distributions risk theory ruin tail-estimation financial time series 
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References

  1. Abraham B, Ledolter J (1983) Statistical methods for forecasting. John Wiley, New YorkGoogle Scholar
  2. Aebi M, Embrechts P, Mikosch T (1992) A large claim index. Mitteilungen der Schweiz. Vereinigung der Versicherungsmathematiker 2:143–156Google Scholar
  3. Asmussen S, Fløe Henriksen L, Klüppelberg C (1992) Large claims approximations for risk processes in a Markovian environment. PreprintGoogle Scholar
  4. Bachelier L (1900) “Theory of speculation”, reprinted in Cootner P (ed), 1964. The random character of stock market prices. MIT Press, Cambridge 17–78Google Scholar
  5. Beirlant J, Teugels JL (1992) Modelling large claims in non-life insurance. Insurance: Mathematics and Economics 11:17–29Google Scholar
  6. Bingham NH, Goldie CM, Teugels JL (1987) Regular variation. Cambridge University Press, CambridgeGoogle Scholar
  7. Clewlow LJ, Xu X (1992) A review of option pricing with stochastic volatility. FORC. Preprint 92/35Google Scholar
  8. Cline DBH (1986) Convolution tails, product tails and domains of attraction. Prob Th Rel Fields 72:529–557Google Scholar
  9. Davis RA, Resnick SI (1985) Limit theory for moving averages of random variables with regularly varying tail probabilities. Ann Prob 13:179–195Google Scholar
  10. Davis RA, Resnick SI (1986) Limit theory for the sample covariance and correlation function of moving averages. Ann Statist 14:533–558Google Scholar
  11. Embrechts P (1983) A property of the generalized inverse Gaussian distribution with some applications. J Appl Prob 20:537–544Google Scholar
  12. Embrechts P (1993) Modelling of catastrophic events in insurance and finance. Published in “Operations Research '92 GMÖOR”. Eds Karmann A, Mosler K, Schader M, Uebe G. Physica-Verlag, Hamburg 544–546Google Scholar
  13. Embrechts P, Goldie CM, Veraverbeke N (1979) Subexponentiality and infinite divisibility. Z Wahrscheinlichkeitstheorie verw Gebiete 49:335–347Google Scholar
  14. Embrechts P, Grandell J, Schmidli H (1993) Finite-time Lundberg inequalities in the Cox case. Scand Actuarial J, to appearGoogle Scholar
  15. Embrechts P, Klüppelberg C (1993) Some aspects of insurance mathematics. Th Prob Appl, to appearGoogle Scholar
  16. Embrechts P, Omey E (1984) A property of longtailed distributions. J Appl Prob 21:80–87Google Scholar
  17. Embrechts P, Pugh D, Smith RL (1985) Statistical extremes and risks. Department of Mathematics. One day course. Lecture Notes Series, Imperial CollegeGoogle Scholar
  18. Embrechts P, Veraverbeke N (1982) Estimates for the probability of ruin with special emphasis on the possibility of large claims. Insurance: Mathematics and Economics 1:55–72Google Scholar
  19. Fama EF(1965) The behaviour of stock market prices. Journal of Business 38:34–105Google Scholar
  20. Fama EF, Roll R (1968) Some properties of symmetric stable distributions. JASA 63:817–836Google Scholar
  21. Feller W (1971) An introduction to probability theory and its applications. Volume II, 2nd ed. Wiley, New YorkGoogle Scholar
  22. Franks JR, Schwartz ES (1992) The stochastic behaviour of market variance implied in the prices of index options. Economic Journal. Also reprinted in Hodges SD (ed) Options. Recent advances in theory and practice. Volume 2. Manchester University PressGoogle Scholar
  23. Geluk JL, de Haan L (1987) Regular variation extensions and Tauberian theorems. CWI Tract 40, AmsterdamGoogle Scholar
  24. Gisler A, Reinhard P (1990) Robust credibility. Paper presented at the XXIIth Astin Colloquium, MontreuxGoogle Scholar
  25. Goovaerts MJ, Kaas R, van Heerwaarden AE, Bauwelinckx T (1990) Effective actuarial methods. North-Holland, AmsterdamGoogle Scholar
  26. Grandell J (1991) Aspects of risk theory. Springer-Verlag, BerlinGoogle Scholar
  27. Häusler E, Teugels JL (1985) On asymptotic normality of Hill's estimator for the exponent of regular variation. Ann Statist 13:743–756Google Scholar
  28. Heyman DP, Sobel MJ (1982) Stochastic models in operations research. Volume I: Stochastic processes and operating characteristics. McGraw-Hill, New YorkGoogle Scholar
  29. Hill BM (1975) A simple general approach to inference about the tail of a distribution. Ann Statist 3:1163–1174Google Scholar
  30. Hogg RV, Klugman SA (1984) Loss distributions. John Wiley, New YorkGoogle Scholar
  31. Hull J, White A (1987) The pricing of options on assets with stochastic volatility. Journal of Finance 42:281–300Google Scholar
  32. Iglehart DL, Stone ML (1983) Regenerative simulation for estimating extreme values. Operations Research 31(6): 1145–1166Google Scholar
  33. Jung J (1982) Association of Swedish Insurance Companies, Statistical Department, StockholmGoogle Scholar
  34. Karatzas I, Shreve SES (1988) Brownian motion and stochastic calculus. Springer-Verlag, New YorkGoogle Scholar
  35. Karplus WJ (1992) The heavens are falling. The scientific prediction of catastrophes in our time. Plenum Press, New YorkGoogle Scholar
  36. Keller B, Klüppelberg C (1991) Statistical estimation of large claim distributions. Mitteilungen der Schweiz. Vereinigung der Versicherungsmathematiker 2/1991:203–216Google Scholar
  37. Klüppelberg C (1989) Subexponential distributions and characterization of related classes. Prob Th Rel Fields 82:259–269Google Scholar
  38. Klüppelberg C, Mikosch T (1992a) Spectral estimates and stable processes. Stoch Proc Appl 37 (1993) to appearGoogle Scholar
  39. Klüppelberg C, Mikosch T (1992b) Some limit theory for the normalized periodogram ofp-stable moving averages. ETH preprintGoogle Scholar
  40. Künsch HR (1992) Robust methods for credibility. Astin Bulletin 22(1): 33–49Google Scholar
  41. Leadbetter MR, Lindgren G, Rootzén H (1983) Extremes and related properties of random sequences and processes. Springer-Verlag, New YorkGoogle Scholar
  42. Locher R (ed) (1992) Risiko, zwischen Chance und Gefahr. Reto Locher und Partner AG, BaselGoogle Scholar
  43. Mandelbrot B (1963) The variation of certain speculative prices. Journal of Business 36:394–419Google Scholar
  44. Øksendahl B (1992) Stochastic differential equations. Springer-Verlag, BerlinGoogle Scholar
  45. Panjer H, Willmot G (1992) Insurance risk models. Society of Actuaries, Schaumburg, ILGoogle Scholar
  46. Pfeifer D (1989) Einführung in die Extremwertstatistik. Teubner, StuttgartGoogle Scholar
  47. Pickands J (1971) The two-dimensional Poisson process and extremal processes. J Appl Prob 8:745–756Google Scholar
  48. Ramlau-Hansen H (1988a) A solvency study in non-life insurance. Part 1: Analysis of fire, windstorm, and glass claims. Scand Actuarial J: 3–34Google Scholar
  49. Ramlau-Hansen H (1988b) A solvency study in non-life insurance. Part 2: Solvency margin requirements. Scand Actuarial J: 35–60Google Scholar
  50. Reiss R-D (1989) Approximate distribution of order statistics (with applications to non-parametric statistics). Springer-Verlag, New YorkGoogle Scholar
  51. Resnick SI (1987) Extreme values, regular variation, and point processes. Springer-Verlag, New YorkGoogle Scholar
  52. Schmidii H (1992) A general insurance risk model. PhD thesis, ETH ZürichGoogle Scholar
  53. Schwert GW (1989) Why does stock market volatility change over time. Journal of Finance 44: 1115–1153Google Scholar
  54. Seber GAF (1984) Multivariate observations. John Wiley, New YorkGoogle Scholar
  55. Sigma (1991) Naturkatastrophen und Grossschäden 1990: Versicherte Schäden erreichen Rekordsumme von 17 Mrd US$. Schweizerische Rückversicherungsgesellschaft 3/91Google Scholar
  56. Smith RL (1987) Estimating tails of probability distributions. Ann Statist 15:1174–1207Google Scholar
  57. Smith RL (1989) Extreme value analysis. Statistical Science 4:367–393Google Scholar
  58. Stam AJ (1973) Regular variation of the tail of a subordinated probability distribution. Adv Appl Prob 5:308–327Google Scholar
  59. Taylor S (1986) Modelling financial time series. John Wiley, ChichesterGoogle Scholar
  60. Turner AL, Weigel EJK (1992) Daily stock market volatility 1928–1989. Management Science 38:1586–1609Google Scholar
  61. Weissman I (1978) Estimation of parameters and large quantiles based on thek largest observations. J Amer Statist Assoc 73:812–815Google Scholar

Copyright information

© Physica-Verlag 1994

Authors and Affiliations

  • Paul Embrechts
    • 1
  • Hanspeter Schmidli
    • 1
  1. 1.Department of MathematicsETH-ZZürichSwitzerland

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