Skip to main content
SpringerLink
Log in

Modelling of extremal events in insurance and finance

  • Articles
  • Published:
Zeitschrift für Operations Research Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Abraham B, Ledolter J (1983) Statistical methods for forecasting. John Wiley, New York

    Google Scholar 

  • Aebi M, Embrechts P, Mikosch T (1992) A large claim index. Mitteilungen der Schweiz. Vereinigung der Versicherungsmathematiker 2:143–156

    Google Scholar 

  • Asmussen S, Fløe Henriksen L, Klüppelberg C (1992) Large claims approximations for risk processes in a Markovian environment. Preprint

  • Bachelier L (1900) “Theory of speculation”, reprinted in Cootner P (ed), 1964. The random character of stock market prices. MIT Press, Cambridge 17–78

    Google Scholar 

  • Beirlant J, Teugels JL (1992) Modelling large claims in non-life insurance. Insurance: Mathematics and Economics 11:17–29

    Google Scholar 

  • Bingham NH, Goldie CM, Teugels JL (1987) Regular variation. Cambridge University Press, Cambridge

    Google Scholar 

  • Clewlow LJ, Xu X (1992) A review of option pricing with stochastic volatility. FORC. Preprint 92/35

    Google Scholar 

  • Cline DBH (1986) Convolution tails, product tails and domains of attraction. Prob Th Rel Fields 72:529–557

    Google Scholar 

  • Davis RA, Resnick SI (1985) Limit theory for moving averages of random variables with regularly varying tail probabilities. Ann Prob 13:179–195

    Google Scholar 

  • Davis RA, Resnick SI (1986) Limit theory for the sample covariance and correlation function of moving averages. Ann Statist 14:533–558

    Google Scholar 

  • Embrechts P (1983) A property of the generalized inverse Gaussian distribution with some applications. J Appl Prob 20:537–544

    Google Scholar 

  • 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 544–546

    Google Scholar 

  • Embrechts P, Goldie CM, Veraverbeke N (1979) Subexponentiality and infinite divisibility. Z Wahrscheinlichkeitstheorie verw Gebiete 49:335–347

    Google Scholar 

  • Embrechts P, Grandell J, Schmidli H (1993) Finite-time Lundberg inequalities in the Cox case. Scand Actuarial J, to appear

  • Embrechts P, Klüppelberg C (1993) Some aspects of insurance mathematics. Th Prob Appl, to appear

  • Embrechts P, Omey E (1984) A property of longtailed distributions. J Appl Prob 21:80–87

    Google Scholar 

  • Embrechts P, Pugh D, Smith RL (1985) Statistical extremes and risks. Department of Mathematics. One day course. Lecture Notes Series, Imperial College

  • Embrechts P, Veraverbeke N (1982) Estimates for the probability of ruin with special emphasis on the possibility of large claims. Insurance: Mathematics and Economics 1:55–72

    Google Scholar 

  • Fama EF(1965) The behaviour of stock market prices. Journal of Business 38:34–105

    Google Scholar 

  • Fama EF, Roll R (1968) Some properties of symmetric stable distributions. JASA 63:817–836

    Google Scholar 

  • Feller W (1971) An introduction to probability theory and its applications. Volume II, 2nd ed. Wiley, New York

    Google Scholar 

  • 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 Press

  • Geluk JL, de Haan L (1987) Regular variation extensions and Tauberian theorems. CWI Tract 40, Amsterdam

  • Gisler A, Reinhard P (1990) Robust credibility. Paper presented at the XXIIth Astin Colloquium, Montreux

  • Goovaerts MJ, Kaas R, van Heerwaarden AE, Bauwelinckx T (1990) Effective actuarial methods. North-Holland, Amsterdam

    Google Scholar 

  • Grandell J (1991) Aspects of risk theory. Springer-Verlag, Berlin

    Google Scholar 

  • Häusler E, Teugels JL (1985) On asymptotic normality of Hill's estimator for the exponent of regular variation. Ann Statist 13:743–756

    Google Scholar 

  • Heyman DP, Sobel MJ (1982) Stochastic models in operations research. Volume I: Stochastic processes and operating characteristics. McGraw-Hill, New York

    Google Scholar 

  • Hill BM (1975) A simple general approach to inference about the tail of a distribution. Ann Statist 3:1163–1174

    Google Scholar 

  • Hogg RV, Klugman SA (1984) Loss distributions. John Wiley, New York

    Google Scholar 

  • Hull J, White A (1987) The pricing of options on assets with stochastic volatility. Journal of Finance 42:281–300

    Google Scholar 

  • Iglehart DL, Stone ML (1983) Regenerative simulation for estimating extreme values. Operations Research 31(6): 1145–1166

    Google Scholar 

  • Jung J (1982) Association of Swedish Insurance Companies, Statistical Department, Stockholm

  • Karatzas I, Shreve SES (1988) Brownian motion and stochastic calculus. Springer-Verlag, New York

    Google Scholar 

  • Karplus WJ (1992) The heavens are falling. The scientific prediction of catastrophes in our time. Plenum Press, New York

    Google Scholar 

  • Keller B, Klüppelberg C (1991) Statistical estimation of large claim distributions. Mitteilungen der Schweiz. Vereinigung der Versicherungsmathematiker 2/1991:203–216

    Google Scholar 

  • Klüppelberg C (1989) Subexponential distributions and characterization of related classes. Prob Th Rel Fields 82:259–269

    Google Scholar 

  • Klüppelberg C, Mikosch T (1992a) Spectral estimates and stable processes. Stoch Proc Appl 37 (1993) to appear

  • Klüppelberg C, Mikosch T (1992b) Some limit theory for the normalized periodogram ofp-stable moving averages. ETH preprint

  • Künsch HR (1992) Robust methods for credibility. Astin Bulletin 22(1): 33–49

    Google Scholar 

  • Leadbetter MR, Lindgren G, Rootzén H (1983) Extremes and related properties of random sequences and processes. Springer-Verlag, New York

    Google Scholar 

  • Locher R (ed) (1992) Risiko, zwischen Chance und Gefahr. Reto Locher und Partner AG, Basel

    Google Scholar 

  • Mandelbrot B (1963) The variation of certain speculative prices. Journal of Business 36:394–419

    Google Scholar 

  • Øksendahl B (1992) Stochastic differential equations. Springer-Verlag, Berlin

    Google Scholar 

  • Panjer H, Willmot G (1992) Insurance risk models. Society of Actuaries, Schaumburg, IL

    Google Scholar 

  • Pfeifer D (1989) Einführung in die Extremwertstatistik. Teubner, Stuttgart

    Google Scholar 

  • Pickands J (1971) The two-dimensional Poisson process and extremal processes. J Appl Prob 8:745–756

    Google Scholar 

  • Ramlau-Hansen H (1988a) A solvency study in non-life insurance. Part 1: Analysis of fire, windstorm, and glass claims. Scand Actuarial J: 3–34

  • Ramlau-Hansen H (1988b) A solvency study in non-life insurance. Part 2: Solvency margin requirements. Scand Actuarial J: 35–60

  • Reiss R-D (1989) Approximate distribution of order statistics (with applications to non-parametric statistics). Springer-Verlag, New York

    Google Scholar 

  • Resnick SI (1987) Extreme values, regular variation, and point processes. Springer-Verlag, New York

    Google Scholar 

  • Schmidii H (1992) A general insurance risk model. PhD thesis, ETH Zürich

    Google Scholar 

  • Schwert GW (1989) Why does stock market volatility change over time. Journal of Finance 44: 1115–1153

    Google Scholar 

  • Seber GAF (1984) Multivariate observations. John Wiley, New York

    Google Scholar 

  • Sigma (1991) Naturkatastrophen und Grossschäden 1990: Versicherte Schäden erreichen Rekordsumme von 17 Mrd US$. Schweizerische Rückversicherungsgesellschaft 3/91

  • Smith RL (1987) Estimating tails of probability distributions. Ann Statist 15:1174–1207

    Google Scholar 

  • Smith RL (1989) Extreme value analysis. Statistical Science 4:367–393

    Google Scholar 

  • Stam AJ (1973) Regular variation of the tail of a subordinated probability distribution. Adv Appl Prob 5:308–327

    Google Scholar 

  • Taylor S (1986) Modelling financial time series. John Wiley, Chichester

    Google Scholar 

  • Turner AL, Weigel EJK (1992) Daily stock market volatility 1928–1989. Management Science 38:1586–1609

    Google Scholar 

  • Weissman I (1978) Estimation of parameters and large quantiles based on thek largest observations. J Amer Statist Assoc 73:812–815

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, ETH-Z, CH-8092, Zürich, Switzerland

    Paul Embrechts & Hanspeter Schmidli

Authors
  1. Paul Embrechts
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hanspeter Schmidli
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Embrechts, P., Schmidli, H. Modelling of extremal events in insurance and finance. ZOR - Mathematical Methods of Operations Research 39, 1–34 (1994). https://doi.org/10.1007/BF01440733

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01440733

Key words

Search

Navigation