Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-23T16:46:55.886Z Has data issue: false hasContentIssue false
  • English
  • Français

A Discrete Time Benchmark Approach for Insurance and Finance

Published online by Cambridge University Press:  17 April 2015

Hans Bühlmann  and
Eckhard Platen
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

This paper proposes a consistent approach to discrete time valuation in insurance and finance. This approach uses the growth optimal portfolio as reference unit or benchmark. When used as benchmark, it is shown that all benchmarked price processes are supermartingales. Benchmarked fair price processes are characterized as martingales. No measure transformation is needed for the fair pricing of insurance policies and derivatives. The standard actuarial pricing rule is obtained as a particular case of fair pricing when the contingent claim is independent from the growth optimal portfolio.

1991 Mathematics Subject Classification: primary 90A12 secondary 60G30, 62P20

JEL Classification: G10, G13

Keywords

Financial and insurance market modelbenchmark approachgrowth optimal portfolionumeraire portfoliofair pricingactuarial pricingunit linked insurance
Type
Articles
Information
ASTIN Bulletin: The Journal of the IAA , Volume 33 , Issue 2 , November 2003 , pp. 153 - 172
Copyright
Copyright © ASTIN Bulletin 2003

Footnotes

1

Department of Mathematics, Eidgenössische Technische Hochschule, 8092 Zürich, Switzerland.

2

University of Technology Sydney, School of Finance & Economics and Department of Mathematical Sciences, PO Box 123, Broadway, NSW, 2007, Australia.

References

Artzner, P. (1997) On the numeraire portfolio. In Mathematics of Derivative Securities, pp. 5358. Cambridge University Press.Google Scholar
Bajeux-Besnainou, I. and Portait, R. (1997) The numeraire portfolio: A new perspective on financial theory. The European Journal of Finance 3, 291–309.CrossRefGoogle Scholar
Becherer, D. (2001) The numeraire portfolio for unbounded semimartingales. Finance Stoch. 5, 327–341.CrossRefGoogle Scholar
Bühlmann, H. (1992) Stochastic discounting. Insurance: Mathematics and Economics 11, 113–127.Google Scholar
Bühlmann, H. (1995) Life insurance with stochastic interest rates. In Ottaviani, G. (Ed.), Financial Risk and Insurance, pp. 1–24. Springer.Google Scholar
Bühlmann, H., Delbaen, F., Embrechts, P. and Shiryaev, A. (1998) On Esscher transforms in discrete finance models. ASTIN Bulletin 28(2), 171–186.CrossRefGoogle Scholar
Constatinides, G.M. (1992) A theory of the nominal structure of interest rates. Rev. Financial Studies 5, 531–552.CrossRefGoogle Scholar
Duffie, D. (1996) Dynamic Asset Pricing Theory (2nd ed.). Princeton, University Press.Google Scholar
Goll, T. and Kallsen, J. (2003) A complete explicit solution to the log-optimal portfolio problem. Adv. in Appl. Probab. 13(2), 774–799.Google Scholar
Heath, D. and Platen, E. (2002) Pricing and hedging of index derivatives under an alternative asset price model with endogenous stochastic volatility. In Yong, J. (Ed.), Recent Developments in Mathematical Finance, pp. 117–126. World Scientific.Google Scholar
Karatzas, I. and Shreve, S.E. (1998) Methods of Mathematical Finance, Volume 39 of Appl. Math. Springer.Google Scholar
Kelly, J.R. (1956) A new interpretation of information rate. Bell Syst. Techn. J. 35, 917–926.CrossRefGoogle Scholar
Korn, R. (2001) Value preserving strategies and a general framework for local approaches to optimal portfolios. Math. Finance 10(2), 227–241.CrossRefGoogle Scholar
Kramkov, D.O. and Schachermayer, W. (1999) The asymptotic elasticity of utility functions and optimal investment in incomplete markets. Ann. Appl. Probab. 9, 904–950.CrossRefGoogle Scholar
Long, J.B. (1990) The numeraire portfolio. J. Financial Economics 26, 29–69.CrossRefGoogle Scholar
Platen, E. (2001) A minimal financial market model. In Trends in Mathematics, pp. 293–301. Birkhäuser.Google Scholar
Platen, E. (2002) Arbitrage in continuous complete markets. Adv. in Appl. Probab. 34(3), 540–558.CrossRefGoogle Scholar
Platen, E. (2004). A class of complete benchmark models with intensity based jumps to appear in J. Appl. Probab. 4(1).Google Scholar
Rogers, L.C.G. (1997) The potential approach to the term structure of interest rates and their exchange rates. Math. Finance 7, 157–176.CrossRefGoogle Scholar