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A Discrete Time Benchmark Approach for Finance and Insurance

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Abstract

This paper proposes an integrated appraoch to discrete time modelling in finance and insurance. This approach is based on the existence of a specific benchmark portfolio, known as the growth optimal portfolio. When used as numeraire, this portfolio ensures that all benchmarked price processes are super-martingales. A fair market is characterized in terms of the type of maximum that the optimal growth rate attains. In general, arbitrage amounts arise due to supermartingale property of benchmarked traded prices. No measure transformation is needed for the pricing of insurance policies and derivatives in a fair market.

Suggested Citation

  • Hans Buhlmann & Eckhard Platen, 2002. "A Discrete Time Benchmark Approach for Finance and Insurance," Research Paper Series 74, Quantitative Finance Research Centre, University of Technology, Sydney.
    • Handle: RePEc:uts:rpaper:74
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    File URL: http://www.qfrc.uts.edu.au/research/research_papers/rp74.pdf
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    References listed on IDEAS

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    1. Constantinides, George M, 1992. "A Theory of the Nominal Term Structure of Interest Rates," The Review of Financial Studies, Society for Financial Studies, vol. 5(4), pages 531-552.
    2. Harrison, J. Michael & Pliska, Stanley R., 1981. "Martingales and stochastic integrals in the theory of continuous trading," Stochastic Processes and their Applications, Elsevier, vol. 11(3), pages 215-260, August.
    3. Platen, Eckhard, 2000. "A minimal financial market model," SFB 373 Discussion Papers 2000,91, Humboldt University of Berlin, Interdisciplinary Research Project 373: Quantification and Simulation of Economic Processes.
    4. I. Bajeux-Besnainou & R. Portait, 1997. "The numeraire portfolio: a new perspective on financial theory," The European Journal of Finance, Taylor & Francis Journals, vol. 3(4), pages 291-309.
    5. Harrison, J. Michael & Kreps, David M., 1979. "Martingales and arbitrage in multiperiod securities markets," Journal of Economic Theory, Elsevier, vol. 20(3), pages 381-408, June.
    6. David Heath & Eckhard Platen, 2001. "Pricing and Hedging of Index Derivatives under an Alternative Asset Price Model with Endogenous Stochastic Volatility," World Scientific Book Chapters, in: Jiongmin Yong (ed.), Recent Developments In Mathematical Finance, chapter 10, pages 117-126, World Scientific Publishing Co. Pte. Ltd..
    7. Norbert Hofmann & Eckhard Platen & Martin Schweizer, 1992. "Option Pricing Under Incompleteness and Stochastic Volatility," Mathematical Finance, Wiley Blackwell, vol. 2(3), pages 153-187, July.
    8. Buhlmann, H., 1992. "Stochastic discounting," Insurance: Mathematics and Economics, Elsevier, vol. 11(2), pages 113-127, August.
    9. Dirk Becherer, 2001. "The numeraire portfolio for unbounded semimartingales," Finance and Stochastics, Springer, vol. 5(3), pages 327-341.
    10. Long, John Jr., 1990. "The numeraire portfolio," Journal of Financial Economics, Elsevier, vol. 26(1), pages 29-69, July.
    11. L. C. G. Rogers, 1997. "The Potential Approach to the Term Structure of Interest Rates and Foreign Exchange Rates," Mathematical Finance, Wiley Blackwell, vol. 7(2), pages 157-176, April.
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    Cited by:

    1. Morten Christensen & Eckhard Platen, 2004. "A General Benchmark Model for Stochastic Jump Sizes," Research Paper Series 139, Quantitative Finance Research Centre, University of Technology, Sydney.
    2. Eckhard Platen & Jason West, 2004. "A Fair Pricing Approach to Weather Derivatives," Asia-Pacific Financial Markets, Springer;Japanese Association of Financial Economics and Engineering, vol. 11(1), pages 23-53, March.
    3. Eckhard Platen, 2004. "A Benchmark Framework for Risk Management," World Scientific Book Chapters, in: Jiro Akahori & Shigeyoshi Ogawa & Shinzo Watanabe (ed.), Stochastic Processes And Applications To Mathematical Finance, chapter 15, pages 305-335, World Scientific Publishing Co. Pte. Ltd..
    4. Eckhard Platen & Renata Rendek, 2012. "Approximating the numéraire portfolio by naive diversification," Journal of Asset Management, Palgrave Macmillan, vol. 13(1), pages 34-50, February.
    5. Eckhard Platen & Jason West & Wolfgang Breymann, 2004. "An Intraday Empirical Analysis of Electricity Price Behaviour," Research Paper Series 140, Quantitative Finance Research Centre, University of Technology, Sydney.
    6. David R. Banos & Giulia Di Nunno & Frank Proske, 2013. "Sensitivity analysis in a market with memory," Papers 1312.5116, arXiv.org, revised Jan 2017.
    7. Renata Rendek, 2013. "Modeling Diversified Equity Indices," PhD Thesis, Finance Discipline Group, UTS Business School, University of Technology, Sydney, number 23, July-Dece.
    8. T. Marquardt & Eckhard Platen & S. Jaschke, 2008. "Valuing Guaranteed Minimum Death Benefit Options in Variable Annuities Under a Benchmark Approach," Research Paper Series 221, Quantitative Finance Research Centre, University of Technology, Sydney.
    9. Kevin Fergusson & Eckhard Platen, 2013. "Real World Pricing of Long Term Cash-Linked Annuities and Equity-Linked Annuities with Cash-Linked Guarantees," Research Paper Series 338, Quantitative Finance Research Centre, University of Technology, Sydney.

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    More about this item

    Keywords

    financial and insurance market model; benchmark approach; growth optimal portfolio; numeraire portfolio; arbitrage amount; fair pricing; unit linked insurance;
    All these keywords.

    JEL classification:

    • G10 - Financial Economics - - General Financial Markets - - - General (includes Measurement and Data)
    • G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing

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