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A Benchmark Framework for Risk Management

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Abstract

The paper describes a general framework for contingent claim valuation for finance, insurance and general risk management. It considers security prices and portfolios with finite expected returns, where the growth optimal portfolio is taken as numeraire or benchmark. Benchmarked nonnegative wealth processes are shown to be supermartingales. Fair benchmarked values are conditional expectations of future benchmarked prices under the real world probability measure. Standard risk neutral and actuarial pricing formulas are obtained as special cases of fair pricing. The proposed benchmark framework covers the infinite time horizon and does not require the existence of an equivalent risk neutral pricing measure.

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  • Eckhard Platen, 2003. "A Benchmark Framework for Risk Management," Research Paper Series 113, Quantitative Finance Research Centre, University of Technology, Sydney.
    • Handle: RePEc:uts:rpaper:113
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    File URL: http://www.qfrc.uts.edu.au/research/research_papers/rp113.pdf
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    References listed on IDEAS

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    1. Wolfgang Breymann & Leah Kelly & Eckhard Platen, 2005. "Intraday Empirical Analysis and Modeling of Diversified World Stock Indices," Asia-Pacific Financial Markets, Springer;Japanese Association of Financial Economics and Engineering, vol. 12(1), pages 1-28, March.
    2. Dirk Becherer, 2001. "The numeraire portfolio for unbounded semimartingales," Finance and Stochastics, Springer, vol. 5(3), pages 327-341.
    3. Stephen A. Ross, 2013. "The Arbitrage Theory of Capital Asset Pricing," World Scientific Book Chapters, in: Leonard C MacLean & William T Ziemba (ed.), HANDBOOK OF THE FUNDAMENTALS OF FINANCIAL DECISION MAKING Part I, chapter 1, pages 11-30, World Scientific Publishing Co. Pte. Ltd..
    4. 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.
    5. David Heath & Eckhard Platen, 2002. "Perfect Hedging Of Index Derivatives Under A Minimal Market Model," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 5(07), pages 757-774.
    6. David Heath & Eckhard Platen, 2002. "Consistent pricing and hedging for a modified constant elasticity of variance model," Quantitative Finance, Taylor & Francis Journals, vol. 2(6), pages 459-467.
    7. 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.
    8. Ralf Korn & Manfred Schäl, 1999. "On value preserving and growth optimal portfolios," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 50(2), pages 189-218, October.
    9. Long, John Jr., 1990. "The numeraire portfolio," Journal of Financial Economics, Elsevier, vol. 26(1), pages 29-69, July.
    10. Eckhard Platen, 2001. "Arbitrage in Continuous Complete Markets," Research Paper Series 72, Quantitative Finance Research Centre, University of Technology, Sydney.
    11. 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.
    12. 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.
    13. Eckhard Platen, 2004. "A class of complete benchmark models with intensity-based jumps," Published Paper Series 2004-5, Finance Discipline Group, UTS Business School, University of Technology, Sydney.
    14. Duffie, Darrell & Huang, Chi-fu, 1986. "Multiperiod security markets with differential information : Martingales and resolution times," Journal of Mathematical Economics, Elsevier, vol. 15(3), pages 283-303, June.
    15. 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..
    16. David Heath & Eckhard Platen, 2003. "Pricing of index options under a minimal market model with log-normal scaling," Quantitative Finance, Taylor & Francis Journals, vol. 3(6), pages 442-450.
    17. Eckhard Platen, 2002. "Benchmark Model with Intensity Based Jumps," Research Paper Series 81, Quantitative Finance Research Centre, University of Technology, Sydney.
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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. Francesca Biagini & Jan Widenmann, 2012. "Pricing Of Unemployment Insurance Products With Doubly Stochastic Markov Chains," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 15(04), pages 1-32.
    3. Kevin Fergusson & Eckhard Platen, 2006. "On the Distributional Characterization of Daily Log-Returns of a World Stock Index," Applied Mathematical Finance, Taylor & Francis Journals, vol. 13(1), pages 19-38.
    4. 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.
    5. Hardy Hulley & Shane Miller & Eckhard Platen, 2005. "Benchmarking and Fair Pricing Applied to Two Market Models," Research Paper Series 155, Quantitative Finance Research Centre, University of Technology, Sydney.
    6. Gabriel Frahm, 2016. "Pricing And Valuation Under The Real-World Measure," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 19(01), pages 1-39, February.
    7. Eckhard Platen, 2006. "On the Pricing and Hedging of Long Dated Zero Coupon Bonds," Research Paper Series 185, Quantitative Finance Research Centre, University of Technology, Sydney.
    8. 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.
    9. Eckhard Platen, 2004. "Diversified Portfolios with Jumps in a Benchmark Framework," Asia-Pacific Financial Markets, Springer;Japanese Association of Financial Economics and Engineering, vol. 11(1), pages 1-22, March.
    10. Eckhard Platen, 2008. "A Unifying Approach to Asset Pricing," Research Paper Series 227, Quantitative Finance Research Centre, University of Technology, Sydney.
    11. David R. Banos & Giulia Di Nunno & Frank Proske, 2013. "Sensitivity analysis in a market with memory," Papers 1312.5116, arXiv.org, revised Jan 2017.
    12. Eckhard Platen, 2011. "A Benchmark Approach to Investing and Pricing," World Scientific Book Chapters, in: Leonard C MacLean & Edward O Thorp & William T Ziemba (ed.), THE KELLY CAPITAL GROWTH INVESTMENT CRITERION THEORY and PRACTICE, chapter 28, pages 409-426, World Scientific Publishing Co. Pte. Ltd..
    13. Eckhard Platen, 2008. "The Law of Minimal Price," Research Paper Series 215, Quantitative Finance Research Centre, University of Technology, Sydney.
    14. Biagini, Francesca & Groll, Andreas & Widenmann, Jan, 2013. "Intensity-based premium evaluation for unemployment insurance products," Insurance: Mathematics and Economics, Elsevier, vol. 53(1), pages 302-316.
    15. Francesca Biagini & Yinglin Zhang, 2016. "Polynomial Diffusion Models for Life Insurance Liabilities," Papers 1602.07910, arXiv.org, revised Sep 2016.
    16. Eckhard Platen, 2006. "A Benchmark Approach To Finance," Mathematical Finance, Wiley Blackwell, vol. 16(1), pages 131-151, January.
    17. Platen, Eckhard, 2006. "Portfolio selection and asset pricing under a benchmark approach," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 370(1), pages 23-29.
    18. Renata Rendek, 2013. "Modeling Diversified Equity Indices," PhD Thesis, Finance Discipline Group, UTS Business School, University of Technology, Sydney, number 23, July-Dece.
    19. Eckhard Platen, 2005. "Investments for the Short and Long Run," Research Paper Series 163, Quantitative Finance Research Centre, University of Technology, Sydney.
    20. Biagini, Francesca & Zhang, Yinglin, 2016. "Polynomial diffusion models for life insurance liabilities," Insurance: Mathematics and Economics, Elsevier, vol. 71(C), pages 114-129.
    21. Renata Rendek, 2013. "Modeling Diversified Equity Indices," PhD Thesis, Finance Discipline Group, UTS Business School, University of Technology, Sydney, number 4-2013.

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    More about this item

    Keywords

    benchmark model; growth optimal portfolio; fair pricing; risk neutral pricing; actuarial pricing;
    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
    • D52 - Microeconomics - - General Equilibrium and Disequilibrium - - - Incomplete Markets

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