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A Benchmark Approach to Investing and Pricing

In: THE KELLY CAPITAL GROWTH INVESTMENT CRITERION THEORY and PRACTICE

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  • Eckhard Platen

Abstract

This paper introduces a general market modeling framework, the benchmark approach, which assumes the existence of the numéraire portfolio. This is the strictly positive portfolio that when used as benchmark makes all benchmarked non-negative portfolios supermartingales, that is intuitively speaking downward trending or trendless. It can be shown to equal the Kelly portfolio, which maximizes expected logarithmic utility. In several ways, the Kelly or numéraire portfolio is the “best” performing portfolio and cannot be outperformed systematically by any other non-negative portfolio. Its use in pricing as numéraire leads directly to the real world pricing formula, which employs the real world probability when calculating conditional expectations. In a large regular financial market, the Kelly portfolio is shown to be approximated by well-diversified portfolios.

Suggested Citation

  • 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..
    • Handle: RePEc:wsi:wschap:9789814293501_0028
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    Cited by:

    1. Eckhard Platen & Renata Rendek, 2017. "Market Efficiency and Growth Optimal Portfolio," Papers 1706.06832, arXiv.org.
    2. Yaroslav Melnyk & Frank Thomas Seifried, 2018. "Small†cost asymptotics for long†term growth rates in incomplete markets," Mathematical Finance, Wiley Blackwell, vol. 28(2), pages 668-711, April.
    3. Claudio Fontana & Wolfgang J. Runggaldier, 2012. "Diffusion-based models for financial markets without martingale measures," Papers 1209.4449, arXiv.org, revised Feb 2013.
    4. Ke Du & Eckhard Platen & Renata Rendek, 2012. "Modeling of Oil Prices," Research Paper Series 321, Quantitative Finance Research Centre, University of Technology, Sydney.
    5. 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.
    6. Gabriel Frahm, 2013. "Pricing and Valuation under the Real-World Measure," Papers 1304.3824, arXiv.org, revised Jan 2016.
    7. Ke Du, 2013. "Commodity Derivative Pricing Under the Benchmark Approach," PhD Thesis, Finance Discipline Group, UTS Business School, University of Technology, Sydney, number 2, July-Dece.
    8. Ke Du, 2013. "Commodity Derivative Pricing Under the Benchmark Approach," PhD Thesis, Finance Discipline Group, UTS Business School, University of Technology, Sydney, number 1-2013, January-A.
    9. Mathias Barkhagen & Jörgen Blomvall & Eckhard Platen, 2016. "Recovering the real-world density and liquidity premia from option data," Quantitative Finance, Taylor & Francis Journals, vol. 16(7), pages 1147-1164, July.
    10. Claudio Fontana, 2013. "Weak and strong no-arbitrage conditions for continuous financial markets," Papers 1302.7192, arXiv.org, revised May 2014.
    11. David Feldman & Xin Xu, 2018. "Equilibrium-based volatility models of the market portfolio rate of return (peacock tails or stotting gazelles)," Annals of Operations Research, Springer, vol. 262(2), pages 493-518, March.
    12. Renata Rendek, 2013. "Modeling Diversified Equity Indices," PhD Thesis, Finance Discipline Group, UTS Business School, University of Technology, Sydney, number 23, July-Dece.
    13. Eckhard Platen & Renata Rendek, 2012. "The Affine Nature of Aggregate Wealth Dynamics," Research Paper Series 322, Quantitative Finance Research Centre, University of Technology, Sydney.
    14. Claudio Fontana, 2015. "Weak And Strong No-Arbitrage Conditions For Continuous Financial Markets," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 18(01), pages 1-34.
    15. Renata Rendek, 2013. "Modeling Diversified Equity Indices," PhD Thesis, Finance Discipline Group, UTS Business School, University of Technology, Sydney, number 4-2013, January-A.

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    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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