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Modeling The Volatility And Expected Value Of A Diversified World Index

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  • ECKHARD PLATEN

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

Abstract

This paper considers a diversified world stock index in a continuous financial market with the growth optimal portfolio (GOP) as reference unit or benchmark. Diversified broadly based indices and portfolios, which include major world stock market indices, are shown to approximate the GOP. It is demonstrated that a key financial quantity is the trend of a world index. It turns out that it can be directly observed since the expected increments of the index equal four times those of the quadratic variation of its square root. Using a world stock index as approximation of the discounted GOP it is shown that, in reality, the trend of the discounted GOP does not vary greatly in the long term. This leads for a diversified world index to a natural model, where the index is a transformed square root process of dimension four. The squared index volatility appears then as the inverse of the square root process. This feature explains most of the properties of an index and its volatility.

Suggested Citation

  • Eckhard Platen, 2004. "Modeling The Volatility And Expected Value Of A Diversified World Index," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 7(04), pages 511-529.
    • Handle: RePEc:wsi:ijtafx:v:07:y:2004:i:04:n:s0219024904002499
    DOI: 10.1142/S0219024904002499
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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. Rama Cont & Jose da Fonseca, 2002. "Dynamics of implied volatility surfaces," Quantitative Finance, Taylor & Francis Journals, vol. 2(1), pages 45-60.
    3. Long, John Jr., 1990. "The numeraire portfolio," Journal of Financial Economics, Elsevier, vol. 26(1), pages 29-69, July.
    4. Eckhard Platen, 2001. "Arbitrage in Continuous Complete Markets," Research Paper Series 72, Quantitative Finance Research Centre, University of Technology, Sydney.
    5. Merton, Robert C, 1973. "An Intertemporal Capital Asset Pricing Model," Econometrica, Econometric Society, vol. 41(5), pages 867-887, September.
    6. Leah Kelly, 2004. "Inference and Intraday Analysis of Diversified World Stock Indices," PhD Thesis, Finance Discipline Group, UTS Business School, University of Technology, Sydney, number 1-2004.
    7. Eckhard Platen, 2003. "Diversified Portfolios in a Benchmark Framework," Research Paper Series 87, Quantitative Finance Research Centre, University of Technology, Sydney.
    Full references (including those not matched with items on IDEAS)

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    More about this item

    Keywords

    Diversified world stock index; stochastic volatility; benchmark model; growth optimal portfolio; square root process; market activity; JEL classification code G10; JEL classification code G13; primary 90A12; secondary 60G30; secondary 62P20;
    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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