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A Benchmark Approach To Finance

Listed author(s):
  • Eckhard Platen

This paper derives a unified framework for portfolio optimization, derivative pricing, financial modeling and risk measurement. It is based on the natural assumption that investors prefer more or less, in the sense that the higher drift is preferred. Each such investor is shown to hold an efficient portfolio in the sense of Markowitz with units in the market portfolio and the savings account of his or her home currency. If the market portfolio is diversified or monetary authorities aim to maximize the growth rates of the portfolios of their market participants through corresponding interest policies, then the market portfolio is the growth optimal portfolio (GOP). In this setup the capital asset pricing model follows without the use of expected utility functions or equilibrium assumptions. The expected increase of the discounted value of GOP is shown to coincide with the expected increase of its discounted underlying value. The discounted GOP has the dynamics of a time transformed squared Bessel process of dimension four. The time transformation is given by the discounted underlying value of the GOP. The squared volatility of the GOP equals the discounted GOP drift, when expressed in units of the discounted GOP. Risk neutral derivative pricing and actuarial pricing are generalized by the fair pricing concept, which uses the GOP as numeraire and the real world probability measure as pricing measure. An equivalent risk neutral martingale measure does not exist under the derived minimal market model.

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Article provided by Wiley Blackwell in its journal Mathematical Finance.

Volume (Year): 16 (2006)
Issue (Month): 1 ()
Pages: 131-151

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Handle: RePEc:bla:mathfi:v:16:y:2006:i:1:p:131-151
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  1. David Heath & Eckhard Platen, 2004. "Understanding the Implied Volatility Surface for Options on a Diversified Index," Research Paper Series 128, Quantitative Finance Research Centre, University of Technology, Sydney.
  2. 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.
  3. Merton, Robert C, 1973. "An Intertemporal Capital Asset Pricing Model," Econometrica, Econometric Society, vol. 41(5), pages 867-887, September.
  4. Eckhard Platen, 2001. "A Minimal Financial Market Model," Research Paper Series 48, Quantitative Finance Research Centre, University of Technology, Sydney.
  5. 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.
  6. Eckhard Platen, 2001. "Arbitrage in Continuous Complete Markets," Research Paper Series 72, Quantitative Finance Research Centre, University of Technology, Sydney.
  7. Eckhard Platen, 2004. "A Benchmark Framework for Risk Management," World Scientific Book Chapters, in: Stochastic Processes And Applications To Mathematical Finance, chapter 15, pages 305-335 World Scientific Publishing Co. Pte. Ltd..
  8. 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.
  9. Eckhard Platen, 2003. "Modeling the Volatility and Expected Value of a Diversified World Index," Research Paper Series 103, Quantitative Finance Research Centre, University of Technology, Sydney.
  10. Ross, Stephen A., 1976. "The arbitrage theory of capital asset pricing," Journal of Economic Theory, Elsevier, vol. 13(3), pages 341-360, December.
  11. Dirk Becherer, 2001. "The numeraire portfolio for unbounded semimartingales," Finance and Stochastics, Springer, vol. 5(3), pages 327-341.
  12. Eckhard Platen, 2003. "Pricing and Hedging for Incomplete Jump Diffusion Benchmark Models," Research Paper Series 110, Quantitative Finance Research Centre, University of Technology, Sydney.
  13. Long, John Jr., 1990. "The numeraire portfolio," Journal of Financial Economics, Elsevier, vol. 26(1), pages 29-69, July.
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