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Paper Review: A Benchmark Approach To Finance Eckhard Platen Mathematical Finance, Vol. 16 No.1, 131-151

Outline. IntroductionContinuous Benchmark ModelMore Rather Than LessExpectation of Discounted GOPConclusion. 1.Introduction. This paper derives a unified framework for portfolio optimization, derivatives pricing, financial modeling, and risk measurement.General Assumption: Investors prefer

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Paper Review: A Benchmark Approach To Finance Eckhard Platen Mathematical Finance, Vol. 16 No.1, 131-151

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    1. Paper Review: A Benchmark Approach To Finance Eckhard Platen (Mathematical Finance, Vol. 16 No.1, 131-151) Zhao, Lu The Mathematical and Computational Finance Lab University of Calgary October 24, 2006

    2. Outline Introduction Continuous Benchmark Model More Rather Than Less Expectation of Discounted GOP Conclusion

    3. 1.Introduction This paper derives a unified framework for portfolio optimization, derivatives pricing, financial modeling, and risk measurement. General Assumption: Investors prefer more rather than less, in the sense that given two portfolios with the same diffusion coefficient value, the one with the higher drift is preferred.

    4. Characterizations of the Benchmark Approach: 1)Growth optimal portfolio (GOP) as a central building block; 2)Without use of expected utility functions in the capital asset pricing model; 3)Fair pricing concept, using the GOP as numeraire and the real-world probability measure as pricing measure.

    5. 2. Continuous Benchmark Model 2.1 Primary Security Accounts We consider a continuous financial market model that comprised d+1 primary security accounts. These include a savings account, which is a locally riskless primary security account whose value at time t is given by

    6. They also include d nonnegative risky primary security account processes. The j-th primary security account value satisfies the following SDE: where the k Wiener processes are independent.

    7. Primary security accounts also follows the assumption:

    8. This allows us to introduce the k-th market price for risk with respect to the k-th Wiener process: and we can reswrite (2.2) in the form:

    9. 2.2 Portfolios We call a predictable stochastic process a strategy (assumed to be self-financing), and denote by the time t value of the portfolio process. Let the portfolio process is nonzero. The j-th fraction of this portfolio is given by

    10. By (2.6) and (2.10), we get for a nonzero portfolio value the SDE: Where the k-th portfolio volatility and its appreciation rate are given by

    11. If we define the discounted portfolio value in the form then it satisfies where the k-th portfolio diffusion coefficient is given by

    12. Obviously, the discounted portfolio process has discounted drift and its aggregate diffusion coefficient and aggregate volatility used to measure the trading uncertainty are given by

    13. 2.3 Growth Optimal Portfolio The GOP, which maximizes expected logarithmic utility from terminal wealth, plays a central role in finance theory. If we apply the Ito formula for the strictly positive portfolio, we get with portfolio growth rate

    15. It is not hard to show that the GOP satisfies

    16. 3. More Rather Than Less 3.1 Optimal Portfolios

    17. Define the total market price for risk as we have the following assumption:

    18. An important investment characteristic is the Sharp ratio, defined as the ratio of the risk premium of the discounted portfolio over its aggregate volatility, that is, We have a nice Theorem of this ratio:

    20. 3.2 Markowitz Efficient Frontier

    21. 3.3 Capital Asset Pricing Model

    22. 3.4 GOP and Market Portfolio

    23. 3.5 Fair Pricing

    26. 4. Expectation of Discounted GOP Let us rewrite (2.25) in the discounted form where

    27. Define the discounted GOP drift we get the total market price for risk in the form and obtain This is a time transformed squared Bessel process of dimension four!

    28. Its transformed time at time t is given by Let us decompose the discounted GOP value at time t as where

    30. 6. Conclusion This paper identifies optimal portfolio as combinations of the market portfolio and the savings account; The Markowitz efficient frontier and Sharpe ratio can be derived naturally; The GOP equals a combination of the market portfolio and the savings account under the additional assumptions that the savings account is in net zero supply;

    31. The discounted GOP can be modeled as a time transformed squared Bessel process of dimension four, which can be interpreted as its underlying value; For the pricing of contigent claims the GOP is nominated as numeraire for fair pricing, with expectations to be taken under the real-world probability measure.

    32. THE END THANK YOU!

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