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Multilinear Superhedging of Lookback Options

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  • Alex Garivaltis

Abstract

In a pathbreaking paper, Cover and Ordentlich (1998) solved a max-min portfolio game between a trader (who picks an entire trading algorithm, $\theta(\cdot)$) and "nature," who picks the matrix $X$ of gross-returns of all stocks in all periods. Their (zero-sum) game has the payoff kernel $W_\theta(X)/D(X)$, where $W_\theta(X)$ is the trader's final wealth and $D(X)$ is the final wealth that would have accrued to a $\$1$ deposit into the best constant-rebalanced portfolio (or fixed-fraction betting scheme) determined in hindsight. The resulting "universal portfolio" compounds its money at the same asymptotic rate as the best rebalancing rule in hindsight, thereby beating the market asymptotically under extremely general conditions. Smitten with this (1998) result, the present paper solves the most general tractable version of Cover and Ordentlich's (1998) max-min game. This obtains for performance benchmarks (read: derivatives) that are separately convex and homogeneous in each period's gross-return vector. For completely arbitrary (even non-measurable) performance benchmarks, we show how the axiom of choice can be used to "find" an exact maximin strategy for the trader.

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  • Alex Garivaltis, 2018. "Multilinear Superhedging of Lookback Options," Papers 1810.02447, arXiv.org, revised Oct 2022.
    • Handle: RePEc:arx:papers:1810.02447
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    References listed on IDEAS

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    1. Farshid Jamshidian, 1992. "Asymptotically Optimal Portfolios," Mathematical Finance, Wiley Blackwell, vol. 2(2), pages 131-150, April.
    2. David P. Helmbold & Robert E. Schapire & Yoram Singer & Manfred K. Warmuth, 1998. "On‐Line Portfolio Selection Using Multiplicative Updates," Mathematical Finance, Wiley Blackwell, vol. 8(4), pages 325-347, October.
    3. Cox, John C. & Ross, Stephen A. & Rubinstein, Mark, 1979. "Option pricing: A simplified approach," Journal of Financial Economics, Elsevier, vol. 7(3), pages 229-263, September.
    4. Erik Ordentlich & Thomas M. Cover, 1998. "The Cost of Achieving the Best Portfolio in Hindsight," Mathematics of Operations Research, INFORMS, vol. 23(4), pages 960-982, November.
    5. Thomas M. Cover, 1991. "Universal Portfolios," Mathematical Finance, Wiley Blackwell, vol. 1(1), pages 1-29, January.
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    Cited by:

    1. Alex Garivaltis, 2019. "A Note on Universal Bilinear Portfolios," Papers 1907.09704, arXiv.org, revised Oct 2022.
    2. Alex Garivaltis, 2021. "A Note on Universal Bilinear Portfolios," IJFS, MDPI, vol. 9(1), pages 1-17, February.

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