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Outperformance Portfolio Optimization via the Equivalence of Pure and Randomized Hypothesis Testing


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  • Tim Leung
  • Qingshuo Song
  • Jie Yang


We study the portfolio problem of maximizing the outperformance probability over a random benchmark through dynamic trading with a fixed initial capital. Under a general incomplete market framework, this stochastic control problem can be formulated as a composite pure hypothesis testing problem. We analyze the connection between this pure testing problem and its randomized counterpart, and from latter we derive a dual representation for the maximal outperformance probability. Moreover, in a complete market setting, we provide a closed-form solution to the problem of beating a leveraged exchange traded fund. For a general benchmark under an incomplete stochastic factor model, we provide the Hamilton-Jacobi-Bellman PDE characterization for the maximal outperformance probability.

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Paper provided by in its series Papers with number 1109.5316.

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Date of creation: Sep 2011
Date of revision: Mar 2013
Handle: RePEc:arx:papers:1109.5316

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  1. Alexander Schied, 2004. "On the Neyman-Pearson problem for law-invariant risk measures and robust utility functionals," Papers math/0407127,
  2. Hans Föllmer & Alexander Schied, 2002. "Convex measures of risk and trading constraints," Finance and Stochastics, Springer, Springer, vol. 6(4), pages 429-447.
  3. Birgit Rudloff, 2007. "Convex Hedging in Incomplete Markets," Applied Mathematical Finance, Taylor & Francis Journals, Taylor & Francis Journals, vol. 14(5), pages 437-452.
  4. Hans FÃllmer & Peter Leukert, 1999. "Quantile hedging," Finance and Stochastics, Springer, Springer, vol. 3(3), pages 251-273.
  5. Erhan Bayraktar & Yu-Jui Huang & Qingshuo Song, 2010. "Outperforming the market portfolio with a given probability," Papers 1006.3224,, revised Aug 2012.
  6. Jarrow, Robert A., 2010. "Understanding the risk of leveraged ETFs," Finance Research Letters, Elsevier, Elsevier, vol. 7(3), pages 135-139, September.
  7. Marc Romano & Nizar Touzi, 1997. "Contingent Claims and Market Completeness in a Stochastic Volatility Model," Mathematical Finance, Wiley Blackwell, Wiley Blackwell, vol. 7(4), pages 399-412.
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Cited by:
  1. Erhan Bayraktar & Gu Wang, 2014. "Quantile Hedging in a Semi-Static Market with Model Uncertainty," Papers 1408.4848,


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