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Shadow prices, fractional Brownian motion, and portfolio optimisation under transaction costs

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  • Christoph Czichowsky
  • R'emi Peyre
  • Walter Schachermayer
  • Junjian Yang

Abstract

We continue the analysis of our previous paper (Czichowsky/Schachermayer/Yang 2014) pertaining to the existence of a shadow price process for portfolio optimisation under proportional transaction costs. There, we established a positive answer for a continuous price process $S=(S_t)_{0\leq t\leq T}$ satisfying the condition $(NUPBR)$ of "no unbounded profit with bounded risk". This condition requires that $S$ is a semimartingale and therefore is too restrictive for applications to models driven by fractional Brownian motion. In the present paper, we derive the same conclusion under the weaker condition $(TWC)$ of "two way crossing", which does not require $S$ to be a semimartingale. Using a recent result of R.~Peyre, this allows us to show the existence of a shadow price for exponential fractional Brownian motion and $all$ utility functions defined on the positive half-line having reasonable asymptotic elasticity. Prime examples of such utilities are logarithmic or power utility.

Suggested Citation

  • Christoph Czichowsky & R'emi Peyre & Walter Schachermayer & Junjian Yang, 2016. "Shadow prices, fractional Brownian motion, and portfolio optimisation under transaction costs," Papers 1608.01415, arXiv.org.
    • Handle: RePEc:arx:papers:1608.01415
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    References listed on IDEAS

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    Cited by:

    1. Czichowsky, Christoph & Schachermayer, Walter, 2017. "Portfolio optimisation beyond semimartingales: shadowprices and fractional Brownian motion," LSE Research Online Documents on Economics 67689, London School of Economics and Political Science, LSE Library.
    2. Miklos Rasonyi, 2017. "On utility maximization without passing by the dual problem," Papers 1702.00982, arXiv.org, revised Mar 2018.

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