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Portfolio optimisation beyond semimartingales: shadowprices and fractional Brownian motion


  • Czichowsky, Christoph
  • Schachermayer, Walter


While absence of arbitrage in frictionlessfinancial markets requires price processes to be semimartingales, non-semimartingales can be used to model prices in an arbitrage-free way, if proportional transaction costs are taken into account. In this paper, we show, for a class of price processes which are not necessarily semimartingales, the existence of an optimal trading strategy for utility maximisation under transaction costs by establishing the existence of a so-called shadow price. This is a semimartingale price process, taking values in the bid ask spread, such that frictionless trading for that price process leads to the same optimal strategy and utility as the original problem under transaction costs. Our results combine arguments from convex duality with the stickiness condition introduced by P. Guasoni. They apply in particular to exponential utility and geometric fractional Brownian motion. In this case, the shadow price is an It^o process. As a consequence we obtain a rather surprising result on the pathwise behaviour of fractional Brownian motion: the trajectories may touch an It^o process in a one-sided manner without reflection.

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  • 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.
  • Handle: RePEc:ehl:lserod:67689

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    References listed on IDEAS

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    Cited by:

    1. Dániel Ágoston Bálint & Martin Schweizer, 2019. "Properly Discounted Asset Prices Are Semimartingales," Swiss Finance Institute Research Paper Series 19-53, Swiss Finance Institute.
    2. Huy N. Chau & Miklós Rásonyi, 2019. "Robust utility maximisation in markets with transaction costs," Finance and Stochastics, Springer, vol. 23(3), pages 677-696, July.
    3. Christoph Belak & Jörn Sass, 2019. "Finite-horizon optimal investment with transaction costs: construction of the optimal strategies," Finance and Stochastics, Springer, vol. 23(4), pages 861-888, October.

    More about this item


    portfolio choice; non-semimartingale price processes; fractional Brownian motion; proportional transaction costs; utilities on the whole real line; exponential utility; shadow price; convex duality; stickiness; optimal trading strategies;

    JEL classification:

    • C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis
    • G11 - Financial Economics - - General Financial Markets - - - Portfolio Choice; Investment Decisions

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