Portfolio optimisation beyond semimartingales: shadowprices and fractional Brownian motion

Christoph Czichowsky and Walter Schachermayer

LSE Research Online Documents on Economics from London School of Economics and Political Science, LSE Library

Abstract: 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.

Keywords: 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 (search for similar items in EconPapers)
JEL-codes: C61 G11 (search for similar items in EconPapers)
Date: 2017-06-01
New Economics Papers: this item is included in nep-upt
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Citations: View citations in EconPapers (6)

Published in Annals of Applied Probability, 1, June, 2017, 27(3), pp. 1414-1451. ISSN: 1050-5164

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