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Portfolio optimisation beyond semimartingales: shadowprices and fractional Brownian motion

Czichowsky, Christoph and Schachermayer, Walter (2017) Portfolio optimisation beyond semimartingales: shadowprices and fractional Brownian motion. Annals of Applied Probability, 27 (3). pp. 1414-1451. ISSN 1050-5164

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Identification Number: 10.1214/16-AAP1234


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.

Item Type: Article
Official URL:
Additional Information: © 2017 Institute of Mathematical Statistics
Divisions: Mathematics
Subjects: H Social Sciences > HG Finance
Q Science > QA Mathematics
JEL classification: C - Mathematical and Quantitative Methods > C6 - Mathematical Methods and Programming > C61 - Optimization Techniques; Programming Models; Dynamic Analysis
G - Financial Economics > G1 - General Financial Markets > G11 - Portfolio Choice; Investment Decisions
Date Deposited: 12 Sep 2016 09:42
Last Modified: 20 Sep 2021 03:03
Projects: PBEZP2 137313, FA506041, MA09-003, P25815
Funders: Swiss National Science Foundation, European Research Council, Vienna Science and Technology Fund, Austrian Science Fund

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