PORTFOLIO OPTIMISATION BEYOND SEMIMARTINGALES: SHADOW PRICES AND FRACTIONAL BROWNIAN MOTION

• 成果类型: Article
• 署名作者: Czichowsky, Christoph; Schachermayer, Walter
• 署名单位: University of London; London School Economics & Political Science; University of Vienna
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/16-AAP1234
• 发表日期: 2017
• 页码: 1414-1451
• 关键词: utility maximization; Optimal investment; transaction costs; fundamental theorem; incomplete markets; consumption; martingale; arbitrage; PROPERTY; Duality

摘要

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 Ito process. As a consequence, we obtain a rather surprising result on the pathwise behaviour of fractional Brownian motion: the trajectories may touch an Ito process in a one-sided manner without reflection.