2015/05/10 by Christoph Czichowsky, Czichowsky, Christoph, Walter Schachermayer +1 · 2 citations
Economics, Econometrics and Finance · #60G22 #60G48 #91G10 #93E20 #Economic theories and models #FOS: Economics and business #Financial Markets and Investment Strategies #Mathematical Finance (q-fin.MF) #Portfolio Management (q-fin.PM) #Stochastic processes and financial applications
paper · pdf · doi:10.48550/arxiv.1505.02416
openalex publication_date 2015/05/10 · openalex created_date 2022/10/03 · openalex updated_date 2026/07/28
While absence of arbitrage in frictionless financial markets requires price\nprocesses to be semimartingales, non-semimartingales can be used to model\nprices in an arbitrage-free way, if proportional transaction costs are taken\ninto account. In this paper, we show, for a class of price processes which are\nnot necessarily semimartingales, the existence of an optimal trading strategy\nfor utility maximisation under transaction costs by establishing the existence\nof a so-called shadow price. This is a semimartingale price process, taking\nvalues in the bid ask spread, such that frictionless trading for that price\nprocess leads to the same optimal strategy and utility as the original problem\nunder transaction costs. Our results combine arguments from convex duality with\nthe stickiness condition introduced by P. Guasoni. They apply in particular to\nexponential utility and geometric fractional Brownian motion. In this case, the\nshadow price is an Ito process. As a consequence we obtain a rather surprising\nresult on the pathwise behaviour of fractional Brownian motion: the\ntrajectories may touch an Ito process in a one-sided manner without reflection.\n