Arbitrage problems with reflected geometric Brownian motion
by Dean Buckner1, Kevin Dowd2 and Hardy Hulley3
1The Eumaeus Project, London, United Kingdom
(email: d.e.buckner@eumaeus.org)
2Durham University Business School, Mill Hill Lane, Durham DH1 3LB, United Kingdom
(email: kevin.dowd@durham.ac.uk)
3Finance Department, UTS Business School, University of Technology Sydney, P.O. Box 123, Broadway, NSW 2007, Australia
(email: hardy.hulley@uts.edu.au)
Abstract
Contrary to the claims made by several authors, a financial market model in which the price of a risky security follows a reflected geometric Brownian motion is not arbitrage-free. In fact, such models violate even the weakest no-arbitrage condition considered in the literature. Consequently, they do not admit numéraire portfolios or equivalent risk-neutral probability measures, which makes them totally unsuitable for contingent claim valuation. Unsurprisingly, the published option pricing formulae for such models violate classical no-arbitrage bounds.
Key words:
Reflected geometric Brownian motion, Arbitrage, Local time, Contingent claim valuation
JEL Classification: C6, G12, G13
Mathematics Subject Classification (2020): 60H10, 91G15, 91G16