Three-Dimensional Brownian Motion and the Golden Ratio Rule

Kristoffer Glover, Hardy Hulley and Goran Peskir

No 295, Research Paper Series from Quantitative Finance Research Centre, University of Technology, Sydney

Abstract: Let X =(Xt)t=0 be a transient diffusion processin (0,8) with the diffusion coeffcient s> 0 and the scale function L such that Xt ?8 as t ?8 ,let It denote its running minimum for t = 0, and let ? denote the time of its ultimate minimum I8 .Setting c(i,x)=1-2L(x)/L(i) we show that the stopping time minimises E(|? - t|- ?) over all stopping times t of X (with finite mean) where the optimal boundary f* can be characterised as the minimal solution to staying strictly above the curve h(i)= L-1(L(i)/2) for i > 0. In particular, when X is the radial part of three-dimensional Brownian motion, we find that where ? =(1+v5)/2=1.61 ... is the golden ratio. The derived results are applied to problems of optimal trading in the presence of bubbles where we show that the golden ratio rule offers a rigourous optimality argument for the choice of the well known golden retracement in technical analysis of asset prices.

Pages: 27 pages
Date: 2011-08-01
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (13)

Published as: Glover, K., Hulley, H. and Peskir, G., 2013, "Three-Dimensional Brownian Motion and the Golden Ratio Rule", The Annals of Applied Probability, 23(3), 895-922.

Downloads: (external link)
https://www.uts.edu.au/sites/default/files/qfr-archive-03/QFR-rp295.pdf (application/pdf)

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:uts:rpaper:295

Access Statistics for this paper

More papers in Research Paper Series from Quantitative Finance Research Centre, University of Technology, Sydney PO Box 123, Broadway, NSW 2007, Australia. Contact information at EDIRC.
Bibliographic data for series maintained by Duncan Ford ().