Three-dimensional brownian motion and the golden ratio rule

Kristoffer Glover, Hardy Hulley, Goran Peskir

Research output: Contribution to journalArticlepeer-review

21 Citations (Scopus)


Let X = (Xt)t≥0 be a transient diffusion process in (0,∞) with the diffusion coefficient σ < 0 and the scale function L such that Xt → ∞ as t → ∞, let It denote its running minimum for t ≥ 0, and let θ denote the time of its ultimate minimum I∞. Setting c(i, x) = 1.2L(x)/L(i) we show that the stopping time [equation presented] minimizes E(|θ-ρ|-θ) over all stopping times τ of X (with finite mean) where the optimal boundary f. can be characterized as the minimal solution to [equation presented] 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 ψ = (1 +□5)/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 rigorous optimality argument for the choice of the well-known golden retracement in technical analysis of asset prices.

Original languageEnglish
Pages (from-to)895-922
Number of pages28
JournalAnnals of Applied Probability
Issue number3
Publication statusPublished - Jun 2013
Externally publishedYes


  • Bessel process
  • Brownian motion
  • Bubbles
  • Constant elasticity of variance model
  • Fibonacci retracement
  • Optimal prediction
  • Strict local martingale
  • Support and resistance levels
  • The golden ratio
  • The maximality principle
  • Transient diffusion


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