THREE-DIMENSIONAL BROWNIAN MOTION AND THE GOLDEN RATIO RULE

• 成果类型: Article
• 署名作者: Glover, Kristoffer; Hulley, Hardy; Peskir, Goran
• 署名单位: University of Technology Sydney; University of Manchester
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/12-AAP859
• 发表日期: 2013
• 页码: 895-922
• 关键词: local time; inequalities; maximum; PRINCIPLE; options; STOCK

摘要

Let X = (X-t)(t >= 0) be a transient diffusion process in (0, infinity) with the diffusion coefficient sigma > 0 and the scale function L such that X-t -> infinity as t -> infinity, let I-t denote its running minimum for t >= 0, and let theta denote the time of its ultimate minimum I infinity. Setting c(i, x) = 1 - 2L(x)/L(i) we show that the stopping time tau(*) = inf{t >= 0 vertical bar X-t >= f(*)(I-t)} minimizes E(vertical bar theta - tau vertical bar - theta) over all stopping times tau of X (with finite mean) where the optimal boundary f(*) can be characterized as the minimal solution to f'(i) = - sigma(2)(f (i)) L' (f(i))/c(i, f(i))[L(f(i)) - L(i)] integral(i) f (i) c'(i)(i, y)[L(y) - L(i)]/sigma(2()y)L'(y) dy 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 tau(*) = inf{t >= 0 vertical bar X-t - I-t/I-t >= phi}, where phi = (1 + root 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.