  Let $X=(X_t)_{t\ge0}$ be a transient diffusion process in $(0,\infty)$ with
the diffusion coefficient $\sigma>0$ and the scale function $L$ such that
$X_t\rightarrow\infty$ as $t\rightarrow \infty$, let $I_t$ denote its running
minimum for $t\ge0$, and let $\theta$ denote the time of its ultimate minimum
$I_{\infty}$. Setting $c(i,x)=1-2L(x)/L(i)$ we show that the stopping time
\[\tau_*=\inf\{t\ge0\vert X_t\ge f_*(I_t)\}\] minimizes
$\mathsf{E}(\vert\theta-\tau\vert-\theta)$ over all stopping times $\tau$ of
$X$ (with finite mean) where the optimal boundary $f_*$ can be characterized as
the minimal solution to
\[f'(i)=-\frac{\sigma^2(f(i))L'(f(i))}{c(i,f(i))[L(f(i))-L(i)]}\int_i^{f(i)}\frac{c_i'(i,y)[L(y)
-L(i)]}{\sigma^2(y)L'(y)}\,dy\] 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 \[\tau_
*=\inf\biggl\{t\ge0\Big\vert\frac{X_t-I_t}{I_t}\ge\varphi\biggr\},\] where
$\varphi=(1+\sqrt{5})/2=1.61\ldots$ 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.
