The well-known absence-of-arbitrage condition NFLVR from the fundamental theorem of asset pricing splits into two conditions, called NA and NUPBR. We give a literature overview of several equivalent reformulations of NUPBR; these include existence of a growth-optimal portfolio, existence of the numeraire portfolio, and for continuous asset prices the structure condition (SC). As a consequence, the minimal market model of E. Platen is seen to be directly linked to the minimal martingale measure. We then show that reciprocals of stochastic exponentials of continuous local martingales are time changes of a squared Bessel process of dimension 4. This directly gives a very specific probabilistic structure for minimal market models.

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.Comment: Published in at http://dx.doi.org/10.1214/12-AAP859 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

The context for this article is a continuous financial market consisting of a riskfree savings account and a single non-dividend-paying risky security. We present two concrete models for this market, in which strict local martingales play decisive roles. The first admits an equivalent risk-neutral probability measure under which the discounted price of the risky security is a strict local martingale, while the second model does not even admit an equivalent risk-neutral probability measure, since the putative density process for such a measure is itself a strict local martingale. We highlight a number of apparent anomalies associated with both models that may offend the sensibilities of the classically-educated reader. However, we also demonstrate that these issues are easily resolved if one thinks economically about the models in the right way. In particular, we argue that there is nothing inherently objectionable about either model.

Age Pension means-testing buffers retired households against shocks to wealth and may influence decumulation patterns and portfolio allocations. Simulations from a simple model of optimal consumption and allocation strategies for a means-tested retired household indicate that, relative to benchmark, eligible and neareligible households should optimally decumulate faster, and choose more risky portfolios, especially early in retirement. Empirical modelling of a Household, Income and Labour Dynamics in Australia panel of pensioner households confirms a riskier portfolio allocation by wealthier retired households. Poorer pensioner households decumulate at around 5 per cent p.a. on average; however, better-off households continue to add around 3 per cent p.a. to wealth, even when facing a steeper implicit tax rate on wealth.

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