difficult to estimate) does not appear in this formula. The final wealth of the rebalancing rule $b$ is now expressed solely in terms of $z_t$, the risk-free rate $r$, the time $t$, and the volatility $\sigma$, which is easily estimated from high-frequency price data.

The Best of Two Asset Allocations in Hindsight {#two}
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Before we can price the rebalancing option with payoff $\max\{V_T(b),V_T(c)\}$, we must characterize the random outcomes under which $b$ will turn out to outperform $c$ over the interval $[0,t]$. Accordingly, we compare the exponents of $V_t(b)$ and $V_t(c)$ to obtain

For two given rebalancing rules $b>c$, b outperforms $c$ over $[0,t]$ if and only if $$\boxed{z_t\geq\frac{b+c}{2}\sigma\sqrt{t}}.$$

The best rebalancing rule (of any kind) in hindsight over $[0,t]$, denoted $b(S,t)$, is $$\boxed{b(S,t):=\arg\max_{b\in\mathbb{R}}V_t(b)=\frac{z(S,t)}{\sigma\sqrt{t}}}.$$Given any closed set $\mathbb{B}$ of rebalancing rules, the best performer in hindsight is the $b\in\mathbb{B}$ that is nearest to $b(S,t)=z(S,t)/(\sigma\sqrt{t})$.

We compute the abscissa of vertex of the parabola $b\mapsto\log\,V_t(b)$. This yields $$b(S,t)=\arg\max_{b\in\mathbb{R}}\log V_t(b)=\frac{-\sigma\sqrt{t}\cdot z_t}{2(-\sigma^2t/2)}=\frac{z_t}{\sigma\sqrt{t}}.$$Because the graph of a parabola is symmetric about its vertex, the $b\in\mathbb{B}$ that maximizes the height of this parabola is whichever element of $\mathbb{B}$ is nearest to the vertex $b(S,t)$.

We proceed to compute the cost of achieving the best 