at time-$0$ is not uniform, even if the $b_i$ themselves are equally spaced. The endpoints $b_1$ and $b_n$ have initial weights $N(\Delta b_1\sigma\sqrt{T}/2)/C_0$ and $N(\Delta b_{n-1}\sigma\sqrt{T}/2)/C_0$, respectively, and the rest of the $b_i$ have initial weights $[N(\Delta b_{i-1}\sigma\sqrt{T}/2)-N(-\Delta b_{i}\sigma\sqrt{T}/2)]/C_0$ for $2\leq i\leq n-1$. If the $b_i$ are equally spaced, then each of the intermediate points ($2\leq i\leq n-1$) gets initial weight $[2N(\delta\sigma\sqrt{T}/2)-1]/C_0$, but the endpoints $b_1,b_n$ get the higher initial weight $N(\delta\sigma\sqrt{T}/2)/C_0$.

For any closed set $\mathbb{B}\subseteq\mathbb{R}$ of rebalancing rules (finite or infinite), the American-style version of Cover’s rebalancing option (with exercise price $K$ and payoff $\max\{\max\limits_{b\in\mathbb{B}}V_t(b)-K,0\})$ will never be exercised early in equilibrium, and its price $C_a(S_t,t)$ equals the price $C_e(S_t,t)$ of the corresponding European-style option.

For simplicity, let $V_t^*:=\max\limits_{b\in\mathbb{B}}V_t(b)$ denote the hindsight-optimized wealth over $[0,t]$, and let $b^*_t:=\arg\max\limits_{b\in\mathbb{B}}V_t(b)$ denote the best (allowable) rebalancing rule in hindsight over $[0,t]$. Consider, from the standpoint of time $t$, the following two trading strategies:

-   Invest $Ke^{-r\tau}$ dollars in the risk-free bond and buy $1$ unit of Cover’s (European-style) rebalancing option at a price of $C_e(S_t,t)$.

-   Invest $V_t^*$ dollars into the rebalancing rule $b_t^*$. That is, take the best rebalancing rule in hindsight over $[0,t]$, and adhere to that same (constant) 