if the practitioner sticks to the Plan for the next $T=25$ years.

The horizon-$T$ replicating strategy for the best of the rebalancing rules $b_1<b_2<\cdot\cdot\cdot<b_n$ in hindsight holds $$\boxed{\Delta=N(-d_{21})\frac{b_1V_t(b_1)}{S_t}+\sum\limits_{i=2}^{n-1}[N(d_{1,i-1})-N(d_{2i})]\frac{b_iV_t(b_i)}{S_t}+N(d_{1,n-1})\frac{b_nV_t(b_n)}{S_t}}$$shares of the stock in state $(S_t,t)$, thereby betting the fraction $\hat{b}(S,t)=\Delta S/C$ of its bankroll on the stock. This amounts to a time- and state-varying convex combination of the $b_i$. As $\tau\to0$, the option price converges to $U_{i^*}:=\max\limits_{1\leq i\leq n}U_i$ and the fraction of wealth bet by the replicating strategy converges to $\arg\max\limits_{b\in\mathbb{B}}V_t(b)$ if this set is a singleton; if $\arg\max\limits_{b\in\mathbb{B}}V_t(b)=U_{i^*}=U_{i^*+1}$ has two distinct points, then $\hat{b}$ converges to the midpoint $(b_{i^*}+b_{i^*+1})/2$ as $\tau\to0$. The horizon-free replicating strategy (corresponding to the perpetual version of the option) bets the performance-weighted average $$\hat{b}(S_t,t)=\frac{\sum\limits_{i=1}^nb_iV_t(b_i)}{\sum\limits_{i=1}^nV_t(b_i)}$$of the rebalancing rules $b_i$, which converges almost surely to $$\arg\max_{b\in\mathbb{B}}\bigg\{(\mu-r)b-\frac{\sigma^2b^2}{2}\bigg\}=\arg\min_{b\in\mathbb{B}}\bigg|b-\frac{\mu-r}{\sigma^2}\bigg|,$$i.e. it converges to whichever element of $\mathbb{B}$ is closest to the continuous time Kelly rule (cf. Luenberger 1998).

Note that the pricing formula (\[combo\]) is a linearly homogeneous function of the underlyings $(U_1,...,U_n)$. By Euler’s theorem for homogeneous functions, we therefore have the relation $$C=\sum\limits_{i=1}^n\frac{\partial C}{\partial U_i}U_i.$$Accordingly, by direct calculation on (\[combo\]) one can (carefully) verify the partial derivatives $$\frac{\partial C}{\partial U_1}=N(-d_{21}),$$ $$\frac{\partial C}{\partial U_i}=N(d_{1,i-1})-N(d_{2i})\text{ for } 2\leq i\leq n-1,$$ $$\frac{\partial 