continuously-rebalanced portfolio over $[t,T]$.

Observe that Strategy 1 has a final payoff of $\max\{V_T^*,K\}$ and Strategy 2 has a final payoff of $V_T(b_t^*)$. Since the payoff at $T$ of Strategy 1 is guaranteed to be at least as great as that of Strategy 2, the initial investment of $Ke^{-r\tau}+C_e(S_t,t)$ dollars into Strategy 1 must be greater or equal to the investment $V_t^*$ that is required for Strategy 2. Thus, we have the inequalities $$C_a(S_t,t)\geq C_e(S_t,t)\geq V_t^*-Ke^{-r\tau}\geq V_t^*-K.$$Hence, since the price of an American rebalancing option always exceeds the exercise value, the option “is worth more alive than dead” and will never be exercised in equilibrium. On account of the fact that early exercise rights are worthless anyhow, we must therefore have $C_a(S_t,t)=C_e(S_t,t)$.

We remark that this is a general model-independent result that applies equally well to rebalancing rules $b\in\mathbb{B}\subseteq\mathbb{R}^d$ over arbitrary $d$-dimensional stock markets. The dominance argument only requires the market (and the set $\mathbb{B}$) to admit a well-defined hindsight-optimized rebalancing rule $b_t^*:=\arg\max\limits_{b\in\mathbb{B}}V_t(b)$. For $\mathbb{B}:=\{1\}$ the best rebalancing rule in hindsight is just $b_t^*=1$ and we get $V_t^*=S_t$; this recovers the proof given by Merton (1973, 1990) of the no-exercise theorem for vanilla call options. The special cases $\mathbb{B}:=\mathbb{R}^d$ and 