was bequeathed to us in sumultaneous papers by Margrabe (1978) and Fischer (1978). Rather than the single underlying $S_t$, one could view the (perfectly correlated) geometric Brownian motions $U_1(t):=V_t(b)$ and $U_2(t):=V_t(c)$ as underlyings of a multi-asset option with payoff $$\max\{U_1,U_2\}=\max\{U_1-U_2,0\}+U_2.$$ This amounts to a $\$1$ deposit into the rebalancing rule $c$, plus the option to exchange the final wealth of $c$ for the final wealth of $b$ at $T$. Substituting the aggregate volatility $\sigma_a:=(b-c)\sigma$ into Margrabe’s Formula (cf. Zhang 1998) yields the same result $$\label{margrabe}
\boxed{C(U_1,U_2,t)=N(d_1)U_1+N(\sigma_a\sqrt{\tau}-d_1)U_2},$$where $$\label{margrabe2}
\boxed{d_1:=\frac{\log(U_1/U_2)}{\sigma_a\sqrt{\tau}}+\frac{\sigma_a\sqrt{\tau}}{2}},$$is in agreement with (\[d1\]). Figure \[fig:iv\] plots the price and intrinsic value of the rebalancing option for different values of $S$ under the parameters $r:=0.03, T:=10, S_0:=100, t:=5, \sigma:=0.7, b:=1.5,$ and $c:=0.5$.

The horizon-$T$ replicating strategy for the best of two rebalancing rules $b>c$ in hindsight holds $$\boxed{\Delta=\frac{N(d_1)bV_t(b)+N(d_2)cV_t(c)}{S_t}}$$shares of the stock in state $(S_t,t)$, which amounts to betting the fraction $$\boxed{\hat{b}(S_t,t)=\frac{\Delta S}{C}=\frac{N(d_1)bV_t(b)+N(d_2)cV_t(c)}{N(d_1)V_t(b)+N(d_2)V_t(c)}}.$$ of wealth on the stock at $t$. Thus, the on-line fraction of wealth bet on the stock is a time- and state-varying convex combination of $b$ and $c$.

First, we note the standard relations $\partial C/\partial U_1=N(d_1)$ and $\partial C/\partial U_2=N(d_2)$, which follow by direct calculation from (\[margrabe\]), (\[margrabe2\]), and 