$C(S,t)$ is equal to the sum of two integrals $I_1+I_2$, where $$I_1:=\frac{\exp(rt-\sigma^2b^2T/2+b\sigma\sqrt{t}\cdot z_t)}{\sqrt{2\pi}}\int_{[(b+c)\sigma T/2-\sqrt{t}\cdot z_t]/\sqrt{\tau}}^\infty\exp(-y^2/2+b\sigma\sqrt{\tau}\cdot y)dy$$and $$I_2:=\frac{\exp(rt-\sigma^2c^2T/2+c\sigma\sqrt{t}\cdot z_t)}{\sqrt{2\pi}}\int_{-\infty}^{[(b+c)\sigma T/2-\sqrt{t}\cdot z_t]/\sqrt{\tau}}\exp(-y^2/2+c\sigma\sqrt{\tau}\cdot y)dy.$$ These integrals simplify to $$I_1=N\bigg(\frac{[b-c]\sigma T/2+\sqrt{t}\cdot z_t-b\sigma t}{\sqrt{\tau}}\bigg)V_t(b)$$ and $$I_2=N\bigg(\frac{[b-c]\sigma T/2-\sqrt{t}\cdot z_t+c\sigma t}{\sqrt{\tau}}\bigg)V_t(c).$$

The general cost $C(S,t)$ of achieving the best of two rebalancing rules $b>c$ in hindsight is $$\label{eq:cost2}
\boxed{C=N(d_1)V_t(b)+N(d_2)V_t(c)},$$where $$\label{d1}
\boxed{d_1:=\frac{(b-c)\sigma T/2+\sqrt{t}\cdot z_t-b\sigma t}{\sqrt{\tau}}}$$ and $$\boxed{d_2:=(b-c)\sigma\sqrt{\tau}-d_1=\frac{(b-c)\sigma T/2-\sqrt{t}\cdot z_t+c\sigma t}{\sqrt{\tau}}}.$$

A perpetual option ($T:=\infty$) on the best of two rebalancing rules $b>c$ costs $C(S,t)=V_t(b)+V_t(c)$ in state $(S_t,t)$. To delta-hedge the perpetual option, one holds $$\boxed{\Delta=\frac{bV_t(b)+cV_t(c)}{S_t}}$$shares of the underlying in state $(S_t,t)$, and therefore bets the fraction $$\boxed{\hat{b}(S_t,t):=\frac{\Delta S}{C}=\frac{bV_t(b)+cV_t(c)}{V_t(b)+V_t(c)}}$$of wealth on the underlying at $t$.

As $T\to\infty$, we see that $d_1,d_2\to+\infty$ and the option price converges to $V_t(b)+V_t(c)$. Next, one can verify by direct calculation from (\[fortune\]) and (\[auxiliary\]) that $$\frac{\partial V(b)}{\partial S}=\frac{\partial V(b)}{\partial z}\frac{\partial z}{\partial S}=\frac{bV_t(b)}{S_t}.$$Alternately, one can observe that the rebalancing rule $b$ keeps (by definition) $bV_t(b)$ dollars in the stock at time $t$, which amounts to $bV_t(b)/S_t$ shares. Thus, to replicate the sum $V_t(b)+V_t(c)$ we must own a total of $\Delta=bV_t(b)/S_t+cV_t(c)/S_t$ shares of the underlying.

We should note that our general pricing formulas could have been obtained differently, by applying the theory of “exchange options” that 