of two rebalancing rules in hindsight, by finding the expected present value of $\max\{V_T(b),V_T(c)\}$ at time-0 with respect to the equivalent martingale measure $\mathbb{Q}$. This cost is the sum of two integrals $I_1+I_2$, where $$I_1:=\frac{\exp(-\sigma^2b^2T/2)}{\sqrt{2\pi}}\int_{\frac{b+c}{2}\sigma\sqrt{T}}^\infty\exp(-z^2/2+b\sigma\sqrt{T}\cdot z)dz$$and $$I_2:=\frac{\exp(-\sigma^2c^2T/2)}{\sqrt{2\pi}}\int_{-\infty}^{\frac{b+c}{2}\sigma\sqrt{T}}\exp(-z^2/2+c\sigma\sqrt{T}\cdot z)dz.$$ In the sequel, we will often use the following general formula (i.e. the appendix to Reiner and Rubinstein 1992): $$\boxed{
\int_{A}^Be^{-\alpha y^2+\beta y}dy=\sqrt{\frac{\pi}{\alpha}}\exp\bigg(\frac{\beta^2}{4\alpha}\bigg)\bigg[N\bigg(B\sqrt{2\alpha}-\frac{\beta}{\sqrt{2\alpha}}\bigg)-N\bigg(A\sqrt{2\alpha}-\frac{\beta}{\sqrt{2\alpha}}\bigg)\bigg]},$$where $\alpha>0$ and $N(\bullet)$ is the cumulative normal distribution function. Simplifying the two integrals, we get $$I_1=I_2=N\bigg(\frac{b-c}{2}\sigma\sqrt{T}\bigg).$$

The time-$0$ cost of achieving the best of two rebalancing rules $\{b,c\}$ in hindsight is $$\boxed{C_0(\delta,\sigma,T)=2N\bigg(\frac{\delta}{2}\sigma\sqrt{T}\bigg)}.$$where $\delta:=|b-c|$ is the distance between the two rebalancing rules.

The equilibrium price at $t=0$ of a perpetual option ($T:=\infty$) on the best of two rebalancing rules $\{b,c\}$ in hindsight is $C_0(\delta,\sigma,\infty)=\$2$.

Note that the horizon-$T$ price is independent of the interest rate $r$, and it is translation invariant in the sense that it depends only on the distance $\delta=|b-c|$ between the two rebalancing rules. We always have $1\leq C_0(\delta,\sigma,T)\leq2$; besides the perpetual version of the option, the maximum $\$2$ price also obtains if $\sigma=\infty$ or $\delta=\infty$. The minimum $\$1$ price obtains if any of the parameters $\delta, \sigma, T$ tends to 0. Since the increasing function $N(\bullet)$ 