is concave over $[0,\infty)$, we see that the option price is increasing and concave separately in each of the parameters $\delta, \sigma, T$.

Given two rebalancing rules $b>c$ with distance $\delta=|b-c|$, an initial $\$1$ deposit into the horizon-$T$ replicating strategy achieves at $T$ a compound growth-rate that is exactly $$\frac{100}{T}\log\bigg\{2N\bigg(\frac{\delta}{2}\sigma\sqrt{T}\bigg)\bigg\}$$percent lower than that of the best of $\{b,c\}$ in hindsight. A $\$1$ deposit into the corresponding horizon-free strategy (that replicates the perpetual version of the option) achieves a compound-growth rate at $T$ that is at most $100\log(2)/T$ percent lower than that of the best of $\{b,c\}$ in hindsight.

The trader’s initial ($\$1$) deposit into the replicating strategy buys him $1/C_0$ units of the option at $t=0$. For the horizon-$T$ option, his wealth at expiration will be $\max\{V_T(b),V_T(c)\}/C_0$, and hence the excess continuously-compounded growth rate will be $$\frac{1}{T}\log[\max\{V_T(b),V_T(c)\}]-\frac{1}{T}\log[\max\{V_T(b),V_T(c)\}/C_0]=\frac{\log C_0(\delta,\sigma,T)}{T}.$$For the horizon-free option, the trader’s initial dollar buys him half a unit of the option at $t=0$. Thus, his wealth at $T$ will be at least half the exercise value of the option, which is $\max\{V_T(b),V_T(c)\}$. Hence, the excess continuously-compounded growth rate of the hindsight-optimized rule at $T$ is at most $$\frac{1}{T}\log[\max\{V_T(b),V_T(c)\}]-\frac{1}{T}\log[\max\{V_T(b),V_T(c)\}/2]=\frac{\log 2}{T}.$$

Consider the following robust scheme for $T:=25$ years of 