$T=25$ years. At the same time, if stocks go through the roof, his strategy will earn the long-run market growth rate minus a $1\%$ “universality cost.”

Would-be practitioners who enjoyed these example can use Figure \[fig:regret\] to inform the choice of horizon: it plots the excess continuously-compounded growth rate for different volatilities and maturities with $\delta:=1$.

![The excess percent growth rate of the best of two rebalancing rules over the replicating strategy, for different horizons and volatilities, with $\delta:=1$.[]{data-label="fig:regret"}](excess2.png){width="375px"}

General Formulas for Pricing and Replication
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Before we can put our on-line schemes for robust asset allocation into actual practice, we must derive general time-$t$ formulas for pricing and replication of the rebalancing option under discrete hindsight optimization. Thus, we proceed to extend the above integration technique to the general situation. To simplify the notation, we let $\tau:=T-t$ denote the remaining life of the option at time $t$. Inspired by Garivaltis (2018), we start with the decomposition $$\boxed{z_T=\sqrt{\frac{t}{T}}\cdot z_t+\sqrt{\frac{\tau}{T}}\cdot y},$$where $$\boxed{y:=\frac{\log(S_T/S_t)-(r-\sigma^2/2)\tau}{\sigma\sqrt{\tau}}}$$is distributed unit normal with respect to the equivalent martingale measure and the information available at $t$. Conditional on the values of time-$t$ variables, $b$ outperforms $c$ at $T$ if and only if $$\boxed{y\geq\frac{\frac{b+c}{2}\sigma T-\sqrt{t}\cdot z_t}{\sqrt{\tau}}}.$$ Thus, the general price 