precise number of shares on every uptick (more precisely, whenever $dS_t/S_t\geq r\,dt$) to restore the target allocation. Similarly, when the risk asset underperforms cash over $[t,t+dt]$ (i.e. when $dS_t/S_t\leq r\,dt$) the trader must buy additional shares to restore the balance. This amounts to a volatility harvesting scheme (cf. Luenberger 1998) that “lives off the fluctuations” of the underlying.

For $b=1$ the trader just buys the stock and holds it; for $b>1$ he carries a margin (debit) balance of $(b-1)V_t(b)$ dollars at time $t$. A levered rebalancing rule $b>1$ must continuously maintain a fixed debt-to-assets ratio of $1-1/b$. Thus, when the stock rises (and debt is now a smaller percentage of assets) the trader will borrow against his new wealth to buy additional shares. Similarly, when the stock falls he must sell some shares to reduce the loan-to-value ratio. This “buy high, sell low” strategy is only appropriate for stocks with relatively high drift and low volatility. Finally, for low quality underlyings one can hold all cash ($b=0$) or a continuously-rebalanced short position $b<0$.

We now imagine a trader who starts with $\$1$ and has two favored rebalancing rules $b>c$, who wants to perform well relative to the best of $\mathbb{B}:=\{b,c\}$ 