in hindsight. Accordingly, we create for him the financial derivative whose final payoff at $T$ is $$\boxed{V_T^*:=\max\{V_T(b),V_T(c)\}}.$$ Ordentlich and Cover (1998) investigated the best unlevered rebalancing rule in hindsight, with payoff $V_T^*:=\max\limits_{0\leq b\leq1}{V_T(b)}$. They found the time-0 price of this contingent claim to be $$C_0=1+\sigma\sqrt{\frac{T}{2\pi}}.$$The owner of this rebalancing option (cf. Garivaltis 2018) will compound his money at the same asymptotic rate as the best unlevered rebalancing rule in hindsight. Indeed, the final excess continuously-compounded growth rate of the best rebalancing rule in hindsight over that of the replicating strategy is $\log\big\{1+\sigma\sqrt{T/(2\pi)}\big\}/T$, which tends to 0 as $T\to\infty$. This growth rate spread obtains deterministically, regardless of the realized price path $(S_t)_{0\leq t\leq T}$.

Garivaltis (2018) extended the Ordentlich-Cover (1998) analysis by computing the general time-$t$ price $C(S,t)$ of Cover’s rebalancing option for both levered and unlevered hindsight optimization. For levered hindsight optimization (with payoff $V_T^*:=\max\limits_{b\in\mathbb{R}}{V_T(b)}$), Garivaltis (2018) found the general pricing formula $$C(S,t)=\sqrt{\frac{T}{t}}\exp\{rt+z_t^2/2\},$$ where $$z_t:=\frac{\log(S_t/S_0)-(r-\sigma^2/2)t}{\sigma\sqrt{t}}$$is an auxiliary variable that is distributed unit normal with respect to the equivalent martingale measure $\mathbb{Q}$. More generally, for a Black-Scholes market with $d$ correlated stocks in geometric Brownian motion, Garivaltis (2018) found that $$C(S,t)=\bigg(\frac{T}{t}\bigg)^{d/2}\cdot\exp\{rt+z_t'R^{-1}z_t/2\},$$where $R:=[\rho_{ij}]_{d\times d}$ is the correlation matrix of instantaneous returns, $$z_{it}:=\frac{\log(S_{it}/S_{i0})-(r-\sigma_i^2/2)t}{\sigma_i\sqrt{t}},$$are 