auxiliary variables, and $\sigma_i$ is the volatility of stock $i$. When we relax the hindsight optimization to include all levered rebalancing rules $b\in\mathbb{R}^d$, replication becomes especially simple. At time $t$, one just looks back at the observed price history $[0,t]$, finds the best ($d$-dimensional) rebalancing rule $b(S,t)$ in hindsight, and bets the fraction $b_i(S,t)$ of wealth on stock $i$ over $[t,t+dt]$. The relation $C(S,t;T)\propto T^{d/2}$ matches the model-independent $\mathcal{O}(T^{d/2})$ super-replicating price calculated by Cover $\&$ Company.

In what follows, we work toward reducing the option price $\sqrt{T/t}\cdot\exp\{rt+z_t^2/2\}$ by replacing $\mathbb{B}=\mathbb{R}$ with $\mathbb{B}:=\{b,c\}$. In order to get the payoff $\max\{V_T(b),V_T(c)\}$ into a more practical form, we note that $V_t(b)$ is a geometric Brownian motion, since $$\label{eq:wealth}
\frac{dV_t(b)}{V_t(b)}=b\frac{dS_t}{S_t}+(1-b)\frac{dB_t}{B_t}=[r+(\mu-r)b]dt+b\sigma dW_t.$$ Solving this stochastic differential equation, we obtain (cf. Wilmott 1998, 2001) $$V_t(b)=\exp\{[r+(\mu-r)b-\sigma^2b^2/2]t+b\sigma W_t\}.$$ In order to get the payoff in terms of the observable variable $S_t$ (rather than the Wiener process $W_t$), we start with the equation $$S_t=S_0\exp\{(\mu-\sigma^2/2)t+\sigma W_t\},$$ and solve for $\sigma W_t$ in terms of $S_t$. Substituting the resulting expression into \[eq:wealth\], we get $$V_t(b)=\exp\{(r-\sigma^2b^2/2)t+b[\log(S_t/S_0)-(r-\sigma^2/2)t]\}.$$ We thus have $$\label{fortune}
\boxed{V_t(b)=\exp\{(r-\sigma^2b^2/2)t+b\sigma\sqrt{t}\cdot z_t\}},$$where $$\label{auxiliary}
\boxed{z_t:=\frac{\log(S_t/S_0)-(r-\sigma^2/2)t}{\sigma\sqrt{t}}}$$is distributed unit normal with respect to the equivalent martingale measure $\mathbb{Q}$. Note that the drift $\mu$ (which is 