the fact that $U_1\phi(d_1)=U_2\phi(d_2)$, where $\phi(\bullet)$ is the standard normal density function. Differentiating the option price, we get $$\frac{\partial C}{\partial S}=\frac{\partial C}{\partial U_1}\frac{\partial U_1}{\partial S}+\frac{\partial C}{\partial U_2}\frac{\partial U_2}{\partial S}=N(d_1)\frac{bU_1}{S}+N(d_2)\frac{cU_2}{S},$$which is the desired result.

Thus, even if the best rebalancing rule (of any kind) in hindsight $b(S,t)=z(S,t)/(\sigma\sqrt{t})$ happens to lie between $b$ and $c$, the replicating strategy will not generally bet the hindsight-optimized fraction $\arg\max\limits_{b\in\mathbb{R}}V_t(b)$ of wealth on the stock. This phenomenon is illustrated in Figure \[fig:comparison\]. Instead, the relative weighting of $b$ and $c$ (which is initially $50/50$ at time-$0$) evolves with the observed performances $V_t(b), V_t(c)$ and the remaining life $\tau:=T-t$ of the option. For a fixed time $t$, if $U_1\to\infty$ or $U_2\to\infty$ then the on-line portfolio weight will converge to $b$ or $c$ accordingly. As $\tau\to 0$, $d_1$ tends to $\pm\infty$ and $d_2$ tends to $\mp\infty$ according as to whether $b$ or $c$ has outperformed over the known price history. Thus, small differences in the observed performances $V_t(b)$ and $V_t(c)$ get amplified in the on-line portfolio weight as $\tau\to0$. Figures \[fig:sim\] and \[fig:sim2\] simulate the performance of the replicating strategy for different parameter values over a $T:=30$ year horizon.

![The fraction of wealth bet by the replicating strategy 