Before revision:  This yields C(S,t)=T/t\cdot\,\exp ( rt+\sigma^2b(S,t)^2  t/2 )\exp(rt+ t\cdot b'\Sigma b  /2), where \Sigma is the covariance of instantaneous returns per unit time.
After revision:  This yields C(S,t)=T/t\cdot\,\exp ( rt+\sigma^2b(S,t)^2 t/2 \} , where b(S,t) is the best rebalancing rule in hindsight over the observed history [0,t]. I show that the replicating strategy amounts to betting the fraction b(S,t) of wealth on the stock over the interval [t,t+dt]. This fact holds for the general market with n correlated stocks in geometric Brownian motion: we get C(S,t)=(T/t)^{n/2\exp(rt+ t\cdot b'\Sigma b /2), where \Sigma is the covariance of instantaneous returns per unit time.