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 )\exp(rt+ t\cdot b'\Sigma b \cdot t /2), where \Sigma is the covariance of instantaneous returns per unit time.