Terminal correlation does not identify adapted stock volatility
= Terminal correlation does not identify adapted stock volatility
The stocks $S_t=e^{B_t-t/2}$ and $S'_t=(1+S_t)/2$ start at one, are true square-integrable zero-rate <martingales>, and have <perfect positive correlation> at each positive time. Their bounded multiplicative volatilities are $1$ and $S_t/(1+S_t)$, which differ throughout. The additive intercept is what allows the same correlation but different stochastic-exponential coefficients.