 Solution

Assume positive deterministic initial stock prices and finite, positive terminal variances, as required to define the correlation coefficient. The explicit stochastic-exponential solutions with constant volatilities giveThe common, possibly random, accumulated interest rate cancels. If $\sigma ={\sigma }^{\prime }$, this ratio is a positive constant, so the prices are perfectly correlated.

Conversely, perfect correlation would give $S_t=c{S}_t^{\prime }+d$ almost surely for constants $c>0,d$. If $\sigma \ne {\sigma }^{\prime }$, the displayed ratio is a nonconstant variable with a lognormal distribution with full support $(0,\infty )$. If $d>0$, the ratio would always exceed $c$, and if $d<0$ it would always be below $c$, both impossible. If $d=0$, the ratio would be constant, also impossible. Therefore $\boxed{\rho (S_t,S_t^{\prime })=1⟺\sigma ={\sigma }^{\prime }}$, whenever the stated terminal variances exist and are positive. No deterministic-interest assumption is needed. Degenerate terminal prices, for example zero volatilities and deterministic rates, have undefined correlation and must be excluded.