Correlation for log-returns vs correlation for prices

Question

So as in the Black-Scholes assumptions if the returns are log-normally distributed and have a specific correlation does this still hold when moving to the price level?. So if I have for example two series of simulated returns with correlation 0.4, if I move to the price level by computing initial stockprice*exp(r) (with mean zero), does this correlation of 0.4 still hold?

Answer 1

In general the correlation of log returns and and price levels will not be the same. Suppose the asset prices $S_1$ and $S_2$ follow geometric Brownian motion, that is

$$S_1(t) = S_1(0)e^{(\mu_1- \frac{1}{2} \sigma_1^2)t}e^{\sigma_1Z_1(t)},\\ S_2(t) = S_2(0)e^{(\mu_2- \frac{1}{2} \sigma_2^2)t}e^{\sigma_2Z_2(t)},$$

where $Z_1$ and $Z_2$ are Brownian motions with $\text{corr}(Z_1(t),Z_2(t)) = E[Z_1(t)Z_2(t)]=\rho t$. The correlation of log returns over an interval of length $\delta t$ is

$$\text{corr}\left(\log \frac{S_1(t+\delta t)}{S_1(t)} , \log \frac{S_2(t + \delta t)}{S_2(t)} \right) = \rho \delta t$$

The price correlation is shown here to be

$$\rho_{S_1S_2}(t) =\frac{E[(S_1(t) - E(S_1(t))(S_2(t) - E(S_2(t))]}{\sqrt{\text{var}(S_1(t))}\sqrt{\text{var}(S_2(t))}}= \frac{e^{\rho\sigma_1\sigma_2t} - 1}{\sqrt{ e^{\sigma_1^2t}-1}\sqrt{ e^{\sigma_2^2t}-1}},$$

which clearly changes with time $t$.