Suppose we have two stocks following GBMs. Drift and volatility are calculated based on historical data. Furthermore the stocks are assumed to be correlated (i.e. they move together, if stock 1 goes up, stock does also). How can I actually simulate their future stock prices so that they stay together? For return I would use Cholesky Decomposition, but since I'm looking directly at price levels and not returns I doubt that would be correct.
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For two GMBs $$ S_i(t)=S_i(0)e^{(r-\delta_i)t+\sigma_i W_i(t)-\sigma_i^2 t/2}\,,\quad i=1,2 $$ with ${\rm Corr}[W_1(t),W_2(t)]=\rho\,t$ we have \begin{align} \mathbb E\left[S_i(t)\right]&=S_i(0)e^{(r-\delta_i)t}\,,\\[2mm] \mathbb E\left[S_i^2(t)\right]&=S_i^2(0)e^{2(r-\delta_i)t+\sigma_i^2t}\,,\\[2mm] {\rm Var}\left[S_i(t)\right]&=S_i^2(0)e^{2(r-\delta_i)t}\left(e^{\sigma_i^2t}-1\right)\,,\\[2mm] \mathbb E\left[S_1(t)S_2(t)\right]&=S_1(0)S_2(0)e^{(2r-\delta_1-\delta_2)t+\sigma_1\sigma_2\rho\,t}\,,\\[2mm] {\rm Cov}\left[S_1(t),S_2(t)\right]&=S_1(0)S_2(0)e^{(2r-\delta_1-\delta_2)t}\left(e^{\sigma_1\sigma_2\rho\,t}-1\right)\,. \end{align} Therefore, $$ \boxed{{\rm Corr}\left[S_1(t),S_2(t)\right]=\frac{e^{\sigma_1\,\sigma_2\,\rho\,t}-1}{\sqrt{e^{\sigma_1^2t}-1}\sqrt{e^{\sigma_2^2t}-1}}\,.} $$ Obviously, the correlation of $S_1$ and $S_2$ is increasing/decreasing/positive/negative when $\rho$ is increasing/decreasing/positive/negative. In other words: for all practical purposes the correlation of the returns, $\rho$ is as good as the correlation of the stocks. I am of the opinion that there is no time series of historical data that will tell you what the true correlation is.
