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Wiener processes are used to model various assets, and I wonder why we are introducing correlations between the Wiener processes and what is the interpretation? Because when the correlations between two Wiener processes is $\rho$, the the correlations between for example stock prices in Black Scholes model will not be equal $\rho$.

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    $\begingroup$ $\rho$ is correlation of the returns $\endgroup$ Commented Mar 19, 2022 at 22:14
  • $\begingroup$ $\rho$ (the correlation of returns) is important when considering the issue of portfolio diversification. The correlation of stock prices is not particularly meaningful or useful AFAIK. $\endgroup$
    – nbbo2
    Commented Mar 20, 2022 at 16:01

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Suppose we model two stocks by \begin{align*} \text{d}S_1 &= \mu_1S_1\text{d}t+\sigma_1S_1\text{d}W_1 \\ \text{d}S_2 &=\mu_2S_2\text{d}t+\sigma_2 S_2\text{d}W_2 \end{align*} where $\text{d}W_1\text{d}W_2=\rho \text{d}t$. Then, $\rho$ indeed measures the correlation of instantaneous returns: \begin{align} \mathbb{C}\text{ov}\left(\frac{\text{d}S_1}{S_1},\frac{\text{d}S_2}{S_2}\right) = \sigma_1\sigma_2\mathbb{C}\text{ov}\left(\text{d}W_1,\text{d}W_2\right)=\sigma_1\sigma_2\rho\text{d}t. \end{align} Thus, \begin{align} \mathbb{C}\text{orr}\left(\frac{\text{d}S_1}{S_1},\frac{\text{d}S_2}{S_2}\right)=\frac{\mathbb{C}\text{ov}\left(\frac{\text{d}S_1}{S_1},\frac{\text{d}S_2}{S_2}\right)}{\sqrt{\mathbb{V}\text{ar}\left[\frac{\text{d}S_1}{S_1}\right]\mathbb{V}\text{ar}\left[\frac{\text{d}S_2}{S_2}\right]}} = \frac{\sigma_1\sigma_2\rho\text{d}t}{\sigma_1\sigma_2\text{d}t}=\rho. \end{align} You can equally consider the correlation between log-returns, $\text{d}\ln(S_1)$ and and $\ln(S_2)$. Note that I did not need to assume that drift or volatility are constant.


This also works for other model set-ups. Consider the Heston (1993) stochastic volatility model \begin{align*} \text{d}S &= \mu S\text{d}t+\sqrt{v}S\text{d}W_1 \\ \text{d}v &=\kappa(\bar{v}-v)\text{d}t+\xi \sqrt{v}\text{d}W_2 \end{align*} where $\text{d}W_1\text{d}W_2=\rho \text{d}t$. Then, the covariance between innovations (changes) in stock prices and variances is \begin{align} \mathbb{C}\text{ov}\left(\text{d}S,\text{d}v\right) = S\xi v\mathbb{C}\text{ov}\left(\text{d}W_1,\text{d}W_2\right)=S\xi v\rho\text{d}t. \end{align} Thus, \begin{align} \mathbb{C}\text{orr}\left(\text{d}S,\text{d}v\right)=\frac{\mathbb{C}\text{ov}\left(\text{d}S,\text{d}v\right)}{\sqrt{\mathbb{V}\text{ar}\left[\text{d}S\right]\mathbb{V}\text{ar}\left[\text{d}v\right]}}= \frac{S\xi v\rho\text{d}t}{S\xi v\text{d}t}=\rho. \end{align}


Thus, $\rho$ measures not only the correlations between changes in Brownian motions, but these correlations typically penetrate through to economic variables which are the actual variables of interest in the model.

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