Volatility matrix
= Volatility matrix
{title2=$\sigma\sigma^T$}
A volatility matrix is the matrix of diffusion coefficients coupling asset returns to Brownian noise. For $n$ positive asset prices driven by $m$ independent standard <Brownian motions>, write
$$
\frac{dS_i(t)}{S_i(t)}=\mu_i(t)\,dt+\sum_{a=1}^m\sigma_{ia}(t)\,dW_a(t).
$$
The $n$-by-$m$ matrix $\sigma$ determines the instantaneous return <covariance matrix> $\sigma\sigma^T$: the Brownian increments satisfy $dW_a\,dW_b=\delta_{ab}\,dt$. The same covariance rate applies to log returns, while their drift includes the Itô correction.
For $\sigma=\begin{pmatrix}0.2&0\\0.1&0.3\end{pmatrix}$, the covariance-rate matrix is $\begin{pmatrix}0.04&0.02\\0.02&0.10\end{pmatrix}$. The two rows share the first noise source, producing a nonzero cross-covariance.
The volatility matrix is not uniquely determined by covariance alone: replacing $\sigma$ by $\sigma Q$ for an orthogonal $m$-by-$m$ matrix leaves $\sigma\sigma^T$ unchanged. The noise-coordinate representation and the observable covariance are therefore different objects.