 Volatility matrix

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, writeThe $n$-by-$m$ matrix $\sigma$ determines the instantaneous return covariance matrix $\sigma {\sigma }^T$: the Brownian increments satisfy $d{W}_{a}d{W}_b={\delta }_{ab}dt$. The same covariance rate applies to log returns, while their drift includes the Itô correction.

For $\sigma =\left(\begin{array}{cc}0.2& 0\\ 0.1& 0.3\end{array}\right)$, the covariance-rate matrix is $\left(\begin{array}{cc}0.04& 0.02\\ 0.02& 0.10\end{array}\right)$. 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.