Linear Algebra for Quantitative Finance

On this post, I am just sharing some linear algebra tools and tricks useful in quantitative finance. This is mainly a post for myself to have a place where I can remember them for when I need them.

Finding the inverse of a 3x3 matrix

• Find the transpose of the cofactor matrix.
  • for each element of the matrix, find its minor (cross the row \(i\) and column \(j\) for element \(ij\), and find the determinant of the square matrix left)
  • alternate the signs (diagonals are positive) \(\begin{pmatrix} + & - & + \\ - & + & - \\ + & - & + \\ \end{pmatrix}\)
• Find the determinant of the 3x3 matrix
  • \(a_{1,1} \cdot (\mbox{ cofactor of } a_{1,1}) - a_{1,2} \cdot (\mbox{ cofactor of } a_{1,2}) + a_{1,3} \cdot (\mbox{ cofactor of } a_{1,3})\)

Going from Correlation to Covariance matrix

How to go from the correlation matrix and standard deviation vector to the covariance matrix?

The standard deviation vector is defined as \(\sigma = \pmatrix{\sigma_1 \\ \sigma_2 \\ \vdots \\ \sigma_n}\)

The correlation matrix is defined as \[ R = \begin{pmatrix} 1 & \rho_{12} & \cdots & \rho_{1n} \\ \rho_{21} & 1 & \cdots & \rho_{2n} \\ \vdots & \vdots & \ddots \\ \rho_{n1} & \rho_{n2} & \cdots & 1 \end{pmatrix} \]

where \(\rho_{ij}\) is the correlation between returns of asset \(i\) and asset \(j\)

we create a diagonal matrix from the standard deviation vector.

\[ S = D(\sigma) = \begin{pmatrix} \sigma_1 & 0 & \cdots & 0 \\ 0 & \sigma_2 & \cdots & 0 \\ \vdots & \vdots & \ddots \\ 0 & 0 & \cdots & \sigma_n \end{pmatrix} \] (aka all other entries being 0)

In R, we use the function diag(x) with x being a vector! Note that S is symmetric, and so \(S = S^T\)

To get the covariance matrix \(\Sigma\), we’ll pre & post-multiply the correlation matrix by the diagonal of standard deviation. Hence: \[ S \cdot R \cdot S = \Sigma = \begin{pmatrix} \sigma_1^2 & \rho_{12} \sigma_1 \sigma2 & \cdots & \rho_{1n} \sigma_1 \sigma_n \\ \rho_{21} \sigma_2 \sigma1 & \sigma2^2 & \cdots & \rho_{2n} \sigma_2 \sigma_n \\ \vdots & \vdots & \ddots \\ \rho_{n1} \sigma_n \sigma_1 & \rho_{n2} \sigma_n \sigma_2 & \cdots & \sigma_n^2 \end{pmatrix} \]