Correlation calibration - 2023.1 English

The instantaneous tenor correlation matrix \(\rho\) is an input to the LMM framework. These correlation matrices should be in the form of symmetric, positive and monotonically decreasing, as that would be expected from real correlations from the market data. Our implementation provides a family of parametric correlation functions that can be chosen. In order to avoid noise in calibration it is recommended to use as few parameters as possible. With that in mind the following functions are available:

One-parametric instantaneous correlation. User needs to specify \(\beta\) with a value \(0 < \beta <= 1\):

\[\rho_{i,j}=e^{-\beta|T_i-T_j|}\]

Two-parametric instantaneous correlation. User needs to specify \(\beta_0,\beta_1\) with values \(0<\beta_0\beta_1<=1\):

\[\rho_{i,j}=\beta_0+(1-\beta_0)e^{-\beta_1|T_i-T_j|}\]

Once a correlation matrix is generated, a Principal Component Analysis will be performed to reduce the dimensionality of the data to \(F\) factors, in order to calculate the reduced factor correlation matrix (\(\bar{\rho}\)) and the pseudo-sqrt of the correlation matrix (\(\eta\)) used in the LMM framework. The dimensionality reduction is applied as follows:

Calculate the \(F\) factors loadings matrix of the correlation matrix \(\rho\):

\[L = pca\_loadings(F, \rho)\]

The \(\eta\) matrix is the loadings matrix normalised by the standard deviation (sqrt of the covariance matrix’s diagonal):

\[\eta_{i,j} = \frac{L_{i,j}}{\sqrt{diag(L\cdot L^T)_i}}\]

From matrix \(\eta\), we can reduce the dimensionality of the original data set to obtain \(\tilde{\rho}\):

\[\tilde{\rho} = \eta\cdot \eta^T\]