Monte Carlo Simulation by Cholesky or PCA?-Part I
Jun 01, 2006
Time and again we are asked by practitioners, be they traders, structurers or derivatives sales professionals, what is the best method to do Monte Carlo simulation for two or more assets. Should they use the Cholesky decomposition or should they use the Principal Components Analysis (PCA) ?
Both these methods have their advantages and disadvantages and their use is pretty much governed by who is doing the simulation and what computing resources are at hand. However, we must add that of the two PCA, which is also called the Eigenvalue decomposition , is a much more robust and stable methodology.
Cholesky decomposition (factorization) and Principal Components Analysis (we will henceforth use the term Eigenvalue decomposition), both, perform one specific job in a multi-asset monte carlo simulation. They both help us to estimate the square root of a variance-covariance (VCV) matrix. Remember that for a single asset the measure of risk is volatility of the asset (and not the variance per se). And volatility is the square root of the variance. In the multi-asset case, variance is replaced by variance-covariance matrix and therefore, for a multi-asset portfolio the measure of risk should be something that is the square root of a variance-covariance matrix.
The purpose of a Cholesky matrix as well as that of an Eigensystem (combination of eigenvalues and eigenvectors) is to make the uncorrelated random numbers (generated for the purpose of Monte Carlo simulation) correlated using the underlying volatilities and correlation (matrix) of the assets. And the way the process is done is by operating the uncorrelated random normal numbers (that are generated by a computer program or an Excel command) with some sort of a "square root matrix".
Cholesky Decompostion (Matrix) method :
Cholesky factorization method is simpler to understand and use (both mathematically and on an Excel spreadsheet) but it is not stable; it can breakdown. Very simply put, Cholesky matrix is the square root matrix of a VCV matrix, i.e. if C is a Cholesky matrix then its decomposition is given by:
From the above it is clear that the Cholesky matrix simply computes the square root of the variance-covariance matrix and is therefore, the measure of the risk of the basket of assets. It is can be visualized as something "similar " to the portfolio volatility. This measure-Cholesky matrix-is then fed into the Monte Carlo simulation for multi-assets; by operating the uncorrelated random normal numbers by the Cholesky to convert them into correlated random numbers.
Eigenvalue Decomposition Method (PCA) :
Eigenvalue decomposition estimates an Eigensystem-an eigenvector matrix (which is of the same dimension as the VCV) and an eigenvalue vector. The eigensystem looks as two matrices, Omega and E such that
The math is slightly more complicated here but we will not go into detail but only state the results. The nature of eigenvector matrix is such that each eigenvector defines a market movement, that is by definition is independent of other movements due to the requirement that transpose of the eigenvector matrix times the eigenvector matrix is equal to the identity matrix.
Anyway, finally we estimate the matrix which gives us the square root of the variance-covariance matrix.
This matrix is then fed into the monte carlo simulation; once again the uncorrelated random numbers are operated by this matrix to convert them into correlated random numbers.
…………………………………………………. To be Continued
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