Cholesky decomposition
Quantitative Finance glossary · Numerical Methods & Optimization
Definition
Factorisation of a symmetric positive-definite matrix boldsymbolΣ = boldsymbolLboldsymbolL^T into a lower-triangular boldsymbolL. If boldsymbolZ sim mathcalN(boldsymbol0, boldsymbolI), then boldsymbolLboldsymbolZ sim mathcalN(boldsymbol0, boldsymbolΣ) — the standard recipe for correlated Gaussian samples in Monte Carlo. When boldsymbolΣ is only positive-semidefinite (rank-deficient, e.g. shrunk correlation matrices), Cholesky fails and one falls back on the eigendecomposition boldsymbolΣ = boldsymbolUboldsymbolLambdaboldsymbolU^T with boldsymbolL = boldsymbolUboldsymbolLambda^1/2.
Cholesky decomposition in plain English
Imagine you have a machine that spits out totally random, unrelated numbers, but you actually want numbers that move together in a realistic way — like how a stock and the overall market tend to rise and fall as a partnership. Cholesky decomposition is the mathematical 'recipe card' that turns independent randomness into correlated randomness. It takes a table that describes how much each pair of things should move together (called boldsymbolΣ) and splits it into a simpler triangular piece boldsymbolL so that boldsymbolΣ = boldsymbolLboldsymbolL^T; multiplying your plain random numbers by boldsymbolL stamps the desired relationships onto them. This is the everyday workhorse behind Monte Carlo simulations, where finance folks generate thousands of fake-but-realistic market scenarios; if the table is 'degenerate' (some info is redundant), the recipe breaks and people switch to a close cousin called the eigendecomposition.
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