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Derivatives Pricing and Computational Finance / Monte Carlo Methods and Stochastic Volatility

Correlated Randoms and Low-Discrepancy Sequences

Two upgrades to the raw random-number stream. First, correlation: multi-asset baskets and the Heston model need normal vectors with a prescribed covariance structure, manufactured from independent normals by a matrix factorisation (Cholesky or spectral) — including the two-line ρ\rho-formula used in every Heston simulation. Second, uniformity: pseudo-random points cluster and leave holes; deterministic low-discrepancy (quasi-random) sequences fill space far more evenly and push the convergence rate from O(N−1/2)O(N^{-1/2}) toward O(N−1)O(N^{-1}) — provided they are used correctly, which is a matter of dimension bookkeeping, inversion, and the Brownian-bridge path construction.
Learning objectives. By the end of this note you should be able to:
  • prove that μ+AZ∼N(μ,Σ)\mu + AZ \sim \mathcal{N}(\mu, \Sigma) whenever AA⊤=ΣAA^\top = \Sigma, and state the resulting algorithm;
  • compute a Cholesky factor by the explicit recursion, and know when to use spectral decomposition instead;
  • write the 2D correlated-normal formula and apply it to correlated Brownian increments;
  • define discrepancy and low-discrepancy sequences, and compare their convergence against Monte Carlo;
  • set up quasi-Monte Carlo correctly: dimension, inversion, and the Brownian-bridge construction.

In this lesson

  1. 2.1 Correlated normal random vectors
  2. 2.2 Discrepancy and quasi-random numbers
  3. 2.3 Brownian bridge construction

The full lesson, with worked examples and exercise solutions, is for members.

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