Lesson 8 of 8 · All lessons in this module
  1. The simulation framework: four steps, three sources of scenarios
  2. Historical simulation and historical VaR
  3. Historical simulation in practice: forecasts, data gaps and bonds
  4. Bootstrap resampling
  5. Bootstrapping in practice: price paths, confidence intervals and limits
  6. Monte Carlo simulation: what it is and what it is for
  7. Monte Carlo valuation with log returns: European and Asian options
  8. Correlated scenarios: Cholesky and multivariate Monte Carlo

This lesson is part of the 2026 and 2027 curriculum. You are following the 2026 curriculum, where it is not taught in this form. Switch if you are sitting the exam under the 2026 and 2027 curriculum.

Lesson 8 of 8 · 13 min

Correlated scenarios: Cholesky and multivariate Monte Carlo

To simulate several correlated risk factors, draw independent standard normals and combine them with the Cholesky decomposition so the outputs have the right volatilities and correlation.

In short

  • Portfolios and multi-asset options need joint scenarios: the variables must move together as they do in reality.
  • Cholesky decomposition factorises the covariance (or correlation) matrix to turn independent normal draws into correlated normal draws.
  • Two variables: X1=σ1Z1X_1 = \sigma_1 Z_1 and X2=σ2(ρZ1+1−ρ2 Z2)X_2 = \sigma_2\left(\rho Z_1 + \sqrt{1 - \rho^2}\,Z_2\right), with Z1,Z2Z_1, Z_2 independent standard normals.
  • The result has means 0, standard deviations σ1,σ2\sigma_1, \sigma_2 and covariance ρσ1σ2\rho\sigma_1\sigma_2. It produces jointly normal variables; it does not add fat tails.
  • Monte Carlo strengths: wide applicability, correlation handling, forward looking, precision, flexibility. Weaknesses: computing cost, model dependency, complexity, data needs.

Unlock this lesson free for 7 days

Create a free account to get 7 days free: every lesson, video and flashcard, and the question bank. No card needed.

Start your 7-day free trialLog in