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FRM Part II · FRM Exam Part II · Financial Correlation Modeling - Bottom-Up Approaches

A risk analyst models the correlation between two equity returns using a bottom-up approach in which each asset follows geometric Brownian motion with dW1 and dW2 linked by a correlation parameter rho. Which statement best describes how the two Wiener processes are linked in this model?

Correlated Wiener processes are built as dW2 = rho*dW1 + sqrt(1 - rho^2)*dZ with dZ independent. This keeps the variance of dW2 equal to one per unit time and gives a correlation of exactly rho with dW1, so the other forms fail.

  1. AdW2 = rho*dW1 + sqrt(1 - rho^2)*dZ, where dZ is an independent Wiener processCorrect
  2. BdW2 = rho*dW1 + (1 - rho)*dZ, where dZ is an independent Wiener process
  3. CdW2 = sqrt(rho)*dW1 + sqrt(1 - rho)*dZ, where dZ is an independent Wiener process
  4. DdW2 = dW1 + rho*dZ, where dZ is an independent Wiener process

Explanation

Correlated Brownian motions are built with a Cholesky-style decomposition: dW2 = rho*dW1 + sqrt(1-rho^2)*dZ. This gives variance rho^2 + (1-rho^2) = 1 and covariance rho with dW1. The other forms give variance not equal to 1 or the wrong correlation.

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