Simulating Correlated Brownian Motions for Forward Rate Curves

Summary

The document describes a simulation question for a collection of forward rates whose dynamics are driven by Brownian motions with instantaneous correlations. Each rate follows a proportional diffusion, and the correlation structure varies over time. The author wants to generate paths for a large set of forward rates while preserving the specified dependence among their shocks.

The question correctly identifies that simulating each Brownian motion independently would omit the covariance structure, but it gives no proposed construction or answer. A simulation must generate joint increments with the stated correlation matrix at each time step; the document itself does not explain how to do this or establish that the supplied correlations form a valid matrix.

Key ideas

• Each forward rate is modeled with a diffusion driven by its own Brownian motion.
• Instantaneous cross-correlations determine the covariance among Brownian increments.
• Independent shock simulation would fail to reproduce the stated dependence.
• The document poses the simulation problem but does not give a method or empirical results.

Tags

# How to simulate from instantaneously correlated Brownian motions?


# How to simulate from instantaneously correlated Brownian motions?












Say I have obtained a distribution for different forward rates F_k such that:

$$ dF_k (t) = \sigma (t) * F_k (t) * dW_k(t) $$ with $$ dW_k(t) * dW_l(t) = \rho_{k,l} (t) dt. $$ From this I want to simulate curves for F for k > 50. How do I go about this? It doesnt seem logical to simulate the brownian motions independently, because then how is their covariance taken into account, but I weirdly enough cannot find anything on multidimensional brownian motion simulation.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.