# Modeling interest rates with correlation

I'm trying to model interest rates, and will use the following equation:

$dr = \mu r dt + \sigma r dW$

I'm also being told that interest rates are 40% correlated to S&P returns. How can I include correlation to the S&P into this equation? (It is pretty weird that interest rates are being correlated to S&P returns)

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First, that is a diffusion equation usually used to model equities. Mainly because interest rates are assume to mean revert to some long range target. A better model would be the Vasicek model. Second, what is the function that is dependent on the diffusion? – strimp099 Dec 4 '11 at 14:40
What short rate model is this that multiplies $r$ with drift and $r$ with diffusion? What probability measure is this specified under? – Jase Dec 7 '12 at 1:55

The model you assume for the interest rate process is a Geometric Brownian Motion.

As strimp099 highlights in his comments it is mainly used to model equities because you most of the time want your interest rate models to be positive and mean reverting.

A few models have been developed: Vasicek, CIR, HW. You could have a pick in there.

As for the correlation, the idea is to make your process $r_t$ rely on a multi-dimensional Brownian motion, for example 2-dimension, where the first one is specific to the interest rate process and the other one is the brownian motion used in the equities model (representing your S&P 500).

Example:

$$dr_t=a(b-r_t)dt+\sigma(dW^1_t+dW^2_t)$$

with

$$dS_t = \mu S_t dt + \sigma S_t dW^2_t$$

This is how you "induce" correlation; by having the same Brownian motion in the dynamics of the two processes. You could also have $r_t$ occuring somewhere in $dS_t$.

In your question, you discuss about the S&P, but it's really important to understand that including the correlation requires you to define a model for S&P as well, which is the $S_t$ in my example.

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For what it's worth, if he is trying to model interest rates with, say, a 1 week horizon then GBM is not a horrible approximation. The mean reversion term would be negligible over such a period. – Brian B Dec 5 '11 at 17:32