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Modeling Correlated Interest Rates and Equity Returns

Article Quant Q&A · Author: sooprise
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Summary

The document considers how to represent a reported correlation between interest rates and S&P returns. It notes that geometric Brownian motion is often used for equities, while interest rate models commonly need to keep rates positive and capture mean reversion. Vasicek, CIR, and Hull–White are named as alternatives for modeling rates.

To create correlation between rates and equities, the answer proposes modeling both processes with multidimensional Brownian drivers and sharing a Brownian component. Its example pairs a mean-reverting rate process with an equity process driven by one of those same components. The discussion is illustrative: it does not calibrate the model to the stated correlation or address estimation, parameter choice, or validation. It also emphasizes that a correlation specification requires an explicit model for the equity index as well as for rates.

Key ideas

  • A geometric Brownian motion is often a poor default for interest rates because it does not capture mean reversion.
  • Vasicek, CIR, and Hull–White are cited as common alternatives for rate dynamics.
  • Shared Brownian motion components can induce dependence between modeled rates and equity returns.
  • Representing the correlation requires specifying dynamics for both the rate and equity processes.

Tags

Fixed incomeEquitiesStatistics
Full text
# Modeling interest rates with correlation


# 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)

## Answer by SRKX (score 5)

https://quant.stackexchange.com/a/2514

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.

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.