How do we model stochastic Correlations?

[Aditya Chitral]

How do we model Stochastic Correlations?


Correlation like volatility can also be stochastic (random) and modelling stochastic correlation is a bit of a challenge. Generally, even in many stochastic volatility models, such as Heston's stochastic volatility model, the correlation is assumed to be constant. However, we all know that correlation is hardly constant. Even over a small interval of time the asset correlation change. Some traders think correlation to be the most unstable of all parameters in option pricing models.



So, if correlation is stochastic, i.e. follows a random process, how do we model it?
Asset correlation, or for that matter correlation between any two variables, is a bounded process and can only take values between -1 and +1. This is the main problem in applying a generalized stochastic differential equation (for a diffusion process) to correlation and then simulating the correlation paths, just as we do for volatility. We need a stochastic process which is bounded between two limits.



We use Jacobi process to model stochastic correlation.
The random evolution of correlation between two assets is described by a square root process which has both a mean reversion term as its drift and a Gaussian random walk as its stochastic part. The random walk is bounded between -1 and +1 and is centered on the equilibrium value.
If

follows a mean-reverting Jacobi process then the stochastic evolution of

is given by the following stochastic differential equation:

In the above equation,

,

, and

are constants and

is a Weiner process given by

. The condition for not breaching the bounds is:

and

To model correlation, which is bounded between +1 and -1 we make the following transformation to the above Jacobi process:

This then gives us the process for correlation:

And the constraint of the process is:

In the above process, the stochastic process is centered around the equilibrium is centered around

. This value is constrained by the above limits.
As an example, we used the following calibrated data from an options trading desk in a bank:

. We have the starting (current period) correlation between two equity indices as 0.65, the long term (equilibrium) correlation is -0.1 and the speed of mean reversion is 10.6. The value of alpha is kept constant at one.
A sample Monte Carlo simulation path is shown below. The process for all paths in the simulation will be bounded between +1 and -1.

The correlation process would need a lot of calibration and research to estimate both the equilibrium value of correlation and the speed of mean reversion.

 

[PatM]

even more

I find this interesting. Can we generalize this to a correlation matrix? There are even more restrictive boundaries on the possible correlation of assets B and C once the correlations of A and B, and A and C, are known. I don't know what a matrix generalization of this would look like? Can you do the n x n case? That would be extremely helpful to me. Thank you.

 

[Aditya Chitral]

I find this interesting. Can we generalize this to a correlation matrix? There are even more restrictive boundaries on the possible correlation of assets B and C once the correlations of A and B, and A and C, are known. I don't know what a matrix generalization of this would look like? Can you do the n x n case? That would be extremely helpful to me. Thank you.

That is an interesting thought. I would be equally interested to know about it. The model would indeed have to be really very robust, as n increases the sample space decreases with a very high rate.

My bad, I forgot to mention the source of the article. You can access it here