Correlated Wiener processes

How to generate two series of random numbers, but construct these series so that they are linked with a specific level of correlation?

Wiener processes are an important part of stochastic models in financial mathematics. In some models, like the SABR model, Wiener processes (also called Brownian process in reference to the Brownian motion in physics) are correlated. In theory it is easy to imagine. But when it comes to build it in practice, when coding a pricer using this model for example, it becomes necessary to build the correlated Wiener processes. So how to use the known characteristics of the correlated Wiener processes to build the expression of one Wiener process according to the other.

Actually this question was asked to me during a very interesting interview, as some sort of a brain teaser. So I found it interesting to re-do the exercise here, and maybe it can be of some help to some of you...

What is the use of correlated brownian motions (or Wiener processes)?

In models like SABR, which is used a lot in derivative products and exotic products, the underlying level and the volatility are modeled jointly. This model of stochastic volatility is very widely spread and useful to know, do refer to the article I made on this SABR model and its principles. Not only is the volatility used to scale the moves on the underlying (which is not very original), the random changes in the volatility and the underlying are also correlated!

Focusing only on the equation describing this Wiener processes:  $ dW1_t .  dW2_t= \Rho. dt$ where W1 and W2 are Wiener processes, and their instantaneous increments $ dW1_t $ and $ dW2_t$ are correlated with the correlation level Rho.

How to define a Wiener process according to the other using their correlation only?

We know that $\Corr(W1,  W2) = Rho$ then what can we deduct for $dW2_t$? Could we write W2 as a function of W1?

First we know that W2 follows a standard normal distribution just like W1. This implies a mean of 0 and a standard deviation of 1. additionally the two must have a correlation of Rho. We need to ensure that these characteristics are true for W2. The rest is mostly intuition guided trial and error.

For simplicity we will assume that W1 follows a standard normal distribution with mean of 0 and standard deviation of 1.

What if $W2 = \Rho . W1$ ? Then we do have E(W2) = Rho . E(W1) = 0 and Corr(W1, W2)  = Rho , but $V(W2) = {Rho}^2  . V(W1) =  {Rho}^2$. In addition we obvisouly see that in practice it doesn't work! If the correlation is 0, then W2 is always equal to 0, which is obvisouly wrong. We need to add an additional random component to W2, following the same distribution as W1 but independent to W1. Lets call this component W3.

What if $W2 =\Rho . W1 + (1-\Rho) . W3 $ ? Again we do have E(W2) = 0, but $V(W2) = {\Rho}^2  . V(W1) + (1- \Rho)^2  . V(W3) = {\Rho}^2 + (1- \Rho)^2  = 1 + 2.  {\Rho}^2 -2. {\Rho} $. So this is wrong again.

What if $W2 =\Rho . W1 + √{1-\Rho^2} . W3 $ ? We do have E(W2) = 0, and $V(W2) = {\Rho}^2  . V(W1) + (1-\Rho^2)  . V(W3) = {\Rho}^2 + 1- \Rho^2  = 1 $

Finally when it comes to correlation :

$$Corr(W1,W2) = {Cov(W1,W2)} / {Std(W1).Std(W2)} = Cov(W1,W2) = E(W1W2) - E(W1).E(W2) = E(W1W2)$$

$$ E(W1W2) = \Rho . E(W1^2) + √{1-\Rho^2} . E(W3.W1) =\Rho . E(W1^2) = \Rho . V(W1) = \Rho $$

 

So we end up with Corr(W1, W2) = Rho

The expression of two correlated Wiener processes

We can then conclude that if W1, W2 and W3 are brownian motions following a standard normal distribution, and correlation between W1 and W2 is Rho, and W1 and W3 are independant, we can express W2 as:

$$W2 =\Rho . W1 + √{1-\Rho^2} . W3 $$

Note that if we generate a large number of these random variables, and we compute the population correlation, this number should get closer to Rho on average as the number of generated numbers increases. On smaller samples though, the observed correlation can be relatively different from Rho.

Graphical analysis of correlated Wiener processes, practical understanding of correlation

We can observe several cases:

If Rho=1, then W1 = W2

If Rho = 0 then W2 = W3 and W1 and W2 are independent and uncorrelated on average.

If Rho=-1, then W2 = -W1.

W2 is somehow a mirrored version of W1.

If Rho = -0.5, W1 and W2 mostly move in opposite directions, but in a semi random way.

If Rho = 0.5, W1 and W2 mostly move in the same direction, but in a semi random way.

I hope this is now clearer!

Matthieu Liatard
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