# Computing Correlation between Forward Rates

I have the feeling this question has an extremely simple answer but I'll put it out to the group anyway.

Imagine I have data for 3M and 6M forward rates following a lognormal process, and that I would like to find out what the correlation coefficient is between the Brownian Motions defining the dynamics of each rate process.

What is the best way of measuring it?

## 1 Answer

Assuming you know volatility of each process, correlation of brownian motions are given by the crochet.

Let $X,Y$ such that: $$\frac{dX_t}{X_t} = \mu^X dt + \sigma^X dW^X_t$$ $$\frac{dY_t}{Y_t} = \mu^Y dt + \sigma^Y dW^Y_t$$ with $d<W^X,W^Y>_t = \rho^{XY}dt$ then: $$d<X,Y>_t = \sigma^X\sigma^Y \rho^{XY} dt$$ thus:

$$\rho^{XY}{\sigma^X\sigma^Y}(t_n-t_0) =<X,Y>_{t_n}-<X,Y>_{t_0}\sim \frac{1}{n}\sum_{i=1}^n \frac{X_{t_i}-X_{t_{i-1}}}{X_{t_{i-1}}}\frac{Y_{t_i}-Y_{t_{i-1}}}{Y_{t_{i-1}}}$$

• great thanks, now it does look obvious (d'oh). If I want to compute this correlation coefficient between two forward rates, how far do I need to look back in time to obtain a meaningful correlation value? Any standard value adopted by practitioners? – Iliana Sep 1 '16 at 13:38
• it depends on how you think this is a stable relation (i.e how $\rho_{XY}$ is constant over time in reality). – MJ73550 Sep 1 '16 at 17:09