# 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?

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