Estimating Brownian Correlation from Forward-Rate Returns

Summary

The document explains how to estimate the correlation between the Brownian motions driving two lognormal rate processes, such as forward rates. Under the stated model, each process has a proportional volatility term, and their instantaneous quadratic covariation equals the product of their volatilities and the Brownian correlation.

For observations over a time interval, the method relates accumulated covariation to the sum of paired relative increments in the two rates. If the volatilities are known, this co-movement estimate can be scaled by their product and elapsed time to infer the correlation. The answer presents the relationship as an approximation based on sampled observations, rather than a complete estimation procedure. It does not discuss data frequency, estimation error, volatility estimation, or adjustments for market features, so practical results depend on the model assumptions and quality of the observations.

Key ideas

• The instantaneous covariation of two lognormal processes depends on both volatilities and their Brownian correlation.
• Paired relative increments provide a sample-based approximation to accumulated covariation.
• Inferring correlation requires the volatilities of both rate processes.
• The stated estimator relies on the assumed diffusion model and sampled data.

Tags

# Computing Correlation between Forward Rates


# 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?

## Answer by M. Jeunesse (score 1, accepted)

https://quant.stackexchange.com/a/29936

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:

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

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.