Pearson Correlation Bounds for Lognormal Assets and Horizon Effects
Summary
The discussion examines why Pearson correlation between two correlated geometric Brownian motion price processes can move toward zero over time, even when the underlying Brownian shocks are highly correlated. The explanation is that Pearson correlation is sensitive to nonlinear transformations: lognormal asset values are nonlinear transformations of normally distributed quantities, so their attainable Pearson correlation need not reach the full range available for linear relationships. The answer gives expressions for upper and lower Pearson-correlation bounds in a lognormal example, with the bounds depending on the variables’ volatilities.
It suggests that measures of concordance such as Kendall’s tau or Spearman’s rho may be useful when Pearson correlation does not capture the dependence of interest. A second response emphasizes that the apparent decay also depends on the sampling frequency and horizon: annual-return inputs over very long spans show the pattern more strongly than daily inputs over shorter periods. These points help interpret spread-option dependence assumptions, but the discussion does not present a practical calibration procedure or establish which dependence measure best fits any particular market.
Key ideas
- Pearson correlation can be constrained when applied to nonlinear transformations such as lognormal asset values.
- The attainable Pearson correlation range in a lognormal setting depends on the volatilities.
- Kendall’s tau and Spearman’s rho are alternative measures of concordance.
- The observed correlation pattern depends on the return sampling frequency and the time horizon.
- The discussion raises implications for spread-option dependence modeling but does not prescribe a calibration method.
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# Correlation decay in lognormal distribution
# Correlation decay in lognormal distribution
I noticed that if you use two correlated geometric brownian motions, the correlation structure decays in time pretty fast even for really high correlation values. I think that is not replicating reality, is it? I wondered how people solve this problem in practice? Specially for pricing of spread options.It overprices long term options.
## Answer by 4pie0 (score 7, accepted)
https://quant.stackexchange.com/a/7531
well, it is absolutely in agreement with theory. the correlation as measured by Pearson's coefficient $\rho$ is linear measure in the sense that the bounds [-1,1] are obtained only when transformations of our variables are linear, so if we have variables $X$ and $Y$ then something like $aX+bY+c$ where $a,b\in\mathbb{R^*}$, $c\in\mathbb{R}$ will have boundaries [-1,1] on correlation coefficient
but
as soon as we drift from linear transformation the boundaries differ and are closer to 0, how close it depends on the type of transformation used. and since brownian motion is not linear transformation of variables of interest the boundaries vanish. as example of this I have attached below a result of my playing with two variables being lognormal distributed:
$X~(0,1)$, and $Y~(0,\sigma^2)$
it can be shown (or here) that low and upper bounds on Pearson $\rho$ in this example are
$\rho_{low}={\frac{e^{-\sigma_X\sigma_Y} -1}{\sqrt{(e^{\sigma_X^2}-1})(e^{\sigma_Y^2}-1)}}$ , $\rho_{high}={\frac{e^{\sigma_X\sigma_Y} -1}{\sqrt{(e^{\sigma_X^2}-1})(e^{\sigma_Y^2}-1)}}$
what is easy to see in my picture and almost identical to your results. how can we deal with this fact? we can use different measures of concordance*, there are many of them, and possibilities are Kendal's tau or Spearmans rho for instance.
- so what measure of concordance is then? just some function satysfying few axioms, I will refer you again to the links above. correlation is NOT one of them since it doesn't satisfy vi) axiom given by Scarsini(1984) (about pointwise convergence: it doesn't converges when the copula (pointwise) does)
## Answer by John (score 3)
https://quant.stackexchange.com/a/7523
It depends on the frequency and the horizon. For instance, I got a similar looking chart when I used annual log returns as the input to the log normal distribution and went out 250 years. With daily log returns over a few years, there isn't nearly as much of a decay. However, when you go out 250 years with daily returns you still see the pattern.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.