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Volatilities and Correlations of Cross Rates, a Geometrical Understanding

juan martinez

01/12/2021

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In this post we will show how the properties of a triangle can be used to intuitively obtain insights about the volatilities and correlations of currency pairs.

Once the dominant branch of mathematics, geometry plays now a secondary role. However, its graphical arguments still seem to be better suited for our brains, as they are often easier to understand and to remember. This is the path followed by Walter & Lopez in “The Shape of Things in a Currency Trio” [1], which we introduce below.

Cross Currency Pairs

Since USD is the most liquid currency in the Foreign Exchange market, exchange rates are mostly quoted against it (with quite a few exceptions which we will ignore) as any other rate can be obtained as a combination of these quotes. For instance, EURCHF (one of those aforementioned exceptions) could be obtained from EURUSD and USDCHF through the following operation:

\(\)

$$ \text{EURCHF} =
\frac{\text{EURUSD}}{\text{CHFUSD}} $$

Where you can think of the USD as cancelling out in the two rates. Note that we could have just multiplied by USDCHF, but soon it will become clear why we chose this approach.

Since we will be dealing with volatilities and correlations, we are more interested in the above relationship expressed in terms of log returns:

$$ r_t^{ \text{EURCHF} } =
\log \left( \frac{ \text{EURCHF}_t}{ \text{EURCHF}_{t-1}} \right) =
\log \left( \frac{ \frac{\text{EURUSD}_t}{\text{CHFUSD}_t} }{ \frac{\text{EURUSD}_{t-1}}{\text{CHFUSD}_{t-1}} } \right) =
$$

$$
\log \left( \frac{ \text{EURUSD}_t}{ \text{EURUSD}_{t-1}} \right) – \log \left( \frac{ \text{CHFUSD}_t}{ \text{CHFUSD}_{t-1}} \right) =
r_t^{ \text{EURUSD} } – r_t^{ \text{CHFUSD} }. \quad (1)
$$

To simplify notation, from now on whenever we are referencing a currency pair (e.g. EURUSD), we will mean its returns.

Using the properties of the variance of the sum of two variables, we can express EURCHF volatility (\(\sigma_{\text{EURCHF}}\)) as a function of the other two rates EURUSD and USDCHF:

$$
\sigma_{\text{EURCHF}}^2 =
\sigma_{\text{EURUSD}}^2 + \sigma_{\text{CHFUSD}}^2 – 2 \sigma_{\text{EURUSD}}^{\phantom{1}} \sigma_{\text{CHFUSD}}^{\phantom{1}} \rho_{\text{EURUSD}, \text{CHFUSD}}^{\phantom{1}} , \quad(2)
$$

where \(\rho\) is the correlation.

A Triangle to Rule Them All

For those familiar with trigonometry the above may ring a bell, in fact it very much looks like the cosine rule!

$$
a^2 = b^2 + c^2 – 2 b c \cos(\alpha), \quad(3)
$$

where \(\alpha\) is the angle formed by \(b\) and \(c\).

Note that even though we did this for the cross rate, we could have obtained a similar expression for any of the other two rates considered. Hence, by identifying terms, we can consider that volatilities are the sides of a triangle and correlations the cosine of its angles. This leads us to the following graphical representation:

Figure 1: An example of a currency triangle representation for EUR, CHF and USD. Data corresponds to 2006Q3.

In the image we can see how EUR and CHF, being both european currencies, are much closer to each other; reflecting also less potential for diversification. This means that, for example, for a EUR investor with USD exposure, CHF would be much more attractive than for a USD investor with EUR exposure. In summary, this approach helps present information in a condensed yet interpretable way.

Translating Triangle Properties Into Financial Insights

Apart from the nice visual representation, how can we make use of this? Well, we can now take advantage of basic triangle properties to help us understand some characteristics of these financial series.

1. Three sides define a triangle, meaning given three volatilities we can obtain the three correlations. This is especially relevant for calculating implied correlations from option prices, but that is a topic for another day.

2. Three angles do not define a triangle. Given three correlations, we cannot obtain the corresponding volatilities.

3. Given two angles, we can obtain the third, as they must add up to 180º. Equivalently, the same applies to correlations. This fact explains point 2., one needs three independent data to build a triangle and the third angle is just a linear combination of the other two.

$$
\pi = \arccos(\rho_{x,y}) + \arccos(\rho_{x,z}) + \arccos(\rho_{y,z}) \quad (4)
$$

We can use this information to place some constraints on the kinds of correlation matrices we can expect. For any three independent variables, the only condition the correlation matrix

\[
\text{Corr} =
\begin{bmatrix}
1 & \rho_{x,y} & \rho_{x,z} \\
\rho_{x,y} & 1 & \rho_{y,z} \\
\rho_{x,z} & \rho_{y,z} & 1 \\
\end{bmatrix}
\]

must meet is to be non-negative definite. This is due to the covariance matrix (the generalization of variance for higher dimensions) from which correlation is obtained, also having to be non-negative definite (which is the equivalent of being non-negative in 1D). A more intuitive explanation is based on the idea that given \(\rho_{x,y}\) and \(\rho_{x,z}\) , \(\rho_{y,z}\) is not completely unconstrained if one wants to preserve consistency. In summary, the need for coherence translates into the following quadratic inequality (non negative determinant):

$$
| \text{Corr} | = 1 – 2 \rho_{x,y} \rho_{x,z} \rho_{y,z} – \rho_{x,y}^2 – \rho_{x,z}^2 – \rho_{y,z}^2 \geq 0. \quad(5)
$$

This represents a 3D volume enclosed by two limiting surfaces, which we can get by solving the quadratic equation corresponding to the case where the determinant is 0. In the plot below, orange (transparent) corresponds to the upper bound and blue to the lower bound.

Figure 2:  Valid values a 3x3 correlation matrix can take. The orange surface defines the upper bound while the blue one sets the lower limit.
Figure 2: Valid values a 3×3 correlation matrix can take. The orange surface defines the upper bound while the blue one sets the lower limit.

Going back to our triangle of currencies, if we now plot equation (4) we will see it’s very similar to the upper bound in the image above:

 Figure 3:  Space of valid values a correlation matrix can take in a currency trio. Instead of being a volume as in the general case, the linear dependency between currency pairs shrinks the space into one of its limiting surfaces.
Figure 3: Space of valid values a correlation matrix can take in a currency trio. Instead of being a volume as in the general case, the linear dependency between currency pairs shrinks the space into one of its limiting surfaces.

In fact, this is not a result of chance. Since in our currency trio variables are linearly dependent (\(z=x-y \)), the correlation matrix is rank 2 and its determinant 0; meaning that the set of possible values shrinks from a volume to just one of its limiting surfaces. This also means that equation (4) is a solution to the quadratic equation (let’s briefly forget it’s actually an inequality) presented in (5).

This constraints may be used, for instance, when we have some estimation of the correlation between two currencies (maybe based on central bank policy or economic interdependence), and we want to set some bounds on the correlations these can have with a third one.

4. Given 3., only one angle can be obtuse (>90º) meaning there can only be one negative correlation.

5. Given two sides, we cannot obtain the third. This nicely explains why we can’t obtain the volatility surface (if the word surface here doesn’t mean anything to you, just ignore it) of cross rates without making additional assumptions on at least one of the correlations between currency pairs.

Wrapping up

So far we have just briefly summarised Walter and Lopez’s paper [1] which gives very nice intuitions into the relationships between volatilities and correlations in a currency trio. In the future, we will focus on the two main topics we left out here, namely time-domain analysis and implied correlations, which deserve a post of its own.

References

[1] Walter, C. and Lopez, J.A., 1999. The Shape of Things in a Currency Trio. Federal Reserve Bank of San Francisco Working Papers4.

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