# Lesson 5: Correlation

Many derivatives contracts depend on more than a single underlying, these are basket options. Many derivatives contracts have multiple sources of randomness, such as stochastic stock price and stochastic volatility. Many derivatives contracts require modelling of an entire forward curve. These situations have something in common; they all, under current theory frameworks, require the input of parameters representing the relationship between the multiple underlyings/factors/forward rates.

And so the quant searches around in his quant toolbox for some mathematical device that can be used to model the relationship between two or more assets or factors. Unfortunately the number of such tools is limited to correlation and, er, ... , correlation. It s not that correlation is particularly brilliant, as I ll explain, but it is easy to understand. Unfortunately, it is also easy to misunderstand.

One problem concerning correlation is its relevance in different applications, we ll look at this next in the context of

Figure 5.4: Two perfectly correlated assets.

timescales, and another is its simplicity, and inability to capture interesting dynamics.

When we think of two assets that are highly correlated then we are tempted to think of them both moving along side by side almost. Surely if one is growing rapidly then so must the other? This is not true. In Figure 5.4 we see two asset paths that are perfectly correlated but going in opposite directions.

And if two assets are highly negatively correlated then they go in opposite directions? No, again not true, as illustrated in Figure 5.5.

If we are modelling using stochastic differential equations then correlation is about what happens at the smallest,

Figure 5.5: Two perfectly negatively correlated assets.

technically infinitesimal, timescale. It is not about the 'big picture' direction. This can be very important and confusing. For example, if we are interested in how assets behave over some finite time horizon then we still use correlation even though we typically don't care about short timescales, only our longer investment horizon (at least in theory). Really we ought to be modelling drift better, and any longer-term interaction between two assets might be represented by a clever drift term (in the stochastic differential equation sense).

However, if we are hedging an option that depends on two or more underlying assets then, conversely, we don't care about direction (because we are hedging), only about dynamics over the hedging timescale. The use of correlation may then be easier to justify. But then we have to ask how stable is this correlation.

So when wondering whether correlation is meaningful in any problem you must answer two questions (at least), one concerning timescales (investment horizons or hedging period) and another about stability.

It is difficult to model interesting and realistic dynamic using a simple concept like correlation. This is illustrated in Figure 5.6 and the following story. In the figure are plotted the share prices against time of two makers of running shoes.

Figure 5.6: Two assets, four regimes.

In Regime 1 both shares are rising, the companies are seen as great investments, who doesn't want to go jogging and get healthy, or hang out in the 'hood wearing their gang's brand? See how both shares rise together (after all, they are in the same business). In Regime 2 company A has just employed a hotshot basketball player to advertise their wares; stock A grows even faster but the greater competition causes share B to fall (after all, they are in the same business). News Flash! Top basketball player in sex and drugs scandal! Parents stop buying brand A and switch to brand B. The competition is on the other foot so to speak. Stock A is not doing so well now and stock B recovers, they are taking away stock A's customers (after all, they are in the same business). Another News Flash! Top advocate of jogging drops dead... while jogging! Sales of both brands fall together (after all, they are in the same business).

Now that is what happens in real life.

In practice there will also be some delay between trading in one stock and trading in the other. If company A is much bigger and better known that company B then A's stocks will trade first and there may be a short delay until people 'join the dots' and think that stock B might be affected as well. This is causality, and not something that correlation models.

And, of course, all this stock movement is based on shifting sentiment. If one company defaults on a payment then there is a tendency for people to think that other companies in the same sector will do the same. This is contagion. Although this could actually decrease the real probability of default even as the perceived probability is increasing! This is because of decreasing competition. Correlation is a poor tool for representing the opposing forces of competition and contagion.

As you can see, the dynamics between just two companies can be fascinating. And can be modelled using all sorts of interesting mathematics. One thing is for sure, and that is such dynamics while fascinating are certainly not captured by a correlation of 0.6!

Is this good news or bad news? If you like modelling then it is great news, you have a blank canvas on which to express your ideas. But if you have to work with correlation on a day-to-day basis it is definitely bad news.

Example In South Korea they are very partial to options on two underlyings. Typically worst-of options, but also with a barrier. The value of such a contract will depend on the correlation between the two assets. But because of the barriers these contracts can have a cross gamma that changes sign. Remember what happens when this happens? It s straight from Lesson 4 above.

In theory there is a term of the form

in the governing equation. If the cross gamma term changes sign, then sensitivity to correlation cannot be measured by choosing a constant p and then varying it.

In Figure 5.7 is a contour map of the cross gamma of one of these two-asset worst-of barrier options. Note the change of sign. There are risk-management troubles ahead for those naive enough to measure |^!

Please remember to plot lots of pictures of values and greeks before ever trading a new contract!

Example Synthetic CDOs suffer from problems with correlation. People typically model these using a copula approach, and then argue about which copula to use. Finally because

Figure 5.7: Contour plot of cross gamma of a two-asset, worst-of, knockout option.

there are so many parameters in the problem they say 'Let's assume they are all the same!' Then they vary that single constant correlation to look for sensitivity (and to back out implied correlations). Where do I begin criticizing this model? Let's say that just about everything in this model is stupid and dangerous. The model does not capture the true nature of the interaction between underlyings, correlation never does, and then making such an enormously simplifying assumption about correlation is just bizarre. (I grant you not

Figure 5.8: Various tranches versus correlation.

as bizarre as the people who lap this up without asking any questions.)[1]

Figure 5.8 is a plot of various CDO tranches versus constant correlation. Note how one of the lines is not monotonic. Can you hear the alarm bells? That tranche is dangerous.[2]

• [1] We know why people do this though, don't we? It's because everyone else does. And people can't bear watching other people getting big bonuses when they're not.
• [2] You know that now do you? And how much did that lesson cost you? The most important lesson in life is to make all your lessons cheap ones. There, I've got you started.

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