# Dispersion Trading For The Uninitiated

Let’s do dispersion trading for the uninitiated.

Imagine selling an index straddle and buying each of the components’ straddles in proportion to the index weights. In practice, liquidity makes this impossible. Instead one settles for a “dirty dispersion” position. The trade is “short correlation”. It wants the average correlation between the stocks in the basket to be as low as possible.

### Why Is Dispersion Trading A Correlation Bet?

Consider a 2 stock index:You own the straddles on the stocks and you are short the index straddle.

Case 1: Low correlation
• The 2 stocks rip in opposite directions.
• The index is unchanged.
That’s a homerun! You win on every leg. You win on the call leg of one stock’s straddle, the put leg of the other stock’s straddle and the index doesn’t go anywhere allowing you to collect on the full short premium.

Now let’s move to the opposite scenario.

Case 2: High correlation

• The stocks move exactly together in a big way.

Why?

The index is cheaper than the sum of the legs in straddle space. To understand why, we will need some simple math.

### An Intuitive Equation

Correlation represents the spread between an index’s vol and the vol of the components.

There are 3 terms:
• Index variance
• Avg stock variance
• Avg cross corr of each stock to every other stock
The equation:

Index variance = avg stock variance x avg cross correlation

We can re-arrange the equation to see the correlation as the ratio of index variance to stock variance:

Avg corr = index variance / avg stock variance

Notice that unless the correlations are 1, index var < stock var!

So if the index variance is trading for 50% of the variance of the avg weighted stock vols then the implied cross correlation is .50

Be careful, you need to take square roots to move from var space to vol space which is how prices are more commonly interpreted. In other words, you must square the ratio of index vol to stock vol you get the implied corr.

Example: If the index vol is 20% and the avg weighted stock vol is 30% then:

implied correlation = (.2/.3)^2

Implied correlation = .44

### The Shape Of Correlation Risk

If stock vols are constant, and index vols increase, implied correlation must be increasing.

Likewise, if correlation surges the spread of index vol to stock vols must be narrowing (at corr = 1 they would converge.)

Here’s index vol relation to corr for a fixed stock vol of 30% If you structure a trade vega neutral or premium neutral you will be short correlation convexity.

• As corr increases: you get shorter vol as the index short will grows faster than the stock vol longs.
• As corr falls: vice versa. You get longer vol as it falls!
Dispersion trades are not just short correlation (notice this is the same risk premia as any risk-on position), but concave:

Your position size has negative gamma with respect to changes in correlation.

Dispersion is tricky. There is a lot of room for creativity in how you structure these trades. A few considerations:

• You may choose to overweight stock long vega to flatten the curvature, but now you increase exposure to owning options.
• What do you want your local gamma/theta profile to be? How do you want your “shocked” portfolio to look (matrix approach would ask “what’s my p/l with spot down 10% and correlation doubling?”)
• How much basis or synthetic basket risk you want to take with names you include or not since this is a “dirty” trade in the first place?

### The Correlation Surface

If you put the 3D options glasses on, you’ll notice that correlation has its own surface!

• Upside implied correlations are cheaper than downside correlations.
• Implied correlation has a term structure as well.

Implied correlation surfaces vary across sectors as well. Energy, biotech, bank etfs. The sector indices have implied correlations between basket components.

Then consider FX vol markets. They care about the rate vols of the individual fiat legs and, you guessed it, the correlation.

How about a US investor trading options on a foreign index of an exporter nation. Like Japan. There’s an implied correlation between the yen and the equity index itself. Google the term quanto if you want to explore that idea.

The risk for any portfolio of long/short trades (either delta one or volatility) is as correlations increase your gross positions become exposed. You can’t hide behind “nets” when corrs explode higher. This is especially dangerous because most “hedged” portfolios are levered.

Imagine a beta neutral trade where you are long 2 units of “alpha stock” and short 1 unit of index (assume they are the same vol, but “alpha” stock is .50 corr).

• When correlations increase towards 100%, you are no longer neutral but long equiv of 1 unit of index into a falling market, increasing correlation mkt.

Relative value books tend to blow up as corrs increase since corrs are used to weight positions.

• A portfolio that wins as correlation increases (which is itself correlated with equity risk premia) should cost carry!

This is, in fact what we find. Implied correlation trades at a premium to realized correlation (and correlation swaps which have linear risks). You pay a premium to hold a long implied correlation position. Those selling correlation via dispersion trades are capturing a risk premia or source of carry correlated with conventional risk premia.

Index options “should” be overpriced because it’s a systematic risk premium. The dispersion traders are the ones who bet when the overpricing is “excessive”. I wouldn’t advise trying that at home though.

### Parting Thoughts

How do implied correlations correlate to systematic risk premia? How do they compare to realized corrs?These types of questions are the start of seeing the world as one big spiderweb of risk premia and cross correlations.

Armed with this understanding, go build the dashboard to find the cheapest hedges, the most efficient basis, or the most levered shot at correlation regimes shifting. In other words, you don’t have to have a view on whether assets are cheap or not. You can look for situations where implied correlations are [over]confident in particular regimes persisting.