Deltas In Standard CDOs

At first glance the delta of a liquid index tranche, whether this be the delta with respect to the underlying index or to each single name, looks like a very straightforward concept. The fact that almost every dealer is quoting different deltas when trading identical tranches perhaps tells a different story. The intuitive definition is simply "How much of the underlying asset is required to hedge the risks in the tranche". A more mathematical definition is "How much does the value of the tranche change, relative to the underlying, when the underlying spreads move by 1 basis point and nothing else changes."

It is those final four innocuous looking words: and nothing else changes that opens up a can of worms.

Even ignoring the fact that every dealer uses proprietary models, which they do not generally make available externally, keeping correlation the same is much more involved than simply keeping interest rates or recovery rates the same. Should you keep a large correlation vector, base correlation or compound correlation constant? Because base correlation is becoming the standard by which the market communicates in the tranche space it would makes sense to keep base correlation constant. However, should you keep the base correlation the same at each loss point when you are explicitly changing the probability of the attachment point being hit? As you shift the underlying risk in the portfolio the 3% point becomes more or less likely to be triggered. It either moves into or out of the money.

For those familiar with other markets, such as equity options and interest rate options, the range of deltas in credit is often bewildering. In other markets most participants agree on a delta. On first appearance these other markets have no issue with keeping everything else constant. These volatility markets, however, are a lot more mature than the current correlation markets and in their infancy they experienced similar issues to those in the credit market. For example, in interest rates the percentage volatility is linked to the level of rates. If rates are shifted up should the volatility of the option be kept constant or should it be shifted to a level consistent with the higher spread? In equities, if the underlying price is shifted the at-the-money point changes. When everything is kept the same should the volatility on the option be kept constant with a static smile or should the ATM volatility be kept constant so there is a floating smile? In fact the concept of an agreed upon delta, even in developed markets, is more a function of an agreed simple model: dealers continue to innovate internally to come up with the best hedge ratios possible.

There is no proof any method is right or wrong. If the tranche value doesn't change by the implied delta for that method then the remaining change is described as correlation movement. As such the usefulness of this correlation number, its stability and its relationship to spreads will also be used to determine which method is preferable.

In the credit markets there is model risk and methodology risk to the deltas. Because there is no standard equivalent to the Black Scholes model in the credit market, even using the same methodology will result in different deltas. However we assume a perfect world in which everyone agrees on a model and the only issue is that of keeping everything constant. Although there are many different methods of doing this we believe there are two main methods with the remaining ones being derivatives of these. The first is simply keeping correlation at each attachment point constant. The second is keeping the correlation at expected loss point, or the at-the-money correlation constant. The first method is generally the default method that is often implemented if the issue of what to keep constant hasn't been considered. The second method makes more sense from a modelling point of view and is similar to the way the delta is calculated in the equity and interest rate volatility markets. In delta terms the first gives a lower delta than the second for equity and mezzanine tranches and a higher delta for the more senior tranches. The table shows a variety of methods and the deltas obtained.

 

Why The Deltas Don't Add Up To One

There is a general misconception that all the spreads on the tranches should add up to the spread on the index and the perfect hedge of all the tranches is the entire underlying portfolio. Equivalently, this would say the deltas of the entire capital structure weighted by tranche size should sum to one. Here we provide an intuitive look why trading the index and trading all the tranches is different and then analyse why this leads to a higher overall delta for the tranches.

Imagine we trade the underlying index against all the tranches, when we take a position in each tranche equivalent to the tranche width (i.e. EUR100m in the index, EUR3 million in the 0-3%, EUR3 million in the 3%-6% etc.). There is little difference in a no default situation: carry will be received on one leg of the trade and paid on the other. The carry on each leg, however, will change significantly after a credit event.

Let us first consider the index and assume it pays 30 basis points. This 30bps is dispersed evenly across the whole portfolio. After a credit event you will continue to receive 30bps, however, this will be on a smaller notional. If an original ticket of EUR100 million is traded, the amounts of carry received will fall by EUR1,440 (assuming 1 of 125 names defaults with a recovery of 40%).

Now we consider a similar scenario for the portfolio of tranches. Unlike the index, the coupons for the tranches are dispersed unevenly across the capital structure with the equity tranche paying the highest coupon in relation to tranche size. Once there is a default some of the high paying equity tranche will be triggered. As such, the average remaining spread on the rest of the portfolio will be lower than before the default. Again, on a total notional of EUR100 millon, the carry received will fall by around EUR45,000 (again assuming 1 of 125 names defaults with a recovery rate of 40% and the whole equity coupon is paid as running spread) a lot more than on the index.

From this reasoning it is clear there are two risks when considering all the tranches. The first is market moves and the second is defaults. In a world in which there are no chances of defaults then any change in risk caused by underlying spreads changing gets passed through directly to the tranches and all the deltas would add up to one. As soon as there are chances of defaults there is the added risk some of the high paying coupon on the equity tranche will not be received, as such a larger delta is required to protect against this risk and the total delta on the portfolio will be greater than 1. If, however, the tranches and the index were all traded as an upfront there is no risk of the high paying coupon being eroded because of a default. Turning the running spread into an upfront removes the risk the coupon will not be paid going forward and the deltas add up to one.

In practice, even by using a variety of methods the delta on the tranches adds up to around 1.1 to 1.4. The difference is driven by the level of spreads, correlation and how much of the equity tranche is paid up front.

 

This week's Learning Curve was written by Mike Harris, credit derivatives strategist at JPMorgan in London.