What does delta hedging a tranche mean?

Some old research from, of all people, Bear Stearns, makes fascinating reading. It is about delta hedging CDX and iTraxx tranches, just about the simplest possible hedging problem in structured credit (in that index itself is the hedge, and both that and the tranches are liquid).

Suppose we have sold a tranche of the CDX. What it the delta with respect to the index? The standard definition would say something like

delta = (price of tranche at index spread plus 1bp - price of tranche at index spread) / 1bp

But there is a hidden correlation assumption: we calculate this delta at constant base correlation. Thus delta hedging will only be P/L minimising if

• spread movements are small;
• rehedging is possible after a small spread movement; and
• base correlation remains constant.

The first two assumptions have not held recently with even the hitherto liquid CDX and iTraxx displaying jumps and illiquidity. But interestingly even back in 2004 the last one was known not to hold either. Here is the tracking error of delta hedging each of the CDX tranches from the Bear's research:
And here are the realised deltas (i.e. I think the best deltas ex post) vs. the calculated ones (ex ante from the model):
And remember, that is the easiest hedge in structured credit. If the simplest position to hedge when the market was not particularly troubled gives you 3% tracking errors, what is it like trying to delta hedge a bespoke hybrid CDO at the moment?