Gents, few observations:
This paper (and it's a firm favourite for reasons I'll elaborate), is very much a bird's eye view. The actual choices of dynamics are much less impactful than (2.3). To mention some of the elements omitted in the paper (and which are 100% priced daily in my desk experience) are for example stochastic intensities (of your choice, Black - Karasinski was productionised where I last traded for example so we have mean reversion and interesting structure to the forwards as spreads vary), credit-rate diffusion correlation (highly significant if we are talking about say large cyclical European counterparties and Euro swaps), and critically bank / counterparty credit correlation driving the first to default intensities - governing who the model thinks will default first. These really are all the elephants in the room. More of this shortly.
On the specific point about swap vols, it's worth mentioning, if we take the ISDA 2002 incarnation of derivative documentation, as the paper rightly points out, there is no mention of uni/bilateral credit risk in the 6E closeout / termination language. It is entirely fair to interpret this as saying (gap and replacement cost risk precipitated by defaults of certain systemic counterparties notwithstanding) then that the CVA defect to collateralised valuation (again in (2.3)) refers to a claim (and hence will apply recovery to a claim) against the
collateralised value, and therefore if you do not use the collateralised vol you are replicating to a payoff on the wrong asset . Counterparty defaults, dealer is owed money, they ask up to 15 banks (more likely 3) to quote to be replacement counterparties on a
collateralised basis. The legal documents really are all important here - the language in the 2002 version is ok, (in the preceding 1999 version it is diabolical, check it out - totally inconsistent and frankly mathematically illogical).
Now back to the credit / risk niceties
T
o give an example say you have a bank highly correlated with a counterparty and trading at the same spread and the bank is long a highly ITM derivative (of any kind such that it is model-unlikely that it will be OTM to the bank at any point over its life). Now say you have some correlation model for the FTD intensities, at high correlation, if the bank spread widens relative to the counterparty then the bank will model wise default first in many models, so there is no incurred CVA loss, but perhaps a smaller funding cost increase. It's likely in this case the bank is short risk its own name since the CVA loss could dominate. If the bank tightens relative to the counterparty, then it's in trouble - the expectation is that counterparty defaults first and potentially here there is a large payoff hit. Hence here again the bank is short risk its own name. If correlation drops between bank and counterparty, then the funding side could start to dominate and so the bank could become long risk its own name owing to funding costs becoming more important than the 'who defaults first' piece. All of this is on the credit delta side but shows I hope some of the interesting credit aspects to the treatment.
Finally for me historically the paper was helpful for 2 reasons: a) the explicit FTD treatment (once you realise that a bank funding spread conditional on joint survival is really the same as the bank FTD) allows for a price to be agreed by both counterparties and b) it removed the DVA double count nonsense since the replication arguments are convincing that any +ve cash balances generated by an unfunded derivative liability do not accrue a funding benefit in addition to the DVA benefit on such a liability without incurring further credit risk.
In short then, the generalisations of dynamics are manifold, but the collateralised vs 'uncollateralised' argument for swaption vols, i.e. think at least for any deriv under 2002 ISDA, are not a central concern, and in fact I think the identity assumption is ok. Be interested in your thought tho, cheers!