Apparently, you refer to this passage from Prof John C. Hull (11th edition, 2021):
It's confusing because Hull is referring to the market conventions before the "Big Bang". This is no longer true, but Hull, or, more precisely, his ghostwriters, never updated his book, although about 15 years have passed.
Before the Big Bang, a CDS quote was literally the fraction of the notional that the protection buyer paid every year to the protection seller, typically quarterly or semi-annually. At inception, the mark to market of a vanilla CDS was close to 0, like for an interest rate swap. During the life of the trade, the counterparties used their internal recovery assumptions to calculate their own separate probabilities of survival and their own separate marks to market. The recovery assumtions might differ. Hence the counterparties' marks to market might differ. Even if the recovery assumptions were the same, the marks to market might differ slightly because of diferent interest rates assumptions. Hull's point is that they did not differ a lot. If you calculated separately the mark to market of the 2 legs of the CDS, they would each have large sensitivities to the recovery assumptions, but these sensitivies almost offset each other. The remaining sensitivity of the swap to the recovery assumption is higher for larger spreads i.e. higher probabilities of default.
However, after the Big Bang, for most credits, the standard CDS quote from MarkIt/S&P Capital IQ/etc observable in the market consists of two parts: an annual spread in basis points (with term structure, although most names only have quotes for the 5 year tenor; for example 250 bps of the notional per year for 2 years or 300 bps per year for 5 years) and a recovery assumption (such as 40%; and, for this purpose, no term structure). A few distressed credits are quoted on upfront fee (please see below).
This recovery assumption in the quote is not a physical number. It's just part of the standard quote. The spread portion of a CDS quote is not fully meaningful without its companion recovery assumption portion of the quote. MarkIt (and the rest of the market) don't really think that a defaulted bond will be worth 40 cents on a dollar after a default, and you shouldn't interpret this number as such.
Post Big Bang, CDSs trade with a standardized annual running spread, such as 100 bps of the notional. So if you were to trade a CDS, you'd plug the standardized quote (spread + recovery assumption) and an interest rate curve for the denominaton currency (also published by MarkIt; doesn't affect the numbers much) into JPMorgan's standard CDS model and calculate a risk-neutral probability of default (with term structure). Then you'd further plug these risk-neutral probabilities and the terms and conditions of your trade (running spread, tenor...) into the same model and calculate a physical upfront fee (as a percentage of the notional, e.g. 15%) that you'd actually pay or receive when trading the CDS contract. The upfront fee is almost equivalent to the swap's mark to market.
Intuitively, the upfront fee is like 100 - the fair price of an imaginary bond that is pari passu with the reference obligation, but has fixed coupon = risk free rate + CDS spread, and the same matutity as the swap. If the CDS spread < running spread, then the bond price would be above par, and the upfront fee is negative.
It would be hard to enter into a new CDS contract with a non-standard annual running spread or maturity. For example, if you see a CDS quote of 50 bps per year, this really means that the protection buyer receives some upfront fee and then pays 100 bps a year to the protection seller. A CDS quote of 600 bps per year, might mean that the protection buyer pays some upfront fee and then pays 500 bps a year to the protection seller. A non-standard swap where the protection buyer really pays 50 bps or 600 bps per year to the protection seller, or that matured on a non-standard date, would be very unusual and probably more expensive. Sometimes people also trade non-standard "zero coupon" swaps, where the protection buyer only pays a (larger) upfront fee, and then pays no running spread. Sometimes people trade amortizing CDSs, whose notionals change with time. Sometimes people trade CDS referencing obscure entities lacking a RED code. Sometimes people trade options on $n$th to default baskets :) While it is still possible to trade almost anything bespoke over the counter, overcoming clearing and swap execution facilities hurdles, these are not the swaps being quoted. Rather, only the standardized swaps, with standard running spreads and maturities, are being quoted.
A few distressed credits are already quoted on upfront fees (with term structure and with a running spread, typically 500 bps), rather than on annual spreads and recovery assumptions. See a very confused Reuters story trying unsuccessfuly to describe Tesla CDS quoted on upfront fee:
This makes no sense at all. Here is a marginally better Bloomberg article describing Russia sovereign CDS quotes switching from spread to upfront:
No, all contracts trade with an upfront fee and a standard running spread. All that changed that day was the quoting convention.
Actually, the contracts assume nothing about the recovery. The new upfront quote contained only the upfront fee and the running spread, did not mention any recovery. One can use some recovery assumption to solve for a risk-neutral probability of default, whose value will depend a lot on the recovery assumption used.
In terms of accounting topic 820, the observable quotes (coventional spread + recovery assumption, or upfront + running) are level 1, while the upfront fees that you calculate from quoted spreads are level 2. If you extrapolated, e.g. the upfront of a 10-year CDS from a 5-year quote, that should probably be binned as level 3.
As long as everyone involved uses the same calculations, assumptions, and inputs, it is no longer possible for two counterparties to have different marks to market on the same CDS, which was possible before the Big Bang.
I will illustrate a common problem that the Big Bang was aiming to solve, and mostly solved. Suppose that bank A sold CDS protection to hedge fund B, referencing C's debt. At inception, both sides see mark to market 0. B promises pay A, say, 300 bps of the notional every year. They have a meeting of minds on the fair spread at inception. However they might disagree on how they arrived at this spread. For example, it's possible (I'm making up some numbers, don't expect them to add up!) that A thinks that the physical recovery would be 45% and the physical probability of default is 20%; while B thinks that the physical recovery would be 35% and the physical probability of default is 10%.
Suppose a few months have passed, and C's CDS now trades at 500 bps in the market. Since A and B have different assumptions, their marks to market on their trade might differ. This wouldn't matter much if they kept the swap until either its maturity or C's credit event, although a margin dispute might arise. But suppose that B now wants to unwind the trade - to tear up the old swap. Since their marks to market are so different, A and B would need some highly paid people to negotiate, and possibly never agree on the fair price for for unwind. So instead of unwinding the old swap with A, B would have do one of the following with some counterparty D (generally, not the same as A) who would buy the protection from B for the same maturity date, either
. at 300 bps a year, to exactly offset the cash flows of the old swap. Since the protection now costs 500 bps a year, this would involve a large upfront fee, which would need to be negotiated.
. at 500 bps a year. Then until the maturity, B would collect 500 bps from D, pass 300 bps to A, and keep 500-300=200 bps.
In both cases, B has counterparty credit risk exposure to A and D.
Post Big Bang, such unwinds, novations, reassignments have become much more practical.
Note now that a view on the physical recovery assumption is not used at all in the post Big Bang calculations. It just doesn't make sence to recalculate probabilities of default by combining the market observable conventional CDS spread quotes and your view on the physical recovery, as you describe in the question.
Consider, for example, this Reuters article:
This is rather misleading. Refinitiv publishes the CDS spreads, and the standard emerging markets recovery assumption = 25%. Together, they imply certain risk-neutral probabilities of default. To present these probabilities as physical shows the lack of understanding by the writers.
You can get some physical recovery assumptions from Moody's or some other data vendor, or you can have your own views. If you think that a defaulted bond will be worth 60 cents on a dollar after a default, and if you can find a counterparty with a different view on this physical recovery assumption, then the instrument you'd use to bet on the physical recovery is not the standard CDS, but a fixed-recovery (typically zero-recovery) CDS, which Hull confusingly calls "binary CDS" in his book, in which if a credit even occurs, the protection seller pays the full notional, but, unlike the standard CDS, the protection buyer delivers some pre-determined fixed recovery (typically just nothing, for simplicity). Hull page 272:
Note that Hull's book still talks about the swap payings 205 bps per year, while in reality, post Big Bang, such a swap would be quoted on upfront fee and would pay a standard running spread, probably 500 bps.
Here's an example Bloomberg article about this product:
If you have Bloomberg Terminal, see also:
You can use the upfronts of standard CDS and your views on the physical recovery, which may have term structure, to calculate the physical probabilities of default, which no longer need to be limited to piecewise constant hazard rates, unlike the standard JPMorgan model, and use these physical probabilities, dependent on your psysical recovery assumptions, to price a zero-recovery CDS. You can also calculate the sensitivity of your CDS mark to markets to bumps in your physical recovery assumptions. Obviously, everything here would be level 3 in the sense of accounting topic 820.
The new CDS quoting conventions can get confusing for people unwilling to invest time into understanding what they mean. Consider, for example, this recent story Reuters story:
what's up with the dollar signs, Mehnaz?
Or this panicky Bloomberg article about Türkiye sovereign CDS:
Being Greek, Tasos doesn't like Turks, we get it, but what he wrote still makes no sense.
Or this Bloomberg article about EUR-denominated quanto CDS on US Treasuries:
I'm not sure what he's trying to say here. For a quanto CDS quote, the currency exchange rate doesn't matter.
I got my first Hull book in 1993. I think it was the 2nd, or even the 1st edition. At that time it was very helpful. It makes me sad that the 11th edition is so woefully out of date.