About the paper of Pan and Singleton 2008 “Default and Recovery Implicit in the Term Structure of Sovereign CDS Spreads”, once the lambdas (hazard rates) for the different tenors of the term structure of the CDS contracts have been obtained, at different time points, I obtain a historical series of the hazard rate for each tenor. the authors hypothesize a Vasicek model for lambda, but I don't understand which tenor they refer to. thanks to anyone who answers.
The paper on Jun Pan's page.
The only quotes readily observable in the market are quotes for a few tenors of the standard CDS contract. Please recall that the standard credit default swap essentially is very simple:
Fixed leg: the protection buyer pays fixed to protection seller.
floating leg: if the reference entity has no credit event happens until the maturity, then the protection seller pays nothing to the protection buyer. But if a credit event does happen, then the protection seller pays to the protection buyer the notional minus the value of a reference obligation ("recovery"). Long ago, the protection buyer would physically deliver a defaulted reference obligation; but now everyone uses cash settlement. There is an auction mechanism to agree on the value of the reference obligation for settling CDS. Historically, the recoveries on senior unsecured have ranged from ~9% (Lehman Brothers) to 99%+ (GSEs), while the recovery on subordinated is usually 0.
Although 5 year tenor (i.e. protection for 5 years) is the most common, for many sovereign and corporate credits quotes are available for other tenors, like 1, 3, 7, or 10 years.
The maturities are standardized (unlike e.g. IR swaps) to March 20, June 20, September 20, and December 20, so "5 years tenor" means "September 20, 2025" and not 5 calendar years exactly from today.
In the past, people tried other variations of CDS, where e.g. the protection buyer paid more fixed, but did not have to deliver the reference entity in case of the credit event. But they have disappeared and you'll never see them quoted.
The most common pricing model for CDS, which goes to mid-1990s JPMorgan, is:
The probability of default today (the first node) is 0.
Make some assumption for the value of recovery (like 40% of the face value). (Note that this assumption should be treated as a "level 3" unobservable model input.) (Note: Many dealers submit their recovery assumption to IHS Markit, who publishes consensus recovery assumption. It is over very limited use.)
Bootstrapping process: loop through the observable quotes. For each tenor (node), solve numerically for the risk-neutral probability of default that explains the CDs quote, given the probability of default at the prior node and the recovery assumption.
If you need to interpolate the probability of default between observable nodes, then assume that the hazard rate is constant between the nodes.
Most shops still use either this model or make small improvements to the way the curve is interpolated between the nodes.
Pan and Singleton paper points out that it is not necessary to use the same recovery assumption for all observable CDS quotes.
Indeed, right after their paper was published in 2008, we observed this with CDS quotes on General Motors (GM). (It's corporate, not sovereign, but GM CDSs were being quoted for many tenors, so everything they wrote applies.) The probability of default was pretty close to 1. People were betting on when, rather than whether GM would default, and what the recovery would be. GM was burning through its once-huge pile of cash. The longer GM waited to file for bankruptcy, the lower the recovery would be. Indeed, if you tried to bootstrap GM CDS quotes using constant recovery assumption, on many days constant recovery assumption would require the risk-neutral probability of default to decrease with time, this admitting arbitrage. This may be necessary to allow under risk scenario, but not when you're bootstrapping unperturbed market rates.
The way around such problems is for any good quant library to allow different recovery assumptions for different nodes (for example, even if you only have one 5-year CDS quote, it would be nice to be able to assume 40% recovery in yers 1-5 and 25% afterwards); and also to allow more sophisicated interpolation than constant hazard rate between nodes.
In terms of the sensitivity to the recovery assumption, good CDS pricing libraries let you see the impact of changing the recovery assumption at one tenor at a time. To make P&L Explain work on the rare occasions when you change the recovery assumption, you should include in it not only recovery delta, but also recovery gamma and the cross gammas to time, to the CDS spread, and possibly to other model inputs.