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It has been my understanding that CDS and CDS indices are traded in the OTC market. Surely, this would mean that if an investor decides to go long a CDS contract, the spread that the investor will be paying will not be public on the market i.e. it will be known only to the investor and the trader who did the transaction.

Then how is it possible that we can actually see the spreads time series (say on bloomberg for example) for a CDS. Does it represent the level at which the last CDS trade was conducted?

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Almost all CDS trades are cleared through DTCC, which knows at what level something has traded. You can see some of their summary data for free https://www.dtcc.com/repository-otc-data .

But the markets sometimes move fast and not every credit is traded every day, on the contrary. Knowing the level at which some credit last traded a few weeks ago may not be a useful prediction of where it might trade now.

Most major market participants submit their indicative quotes for a lot of CDS spreads to services such as IH Markit, who publish some average levels: https://ihsmarkit.com/products/pricing-data-cds.html .

Of course one has to pay to access this data.

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As in case of any other OTC instrument (bonds being fine example), some data are publicly disclosed despite there can be some negotiation between counterparties about a price.

In case of CDS, the spread is also driven by news and events (the same is true for bonds, although in lesser extent, especially developed governments ones). When there is a presure on credit quality (see for example recent COVID-19 pandemic and its influence on CDS spread of South part of Eurozone), the CDS spreads go up. Converselly in case a credit conditions getting better (e.g. when Greece abadonned capital flows restrictions and returned to bond market). These events influence majority of investors in similar way and as a result there is a consensus on CDS spread (or bond price etc.).

Moreover, some counterparties are willing to disclose their actual or indicative prices and these are publicly available on for example Bloomberg or Reuters.

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