Interpolating CDS Spreads Through Hazard Rate Curves
Summary
The exchange considers how to estimate a CDS spread between quoted maturities, such as a spread for an intermediate tenor between shorter and longer quotes. It asks whether hazard rates calibrated to quoted CDS contracts can support a direct spread interpolation, and whether a piecewise constant hazard rate offers a simple formula. The answer stresses that there is no uniquely correct spread interpolation: the key constraint is avoiding arbitrage, and spreads between adjacent maturity quotes can have some latitude while remaining arbitrage-free.
Interpolating hazard rates is presented as a practical way to obtain smoother curves and to support pricing more complex instruments. It is also useful when single-name modeling is needed, since working directly with spreads can make arbitrage control more difficult and computation more intensive. The discussion is qualitative and gives no explicit formula or calibration procedure; its guidance depends on the chosen modeling assumptions and the instruments being priced.
Key ideas
- There is no uniquely correct interpolation of CDS spreads; arbitrage avoidance is the central constraint.
- An intermediate maturity spread need not be fixed by simple linear interpolation between quoted spreads.
- Interpolating hazard rates can produce smoother curves.
- Hazard rate modeling can be convenient for pricing complex instruments and single-name credit risk.
- The document does not provide a direct formula for the requested intermediate spread.
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Full text
# Interpolation on CDS rates # Interpolation on CDS rates I am just wondering if there is any way we could calculate a CDS Spread (not harzard rate) on a CDS curve. Most of the papers that I have come across so far discuss about interpolating the hazard rates using numerical root-finding algorithms by setting the break-even spread to 0. However can I assume that the hazard rates derived this way will give me a "proper" interpolation on the CDS spread? i.e., say if I have the 3 month and 6 month CDS spread, and I need to price the spread for a 4 month CDS, would it be OK if I simply assume the risky PV for the first 3 months is already 0 given the hazard rates that I have derived and just working on figuring out the final one month spread? Is there a direct, nice formula to compute for such a spread given hazard rate using piecewise constant hazard rate assumption between 3 month and 4 month? Thanks! ## Answer by Yulia V (score 3) https://quant.stackexchange.com/a/14942 There is no such thing as a "proper" interpolation of CDS spreads. The only criterium your interpolation must obey is the absence of arbitrage. Note that, assuming that $spread(3M) < spread(6M)$, $spread(4M)$ can take any value between $spread(3M)$ and $spread(6M)$ without creating an arbitrage opportunity (actually it can be even slightly less than $spread(3M)$ or slightly higher than $spread(6M)$) without creating an arbitrage opportunity). Interpolation of hazard rate helps to create smoother curves - this is a desirable property. The other reason why the hazard rate interpolation is popular is because it is convenient to use when pricing more complex instruments. Some instrument require single names modelling, and in this case one cam only use model hazard rate because otherwise it will be difficult to ensure the absence of arbitrage and it would be more computationally intense.
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