Interpolation on CDS rates

Question

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 1

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.