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I'm interested in the term structure of CDS spread.

It's known that the Market CDS rate (fair CDS spread or T-maturity spread) of a CDS contract initiated at $s$, maturity $T$ and recovery function $R$ is given by

$$ S(s,T) = \frac{-\int_s^T R(u) dG(u)}{\int_s^T G(u)du}$$

where $G(t)=\mathbb P[\tau>t]$ is the survival probability associated to default with default time $\tau$. Also one has that $G(t)= \exp({-\int_0^t h_u du})$ where $h$ is the hazard rate (HR).

My first question is what is the easy and market practice to get the hazard ratio from CDS spread.

Assume that the CDS spread is given by a know stochastic process. Let's say we have even a analytical tractability for the stochastic CDS spread as in the SRMR model.

How to obtain the survival probability $G$ ?

Another doubt I have is concerning the SRMR model and the role of maturity. Unless I missed something he gives us model to the dynamic of the market spread (for the log return so for the rate itself) for a fixed maturity $T$ which is not explicated anywhere in this model. Assuming we how to solve the integral equation $ \int_s^T R(u) dG(u)+ S(t, T)\int_s^T G(u)du =0 $ I would be treating every T-maturity as they had the same dynamic.

That may be ok and the distinction between their behaviors must comes from the different parameter obtained by proper calibration for each different maturity.

I would like to have some opinions about that please.

Now assume the HR to be stochastic modeled for instance by a exponential Ornstein-Uhlenbeck process as following:

$$ dH_t= -\mu_t H_t ~dt+\sigma_tdW_t$$ with $H_t := \ln(H_t)$.

How describe the spread dynamics ?

Given that the spread seems to have a stochastic nature (as you see in this paper for instance), how would that be coherent with the first formula ?

Many thanks for you thoughts.

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    $\begingroup$ The market practice for almost all CDS calculations these days use the ISDA standard model. The documentation is pretty wonderful actually and I believe will answer your first two questions. There should be not only explanations but code as well for calculating hazard ratios and survival probabilities. cdsmodel.com/cdsmodel/documentation.html? $\endgroup$ – rhaskett Nov 14 '14 at 22:02

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