Connecting Stochastic Hazard Rates to CDS Spread Term Structures
Summary
The document introduces the relationship between a credit default swap’s fair spread, the survival probability of the reference entity, and its hazard rate. It gives an integral expression for the spread across a contract’s term and notes that survival can be represented through the accumulated hazard rate. The author asks how practitioners infer hazard rates from quoted spreads and how survival probabilities can be recovered when spreads are modeled stochastically.
It also raises a modeling question about maturity: a spread process specified for one maturity may not describe the behavior of the whole term structure, unless each maturity is calibrated separately. Finally, it asks how spreads behave when the hazard rate follows a stochastic log process. The document presents these as open questions rather than providing solutions, empirical evidence, or calibration guidance, so it is useful as a map of modeling issues but not as a practical estimation procedure.
Key ideas
- CDS fair spreads are linked to survival probabilities and recovery through an integral relationship.
- Survival probability can be expressed using the accumulated hazard rate.
- A spread model for a single maturity may not specify dynamics for the entire CDS term structure.
- The document asks how stochastic hazard-rate assumptions translate into spread dynamics.
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# A model to stochastic hazard rate and CDS spread term structure
# A model to stochastic hazard rate and CDS spread term structure
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.