Interpolating Credit Survival Probabilities on a Log Scale
Summary
The document derives the survival probability between two credit-curve nodes by linearly interpolating the logarithms of the endpoint probabilities. Since each node’s log survival probability equals minus its time multiplied by its cumulative hazard, substituting those endpoint values into the interpolation formula and exponentiating gives the stated expression for an intermediate time.
It also describes the related convention of assuming a constant hazard rate between nodes and extending the final rate beyond the last node. The discussion notes that this convention is used for consistency with standard CDS models, while some practitioners use smoother hazard curves internally. The examples involving observable maturities illustrate how interpolation choices can differ from market quotes. The document gives a derivation and market-practice context, but no numerical comparison or empirical test of alternative methods.
Key ideas
- Linear interpolation of log survival probabilities produces the stated between-node formula.
- At each curve node, the log survival probability is the negative product of time and cumulative hazard.
- A flat hazard rate between nodes is a conventional credit-curve interpolation assumption.
- Alternative hazard interpolations may be used to produce smoother curves.
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# Survival probability interpolation between two time nodes
# Survival probability interpolation between two time nodes
In the Open Gamma paper describing the ISDA CDS pricing model, it is mentioned that given the time notes of the credit curve $T^c=\{t_{1}^{c},...,t_{n_{c}}^{c}\}$ and that the survival probability for the time node $i$ is $Q_{i}=e^{-t_{i}^{c}\Lambda_{i}}$, then for a time point $t\in(t_{i}^{c},t_{i+1}^{c})$ the corresponding survival probability is given by the following equation:
$$Q(t)=\exp\Big({-\frac{t_{i}^{c}\Lambda_{i}(t_{i+1}^{c}-t) + t_{i+1}^{c}\Lambda_{i+1}(t-t_{i}^{c})}{t_{i+1}^{c}-t_{i}^{c}}}\Big)$$
My question is how this equation is derived?
## Answer by Whitebeard13 (score 0, accepted)
https://quant.stackexchange.com/a/78008
It's straightforward, yet I missed it. It is just the simple linear interpolation applied on the logs of survival probabilities. I will use the notation $Q(t_{i})\equiv Q_{i}$. Here is the detailed derivation:
$$\ln Q(t) = \ln Q(t_{i}) + \frac{\ln Q(t_{i+1}) - \ln Q(t_{i})}{t_{i+1}-t_{i}}(t-t_{i}) \Leftrightarrow$$ $$\ln Q(t) = \frac{\ln \overbrace{Q(t_{i}}^{e^{-\Lambda_{i}t_{i}}})(t_{i+1}-t_{i}) + (\ln Q(t_{i+1}) - \ln Q(t_{i}))(t-t_{i})}{t_{i+1}-t_{i}} \Leftrightarrow$$ $$\ln Q(t) = -\frac{t_{i}\Lambda_{i}(t_{i+1}-t) + t_{i+1}\Lambda_{i+1}(t-t_{i})}{t_{i+1}-t_{i}}$$
## Answer by Dimitri Vulis (score 1)
https://quant.stackexchange.com/a/78007
For the purpose of matching the standard model with other parties, we assume that the hazard rate is piecewise constant (flat) between nodes. We also extrapolate the last hazard rate beyond the last node.
This assumption dates back to the JPMorgan models from the 1990s, which became industry standard (founder's effect), but leads to the same problems as, for example, interpolating intrest rates assuming that forward rats are piecewise constant between nodes.
To illustrate, pick a curve with observable 3Y, 4Y, and 5Y quotes, and compare the observed 4Y quote with the one interpolated between 3Y and 5Y assuming flat hazard rate. Or if 4.5 years is observable near roll date, compare that with the interpolation.
Credit term structure is much harder to arbitrage than interest rates, so this assumption continues to be used. Internally, however, many use various alternative interplations with smooth hazard rates.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.