Survival probability interpolation between two time nodes

Question

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 1

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.

Answer 2

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:

$$\ln Q(t) = \ln Q(t_{i}) + \frac{\ln Q(t_{i+1}) - \ln Q(t_{i})}{t_{i+1}-t_{i}}(t-t_{i}) \Leftrightarrow$$ $$\ln Q(t) = \frac{\ln \overbrace{Q(t_{i}}^{e^{-\Lambda_{i}t_{i}}})(t_{i+1}-t_{i}) + (\ln Q(t_{i+1}) - \ln Q(t_{i}))(t-t_{i})}{t_{i+1}-t_{i}} \Leftrightarrow$$ $$\ln Q(t) = -\frac{t_{i}\Lambda_{i}(t_{i+1}-t) + t_{i+1}\Lambda_{i+1}(t-t_{i})}{t_{i+1}-t_{i}}$$