CDS Par Spread Derivation Under Constant Hazard Rate
Consider a par Credit Default Swap (CDS) with maturity $T$. The premium leg pays continuously at rate $s$ until default or maturity. The protection leg pays
- R$ at the time of default. Assume a constant hazard rate $h$, constant recovery rate $R$, and constant risk-free rate $r$, all under the risk-neutral measure.
(a) Derive the par spread $s$ in closed form as a function of $r$, $h$, $R$, and $T$.
(b) Show that as $T \to \infty$, the par spread satisfies $s \to (1 - R)h$. State any assumptions you use.
(c) Given $R = 40\%$ and an observed par spread $s = 180$ bps for a long-maturity CDS, estimate the hazard rate $h$.
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