Continuous-premium credit-default swap spread ($s=(1-\mathcal{R})\frac{{\int }_0^{T}B(t)\lambda (t)G(t)dt}{{\int }_0^{T}B(t)G(t)dt}$)

Assume deterministic discount factors, deterministic risk-neutral default intensity $\lambda$, fixed fractional recovery $\mathcal{R}$, and continuous premium until default or maturity. Survival is $G(t)=\exp (-{\int }_0^t\lambda (u)du)$. Expected discounted protection is $N(1-\mathcal{R})\int B\lambda G$, while premium is $Ns\int BG$. Equating them gives the par spread. Constant intensity gives $s=(1-\mathcal{R})\lambda$. Discrete premiums, default accrual, stochastic recovery and counterparty losses require changing the corresponding cash-flow expectations.