= Continuous-premium credit-default swap spread
{title2=$s=(1-\mathcal R)\frac{\int_0^T B(t)\lambda(t)G(t)dt}{\int_0^T B(t)G(t)dt}$}
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>.
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