Credit default swap ($\text{premium payments}\leftrightarrow N(1-\mathcal{R})\text{ at default}$)

The protection buyer pays a contractual premium in exchange for compensation upon a defined credit event of the reference entity. In a simple cash-settled contract of notional $N$, a default loss with fractional recovery $\mathcal{R}$ gives payment $N(1-\mathcal{R})$. Physical settlement instead exchanges an eligible defaulted obligation for par. Payments, accrued premium, maturity and eligible obligations depend on the contract. The protection buyer need not own the reference debt; the seller earns premium while taking the specified credit risk.

A basket credit contract whose protection payment is triggered by the first specified default among its reference entities. The loss and settlement terms refer to the defaulting name according to the contract. Premium generally stops at that first default or maturity. Pricing requires the joint default-time distribution; separate marginal default probabilities alone do not determine the first-default law.

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.