Hazard Rate

Consider a credit default swap (CDS) (see Credit Default Swaps), where the premium payments are periodic and the terminal payment is a digital cash settlement of recovery rate 1 − images. For simplicity, we assume that the current time is normalized to t = 0, the risk-free rate r is constant throughout the maturity of the contract, and spreads are already given at standard interperiod rates (allowing us to ignore day-count fractions and division by period length). The cash flows of a CDS can be decomposed into the default leg and the premium leg. The default leg is a single lump sum compensation for the loss images on the face value of the reference asset made at the default time by the protection seller to the protection buyer, given that the default is before the expiration date T of the contract. The premium leg consists of the fees, called the CDS spread, paid by the protection buyer at dates tm (assumed to be equidistant e.g., quarterly) until the default event or T, whichever is first. The spread S is given as a fraction of the unit notional.

A concise mathematical expression for both legs can be obtained via a point process representation. Suppose we have a filtered probability ...

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