## Hazard Rates

A discrete forward hazard rate in some time interval given survival at the start of the interval is:

The instantaneous hazard rate (the limit as ) is:

Which allows us to write:

### Value of payments on default

Consider a cashflow paying at time . If default occurs at time , and , then the value at is:

This is approximated by assuming a constant forward hazard rate and forward rate between and so that

where

For a unit cashflow paid on default, and

For credit derivatives, define the following:

- is the deal effective date
- are the accrual period end dates
- are the coupon payment dates
- is the principal for the period to
- is the accrual year fraction for the period to
- is coupon paid at time
- and , is the valuation date
- is the recovery rate value on the survival probability price factor and is for

## DealDefaultSwap

This is a bilateral agreement where the buyer purchases protection from the seller against default of a reference entity with period fixed payments. Should default of the reference entity occur, the seller pays the buyer the default amount and payment ceases. The default amount is , where is the principal amount. Note that there could also be an accrued fee (but is currently ignored).

Assuming the default payment does not occur prior to the effective date of the swap, the value of this deal at is

where is the fixed payment rate and

Note that this is further approximated by

which is accurate for low to moderate rates of default.