## default density

The

*default density***(t,T) for some predefined stochastic event E (often thought as a "default" event) and times t, T such that t <= T is defined as the probability that the event E will occur in the interval (T,T+dt) divided by dt.**

*g*It represents the unconditional instantaneous rate of default probability at time T.

In Finance one usually sets the time origin t = 0 to mean today's date, or more precisely the current instant, when all market information is assumed to be known with certainty.

Then

**(0,T), for any fixed T, is a known number, but**

*g***(t,T), for 0 < t < T is a random variable, which may assume any positive number at the future time t.**

*g*Intuitively one can make the simplifying assumption that the likelihood of default stays constant within one year from T, i.e. stays constant in the interval (T,T+1), in which case

**(t,T) would simply represent the probability of defaulting within on year from T.**

*g*So for example

**(0,5) = 0.07 would be interpreted as the existence of a 7% chance that a default will occur during the sixth year, i.e. in the time interval (5,5+1).**

*g*This interpretation breaks down if the likelihood of default does not stay constant between T and T+1, in which case

**(0,T) may even exceed 1.**

*g*The following mathematical relations hold between the survival probability

**, default density**

*P(T)***and hazard rate**

*g(T)***, where**

*h(T)***stands for**

*P(T)***(t,T) with the first time parameter dropped for notational simplicity and similarly for**

*P***and**

*g(T)***:**

*h(T)***=**

*P(T)*

*e⁻ᴵ⁽ᵀ⁾***=**

*g(T)*

*h(T)*

*e⁻ᴵ⁽ᵀ⁾*where

**is the time integral of h(t,u) over the second parameter u from t to T.**

*I(T)*It turns out that the knowledge of any one of

**,**

*P***or**

*g***as a function of time suffices to compute the value of the other two as well.**

*h*