Deriscope

## The Excel Derivatives Periscope

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default_density

The default density g(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.
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 g(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.
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 g(t,T) would simply represent the probability of defaulting within on year from T.
So for example g(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).
This interpretation breaks down if the likelihood of default does not stay constant between T and T+1, in which case g(0,T) may even exceed 1.
The following mathematical relations hold between the
survival probability P(T), default density g(T) and hazard rate h(T), where P(T) stands for P(t,T) with the first time parameter dropped for notational simplicity and similarly for g(T) and h(T):
P(T) = exp(-I(T))
g(T) = h(T)exp(-I(T))
where I(T) is the time integral of h(t,u) over the second parameter u from t to T.
It turns out that the knowledge of any one of P, g or h as a function of time suffices to compute the value of the other two as well.