Deriscope ## The Excel Derivatives Periscope

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survival_probability

The *survival probability* *P*(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 not have occurred by 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 *P*(0,T), for any fixed T, is a known number that represents the probability that the event E will not occur until T.

It follows that *P*(0,T) is always a decreasing function of T with *P*(0,T) -> 0 for T -> infinity.

Also *P*(0,0) = 1 provided that E has not occurred so far, otherwise *P*(0,0) = 0.

Note that *P*(t,T), for 0 < t < T is not a simple number, but rather a random variable, which may assume any possible number between 0 and 1 at the future time t.

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.