Hey rukimaru.

If it wants probabilities, then you need a probability density function given your variables which include time (t), some constant k, a loss of 20% and a lifetime of four years.

Usually the way these things are done is to use stochastic calculus, PDE's, or probability functions to derive the expectation function and/or the probability functions. You can model general pay-out functions which include random variables (like binary 0/1 random variables) which can be used to model defaults if you assume probabilities.

The start to this kind of thing in stochastic calculus is the Ito model which assumes volatility models that resemble normal distributions. There are ways to use general functions and that is what you may have to look for if you are to derive such models.

If you can't do this you can still resort to simulation. The simulation is actually very easy intuitively - all you do is to use something like R (google it if you don't know what it is) and you generate a bunch of random vectors (values are generated through random number generators) and use these values to generate random variables at all points of time. You can decide how many observations you need and modern computers will do it very quick.

So as an example, if you wanted to simulate f(A,B,C) = a^2 + 2^b +abc you would simulate vectors a,b,c and then compute f(a,b,c) for these vectors. This will form a distribution with a mean, variance, quartiles, and even 95% intervals.

If you need to leave answers in variable form, then you won't be able to simulate. If you fix all constants however (including k) then you will be able to simulate.

These are a few ideas of how to approach this problem. Also if you have someone work out the PDF function then you can use that too.