Financial Simulation Modeling in Excel

BOUNDARY CONDITIONS

Let's return to equation A.9 and examine its solution more closely. We determined that the solution to this particular equation is 4x because the derivative of 4x is indeed 4. However, is this the full story? Recall that the derivative of a constant is zero. Therefore our general solution must be of the form in equation A.12:

where C is a constant. What does this mean, and why do we need to complicate matters even further? Moreover, how do we even figure out what C is, given that at this point it could be anything! We will discuss its importance a little further on, but C is determined by the boundary condition, or initial condition, of the system. This means that to determine C for equation A.12 we must first know the value of f(0).

I realize this all sounds very strange, so let's take as a real-life example, something that we are all familiar with. This example should also convey the importance of what a boundary condition is in a very physical sense. Just as in the discussion about Brownian motion, let's look at bond values that accumulate with a constant interest rate. At any time, t, its value will be (equation A.13):

Now we ask ourselves, if we were to invent a toy model, just for demonstration purposes, that were to relate dB, the differential of B, to ...