Model Calibration

In this lesson, we are taking the previously developed binomial lattice model and calibrating it so that the prices in the model agree with the corresponding market prices.  There are too many free parameters in the model, so we fix some parameters: q = 1-q = 0.50. and some some parametric form for for r(i,j) short-terms.  We will focus on the Black-Derman-Toy (BDT) Model.

The BDT model assumes that the interest rate at node N(i,j) is given by r(i,j) = a(i)*e^(b(i)*j) or in log terms: log(r(i,j)) = log(a(i)) + b(i)*j where log(ai) is a drift parameter for log(r) and b(i) is a volatility parameter for log(r).  Now we need to calibrate the model to the observed term-structure in the market.  This is done by choosing different a(i)'s and b(i)'s to match market models.  We can do this by using the Solver add-in in MS Excel, but we can also do this in Matlab or R.

To start an example, let us assume that we have an n-period binomial lattice, as usual.  We will let s(1)...s(n) be the term-structure of interest rates observed in the market.  We will also assume (for now) that b(i) = b for all i.  This is a very strong assumption and we will change it in the future.

We know that:

since this is just the definition of elementary prices of a zcb.
We can replace the right hand side of this equation with the forward equations from the last lesson:

where the first term is equal to P(i,0,e), the second term is equal to P(i,j,e) when j is between 1 and (i-1) and the third term is equal to P(i,i,e).  We can then begin solving for all the a(i)'s.  We could simply plug in i=1 then i=2 all the way up to i=n.  After simplifying and solving, we would have the formula for a(i) and we would be able to get the spot rate from the formula above: log(r(i,j)) = log(a(i)) + b(i)*j.  We can also use MS Excel Solver add-in to do this for us.  We then did exactly that in this module but I have omitted it from this blog post as an exercise for the reader.