Single Factor Model Implementation
Overview
The generic model is a generic functional form of various short rate models listed below.
{% dr = u(r, t) dt + v(r, t) dW %}
Mathematical Tools
Short rate models are typically expressed in their most general form as differential equations. However, from
a computational perspective, implementing short rate models can take serveral forms. In the simplest form,
a short rate can be modeled as a binomial tree. It should be recognized though that a binomial tree is just
a discrete approximation to the more general continuous time form.
Calibration
Once the model has been specified, it needs to be calibrated to the market in order to be useful.
The calibration of the model will depend on what its to be used for. Most models were designed to be
used as risk neutral models of the short rate. That is, they assume the measure being used is
the risk neutral measure, not the real world measure, so they cannot be calibrated by just estimating
the parameters straight from real world data
On the other hand, the short rate models can be used using a real world measure. However, they cant use the equation
above to recover the term structure, only the risk neutral measure can do that. This does not make short rate
real world modeling useless, it only just limits it to modeling floating rate instruments.
For models which have only a few constant parameters, it is impossible to fit the entire curve. That is,
a model with only 3 parameters, say, could only fit 3 bond prices. As such, models that fit the curve
need to have a parameter which is function of time in order to have enough degrees of freedom to fit
the curve. (see
extended vasicek model for example)
