Principal Component Rate Models
Overview
One of the problems with interest rate risk models is the number of degrees of freedom inherent in the
problem. (number of factors influencing the curve) The simplest models try to model the curve as being the
result of a single factor. Models that fall into this category include:
- Short Rate Models - used modstly for derivative risk, these models take the short rate to the
stochastic factor and then derive the rest of the curve from it.
- Duration and Convexity
- model the curve from moves of all rates on the curve up and down
Single factor models can be useful, but they fail to capture the complexity of the curve.
Multi Factor Models
such as Heath Jarrow Morton
were designed to incorporate additional factors for derivative risk.
Key Rate Duration
models were developed to extend the normal duration and convexity models to deal with multiple degrees of freedom, however,
they require a large number of factors before they become useful, and their interpretation can also be complex.
The technique of
principal components
when applied to the rate curve can yield tractable models with only 2 or 3 factors.
Derivation
(Following
Back pg 245)
{% \Delta_{i,j} = y(t_i, \tau_j) - y(t_{i-1}, \tau_j) %}
Then, all the changes in a single period j, can be gathered in a vector {% \vec{\Delta_j} %}.
the
covariance of each item
in the vector with every other item can be computed, and denoted as {% \sigma %}.
Following the logic given in
principal-components,
we can identify a set of vectors that represent the directions of largest variation. That is, compute the
eigenvectors and eigenvalues
of the covariance matrix.
