Hull White Model

Theta - Critical Note

In a previous blog we presented an implementation of the Generalised Hull-White model (2014). This implementation relies on a numerical root-finding routine to determine model parameters so that model prices match actual market prices. The root finding is far from trivial as different algorithms will give different results. Fortunately the quality of the result can be easily monitored as the difference between model prices and actual marked prices is a clear indicator. As it turns out in most cases, the root finding for theta is not required. Instead an analytical formula can be used. Mike Staunton very kindly pointed us to a paper by Grant & Vora (2001) that shows that the approximation used by Hull-White is not necessary. The equations explained in the paper are implemented in the attached spreadsheet by Mike Stauton for the first 3 steps of the lattice giving a good example of how the process for finding theta works in practice.

 

blog theta

 


The spreadsheet is available here:

MikeHWTriTreeGV


 

 

References:

Grant & Vora (2001) , An Analytical Implementation of the Hull and White Model

http://janroman.dhis.org/finance/Hull%20&%20White/analytical-approximation-hull-and-white.pdf

 

Hull & White (2014), A Generalized Procedure for Building Trees for the Short Rate and its

Application to Determining Market Implied Volatility Functions

http://www-2.rotman.utoronto.ca/~hull/downloadablepublications/TreeBuilding.pdf

Generalized Procedure for Building Short Rate Trees in Excel / VBA

In their 2014 paper John Hull and Alan White derive generalized method for the construction of short rate trees. This generalization is interesting as it allows for one tree (or lattice) construction algorithm for all one factor short rate models. The only difference between the various models is the function H(x,t), which is explained briefly here and in detail in the paper. Continue reading