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.
The spreadsheet is available here:
Grant & Vora (2001) , An Analytical Implementation of the Hull and White Model
Hull & White (2014), A Generalized Procedure for Building Trees for the Short Rate and its
Application to Determining Market Implied Volatility Functions
While short rate models have lost ground to libor market models for valuation of (complex) interest rate derivatives, they remain popular for Asset and Liability Management. As a result you may find yourself implementing a short rate model. Often the choice is to implement a lattice approach, perhaps in combination with a Monte Carlo framework. One critical step in these models is to match the model discounts to the ones observed in the market. This makes to model (almost) arbitrage free. To match the market a numerical root finding algorithm needs to be used, for example Newton Raphson. This poses the challenge that mistakes in the model may not be obvious as the root finder can still find parameters to match the market even though there are mistakes in the model. Our suggestion is to check the root finder solution to known analytical solution of the Hull White model using functional form for the zero curve that is continuous and two times differentiable.
In a previous we blog we reproduced example 1 of the 2014 paper John Hull and Alan White “Generalized Procedure for Building Short Rate Trees”. In the paper the authors derive a 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. Continue reading
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 , which is explained briefly here and in detail in the paper. Continue reading
In a recent (2014) paper John Hull and Alan White demonstrate a generalized method for the construction of short rate trees. Keen to understand the model we tried to reproduce the results of the first example mentioned in the paper on page 10. The example considers the short rate model:
which is transformed using