# Hull-White Model¶

## Overview¶

In financial mathematics, the Hull-White model is a model of future interest rates. In its most generic formulation, it belongs to the class of no-arbitrage models that are able to fit today’s term structure of interest rates. It is relatively straightforward to translate the mathematical description of the evolution of future interest rates onto a tree or lattice and so interest rate derivatives such as Bermudan Swaptions can be valued in the model. The first Hull-White model was described by John C. Hull and Alan White in 1990. The model is still popular in the market today (from Wiki).

## Implementation¶

This section mainly introduces the implementation process of short-rate and discount, which is core part of option pricing, and applied in Tree Engine and FD (finite-difference method) Engine.

As an important part of the Tree/ FD Engines, the class $$HWModel$$ implements the single-factor Hull-White model to calculate short-rate and discount by using continuous compounding, including 4 functions (treeShortRate, fdShortRate, discount, discountBond). The implementation process is introduced as follows:

1. Function treeShortRate:
1. The short-rates is calculated at time point $$t$$ with the duration $$dt$$ from 0 to N point-by-point. As in the calculation process, the variable $$value$$ needs to be calculated first. To improve the initiation interval (II), an array $$values16$$ is implemented to store the intermediate results from each iteration. Then, an addition tree is performed subsequently to achieve an II = 1 for the whole process. Finally, the short rate is calculated using variable $$value$$.
2. For implementing the generic Tree framework, the $$state\_price$$ calculating process is moved from Tree Lattice to this Model.
2. Function fdShortRate: The short-rate is calculated at time point $$t$$.
3. Function discount: The discount is calculated at time point $$t$$ with the duration $$dt$$ based on the short-rate.
4. Function discountBond: The discount bond is calculated at time point $$t$$ with the duration $$dt=T-t$$ based on the short-rate.