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 most commonly accepted naming convention is the following: The two-factor Hull–White model (Hull 2006:657–658) contains an additional disturbance term whose mean reverts to zero, and is of the form: where
is a deterministic function, typically the identity function (extension of the one-factor version, analytically tractable, and with potentially negative rates), the natural logarithm (extension of Black–Karasinski, not analytically tractable, and with positive interest rates), or combinations (proportional to the natural logarithm on small rates and proportional to the identity function on large rates); and
has an initial value of 0 and follows the process: For the rest of this article we assume only
θ is calculated from the initial yield curve describing the current term structure of interest rates.
Typically α is left as a user input (for example it may be estimated from historical data).
σ is determined via calibration to a set of caplets and swaptions readily tradeable in the market.
By selecting as numeraire the time-S bond (which corresponds to switching to the S-forward measure), we have from the fundamental theorem of arbitrage-free pricing, the value at time t of a derivative which has payoff at time S. Here,
Moreover, standard arbitrage arguments show that the time T forward price
, thus Thus it is possible to value many derivatives V dependent solely on a single bond
is lognormally distributed, the general calculation used for the Black–Scholes model shows that where and Thus today's value (with the P(0,S) multiplied back in and t set to 0) is: Here
A fairly substantial amount of algebra shows that it is related to the original parameters via Note that this expectation was done in the S-bond measure, whereas we did not specify a measure at all for the original Hull–White process.
Because interest rate caps/floors are equivalent to bond puts and calls respectively, the above analysis shows that caps and floors can be priced analytically in the Hull–White model.
In the event that the underlying is a compounded backward-looking rate rather than a (forward-looking) LIBOR term rate, Turfus (2020) shows how this formula can be straightforwardly modified to take into account the additional convexity.
However, valuing vanilla instruments such as caps and swaptions is useful primarily for calibration.
The real use of the model is to value somewhat more exotic derivatives such as bermudan swaptions on a lattice, or other derivatives in a multi-currency context such as Quanto Constant Maturity Swaps, as explained for example in Brigo and Mercurio (2001).
The efficient and exact Monte-Carlo simulation of the Hull–White model with time dependent parameters can be easily performed, see Ostrovski (2013) and (2016).
An open-source implementation of the exact Monte-Carlo simulation following Fries (2016)[1] can be found in finmath lib.
In Orlando et al. (2018,[3] 2019,[4][5]) was provided a new methodology to forecast future interest rates called CIR#.
The ideas, apart from turning a short-rate model used for pricing into a forecasting tool, lies in an appropriate partitioning of the dataset into subgroups according to a given distribution [6]).
In there it was shown how the said partitioning enables capturing statistically significant time changes in volatility of interest rates.