The model is a short-rate model. In general, it has the following dynamics: There is a degree of ambiguity among practitioners about exactly which parameters in the model are time-dependent or what name to apply to the model in each case. The most commonly accepted naming convention is the following:
has t dependence — the Hull–White model.
and are both time-dependent — the extended Vasicek model.
Two-factor model
The two-factor Hull–White model contains an additional disturbance term whose mean reverts to zero, and is of the form: where has an initial value of 0 and follows the process:
Analysis of the one-factor model
For the rest of this article we assume only has t-dependence. Neglecting the stochastic term for a moment, notice that for the change in r is negative if r is currently "large". σ is determined via calibration to a set of caplets and swaptions readily tradeable in the market. When,, and are constant, Itô's lemma can be used to prove that which has distribution where is the normal distribution with mean and variance. When is time-dependent, which has distribution
It turns out that the time-S value of the T-maturity discount bond has distribution where Note that their terminal distribution for is distributed log-normally.
Derivative pricing
By selecting as numeraire the time-S bond, 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, is the expectation taken with respect to the forward measure. Moreover, standard arbitrage arguments show that the time Tforward price for a payoff at time T given by V must satisfy, thus Thus it is possible to value many derivatives V dependent solely on a single bond analytically when working in the Hull–White model. For example, in the case of a bond put Because is lognormally distributed, the general calculation used for the Black–Scholes model shows that where and Thus today's value is: Here is the standard deviation of the log-normal distribution for. 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. This does not matter — the volatility is all that matters and is measure-independent. 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. Jamshidian's trick applies to Hull–White. Thus knowing how to price caps is also sufficient for pricing swaptions. In the even that the underlying is a compounded backward-looking rate rather than a LIBOR term rate, Turfus shows how this formula can be straightforwardly modified to take into account the additional convexity. Swaptions can also be priced directly as described in Henrard. Direct implementations are usually more efficient.
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. The efficient and exact Monte-Carlo simulation of the Hull–White model with time dependent parameters can be easily performed, see Ostrovski and.
