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.
The model
One-factor model The model is a short-rate model. In general, it has the following dynamics:
d r ( t ) = [ θ ( t ) − α ( t ) r ( t ) ] d t + σ ( t ) d W ( t ) . {\displaystyle dr(t)=\left[\theta (t)-\alpha (t)r(t)\right]\,dt+\sigma (t)\,dW(t).}
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:
θ {\displaystyle \theta } has t (time) dependence — the Hull–White model.
θ {\displaystyle \theta } and α {\displaystyle \alpha } are both time-dependent — the extended Vasicek model.
Two-factor model 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:
d f ( r ( t ) ) = [ θ ( t ) + u − α ( t ) f ( r ( t ) ) ] d t + σ 1 ( t ) d W 1 ( t ) , {\displaystyle d\,f(r(t))=\left[\theta (t)+u-\alpha (t)\,f(r(t))\right]dt+\sigma _{1}(t)\,dW_{1}(t),}
where f {\displaystyle \displaystyle f} 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 the Black–Karasinski model, 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 u {\displaystyle \displaystyle u} has an initial value of 0 and follows the process:
d u = − b u d t + σ 2 d W 2 ( t ) {\displaystyle du=-bu\,dt+\sigma _{2}\,dW_{2}(t)}
Analysis of the one-factor model For the rest of this article we assume only θ {\displaystyle \theta } has t-dependence. Neglecting the stochastic term for a moment, notice that for α > 0 {\displaystyle \alpha >0} the change in r is negative if r is currently "large" (greater than θ ( t ) / α ) {\displaystyle \theta (t)/\alpha )} and positive if the current value is small. That is, the stochastic process is a mean-reverting Ornstein–Uhlenbeck process. θ is calculated from the initial yield curve describing the current term structure of interest rates. For constant α {\displaystyle \alpha } and σ {\displaystyle \sigma } , the time-dependent drift can be determined explicitly from the initial term structure. Let P M ( 0 , T ) {\displaystyle P^{M}(0,T)} denote the market price at time 0 of a zero-coupon bond maturing at T {\displaystyle T} , and define the market instantaneous forward rate by
f M ( 0 , T ) = − ∂ ∂ T ln P M ( 0 , T ) . {\displaystyle f^{M}(0,T)=-{\frac {\partial }{\partial T}}\ln P^{M}(0,T).}
An exact fit to the initial term structure is obtained by choosing
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