math lessons - Hull-White model

The Hull-White model is a mathematical model of future interest rates. It is relatively straight-forward to translate the mathematical description of the evolution of future interest rates on to 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 Hull and Alan White in 1990. The model that is still popular in the market today with practitioners was described by them in 1993.

The model

The model is a short-rate model. In general, it has dynamics

$dr_t = (\theta(t) - \alpha(t) r_t)\,dt + \sigma(t)\, dW_t\,\!$

There is a degree of ambiguity amongst practitioners about exactly which parameters in the model are time-dependent on what name to apply to the model in each case. The most commonly accepted hierarchy has

θ constant - the Vasicek model

θ has t dependence - the Hull-White model

θ and α also time-dependent - the extended Vasicek model

For the rest of this article we assume only theta has t-dependence. Neglecting the stochastic term for a moment, notice that the change in r is negative if r is currently "large" (greater than θ(t)/α) and positive if the current value is small. That is, the stochastic process is mean-reverting .

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.

Itô's lemma can be used to prove that

$r_S = \frac{\theta}{\alpha} + \exp(-\alpha S)(r_0 - \frac{\theta}{\alpha}) + \sigma\exp(-\alpha t)\int_0^S e^{\alpha x}\,dW_x\,\!$

which is normally distributed. It turns out that the time-S value of the T-maturity discount bond has distribution

$P(S,T) = A(S,T)\exp(-B(S,T)r(S))\!$

where

$B(S,T) = {1-\exp(-\alpha(T-S))}{\alpha} \,$

$A(S,T) = \frac{P(0,T)}{P(0,S)}\exp( \,$

$-B(S,T) \frac{\partial\log(P(0,t))}{dt} - \frac{\sigma^2(\exp(-\alpha T)-\exp(-aS))^2(\exp(2aS)-1)}{4a^3}) \,$

Note that their terminal distribution for P(S,T) is distributed log-normally. By selecting as numeraire the time-S bond, we have from the fundamental theorem of finance , the value of a derivative security at time S.

$V_0 = P(0,S)\mathbb{E}[V_S]\,$

Moreover that standard arbitrage arguments show that the forward price F for a contract V must satisfy F = P(0,S)V, thus

$F = \mathbb{E}[V_t].\,$

Thus it is possible to value many securities V dependent solely on a single bond P(S,T) analytically when working in the Hull-White model. For example a in the case of bond put

$V_t = (K - P(S,T))^+.\,\!$

Because P(S,T) is lognormally distributed, the general calculation used for Black-Scholes shows that

${E}[(K-P(S,T))^{+}] = KN(-d_2) - FN(d_1)\,$

where

$d_1 = \log(F/K) + \sigma_P^2S/2\,$

and

$d_2 = d_1 - \sigma_P \sqrt(S)\,$

Thus today's value (with the P(0,S) multiplied back in) is:

$P(0,S)KN(-d_2) - P(0,T)N(-d_1)\,$

Here σ_P is the standard deviation of the log-normal distribution for P(S,T). A fairly substantial amount of algebra shows that it is related to the original parameters via

$\sqrt{S}\sigma_P =\frac{\sigma}{\alpha}(1-\exp(-\alpha(T-S)))\sqrt{\frac{1-\exp(-2\alpha S)}{2a}}\,$

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 (as today's value of a swaption in HW is a monotonic function of today's short rate). Thus knowing how to price caps is also sufficient for pricing swaptions.

Trees and lattices

However valuing vanilla instruments such as caps and swaptions is useful primarily for calibration. The real use of the model is value somewhat more exotic options such as bermudan swaptions on a lattice.

References

John Hull and Alan White, "Using Hull-White interest rate trees," Journal of Derivatives, Vol. 3, No. 3 (Spring 1996), pp 26-36
John Hull and Alan White, "Numerical procedures for implementing term structure models I," Journal of Derivatives, Fall 1994, pp 7-16
John Hull and Alan White, "Numerical procedures for implementing term structure models II," Journal of Derivatives, Winter 1994, pp 37-48
John Hull and Alan White, "The pricing of options on interest rate caps and floors using the Hull-White model" in Advanced Strategies in Financial Risk Management, Chapter 4, pp 59-67.
John Hull and Alan White, "One factor interest rate models and the valuation of interest rate derivative securities," Journal of Financial and Quantitative Analysis, Vol 28, No 2, (June 1993) pp 235-254
John Hull and Alan White, "Pricing interest-rate derivative securities", The Review of Financial Studies, Vol 3, No. 4 (1990) pp. 573-592

Categories: Derivatives | Finance theories | Financial mathematics

Mathematics Encyclopedia and Lessons

Lessons

Popular
Subjects

References

Hull-White model

The model

Trees and lattices

References

Mathematics Encyclopedia and Lessons

Lessons

Popular Subjects

References

Hull-White model

The model

Trees and lattices

References

Popular
Subjects