Hull-White model - Citizendia

In financial mathematics, the Hull-White model is a model of future interest rates. Mathematical finance is the branch of Applied mathematics concerned with the Financial markets. Note The term model has a different meaning in Model theory, a branch of Mathematical logic. Interest is a fee paid on borrowed capital Assets lent include Money, Shares, Consumer goods through Hire purchase, major assets 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 straight-forward 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. An interest rate derivative is a derivative where the underlying asset is the right to pay or receive a (usually notional amount of Money at a given Interest A swaption is an option granting its owner the right but not the obligation to enter into an underlying swap.

The first Hull-White model was described by John Hull and Alan White in 1990. John C Hull is a Professor of Derivatives and Risk Management at the Rotman School of Management at the University of Toronto. The model is still popular in the market today.

1 The model
2 Bond Pricing Using The Hull-White Model
3 Derivative Pricing
4 Trees and lattices
5 See also
6 References

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

The two-factor Hull-White model contains an additional disturbance term which mean reverts to zero. In finance, the Vasicek model is a Mathematical model describing the evolution of Interest rates It is a type of "one-factor model" ( Short In finance, the Vasicek model is a Mathematical model describing the evolution of Interest rates It is a type of "one-factor model" ( Short

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 a mean-reverting Ornstein-Uhlenbeck process. Mean reversion is a mathematical methodology commonly used for stock investing but it can be applied to other processes

θ is calculated from the initial yield curve describing the current term structure of interest rates. In Finance, the yield curve is the relation between the Interest rate (or cost of borrowing and the time to maturity of the debt for a given borrower 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. Calibration is the process of establishing the relationship between a measuring device and the units of measure A swaption is an option granting its owner the right but not the obligation to enter into an underlying swap.

When $α,θ$ and $σ$ are constant, Itô's lemma can be used to prove that

$r(t) = e^{-\alpha t}r(0) + \frac{\theta}{\alpha} \left(1- e^{-\alpha t}\right) + \sigma e^{-\alpha t}\int_0^t e^{\alpha u}\,dW(u)\,\!$

which has distribution

$r(t) \sim N(e^{-\alpha t} r(0) + \frac{\theta}{\alpha} \left(1- e^{-\alpha t}\right), \frac{\sigma^2}{2\alpha} \left(1-e^{-2\alpha t}\right)).$

where $N (. In Mathematics, Itō 's lemma is used in Itō stochastic calculus to find the differential of a function of a particular,. )$ is the normal distribution. The normal distribution, also called the Gaussian distribution, is an important family of Continuous probability distributions applicable in many fields

Bond Pricing Using The Hull-White Model

It turns out that the time-S value of the T-maturity discount bond has distribution (note the affine term structure here!)

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

where

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

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

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

Note that their terminal distribution for P(S,T) is distributed log-normally. A Zero coupon bond (also called a discount bond or deep discount bond) is a bond bought at a price lower than its Face value, with the face value In Probability and Statistics, the log-normal distribution is the single-tailed Probability distribution of any Random variable whose

Derivative Pricing

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 0 of a derivative which has payoff at time S. Numéraire is a basic standard by which values are measured such as gold in a monetary system In a general sense the fundamental theorem of arbitrage/finance is a way to relate Arbitrage opportunities with risk neutral measures that are equivalent to the original probability

$V(t) = P(t,S)\mathbb{E}_S[V(S)| \mathcal{F}(t)]\,$ .

Here, $\mathbb{E}_S$ is the expectation taken with respect to the forward measure. A T-forward measure is a pricing measure absolutely continuous with respect to a Risk-neutral measure but rather than using the money market as Numeraire, it uses a bond with Moreover that standard arbitrage arguments show that the time T forward price $F V (t, T)$ for a payoff at time T given by V(T) must satisfy $F V (t, T) = V (t) / P (t, S)$ , thus

$F_V(t,T) = \mathbb{E}_T[V(T)|\mathcal{F}(t)].\,$

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

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

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

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

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). Example of a put option on a stock Buy a Put A Buyer thinks price of a stock will decrease A fairly substantial amount of algebra shows that it is related to the original parameters via

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

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. Interest rate cap An interest rate cap is a derivative in which the buyer receives payments at the end of each period in which the interest rate exceeds the agreed 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 to value somewhat more exotic options such as bermudan swaptions on a lattice. A swaption is an option granting its owner the right but not the obligation to enter into an underlying swap.

References

John Hull and Alan White, "Using Hull-White interest rate trees," Journal of Derivatives, Vol. In finance, the Vasicek model is a Mathematical model describing the evolution of Interest rates It is a type of "one-factor model" ( Short The Cox-Ingersoll-Ross model in finance is a Mathematical model describing the evolution of Interest rates It is a type of "one factor model" 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

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Contents

The model

Bond Pricing Using The Hull-White Model

Derivative Pricing

Trees and lattices

See also

References