Hull–White model

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 dynamics

dr(t)=\left[\theta (t)-\alpha (t)r(t)\right]\,dt+\sigma (t)\,dW(t)\,\!

There is a degree of ambiguity amongst 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 hierarchy has

θ has t dependence – the Hull-White model

θ and α also 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))=\left[\theta (t)+u-\alpha (t)\,f(r(t))\right]dt+\sigma _{1}(t)\,dW_{1}(t)\!

where $\displaystyle u$ has an initial value of 0 and follows the process:

du=-bu\,dt+\sigma _{2}\,dW_{2}(t)

Analysis of the one-factor 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 a mean-reverting Ornstein–Uhlenbeck process.

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.

When $\alpha$ , $\theta$ , and $\sigma$ 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 {\mathcal {N}}\left(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)\right).

where ${\mathcal {N}}(\mu ,\sigma ^{2})$ is the normal distribution with mean $\mu$ and variance $\sigma ^{2}$ .

When $\theta (t)$ is time dependent,

r(t)=e^{-\alpha t}r(0)+\int _{0}^{t}e^{\alpha (s-t)}\theta (s)ds+\sigma e^{-\alpha t}\int _{0}^{t}e^{\alpha u}\,dW(u)\,\!

which has distribution

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

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 \left(\,-B(S,T){\frac {\partial \log(P(0,S))}{\partial S}}-{\frac {\sigma ^{2}(\exp(-\alpha T)-\exp(-\alpha S))^{2}(\exp(2\alpha S)-1)}{4\alpha ^{3}}}\right)\,

Note that their terminal distribution for P(S,T) is distributed log-normally.

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.

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. 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 $F_{V}(t,T)=V(t)/P(t,T)$ , 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}={\frac {\log(F/K)+\sigma _{P}^{2}S/2}{\sigma _{P}{\sqrt {S}}}}\,

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)}{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. 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.

The swaptions can also be priced directly as described in Henrard (2003). The direct implementation is usually more efficient.

Monte-Carlo simulation, 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 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).

References

Primary 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

Other references

Hull, John C. (2006). "Interest Rate Derivatives: Models of the Short Rate". Options, Futures, and Other Derivatives (6th ed.). Upper Saddle River, N.J: Prentice Hall. pp. 657–658. ISBN 0-13-149908-4. LCCN 2005047692. OCLC 60321487.
Damiano Brigo, Fabio Mercurio (2001). Interest Rate Models — Theory and Practice with Smile, Inflation and Credit (2nd ed. 2006 ed.). Springer Verlag. ISBN 978-3-540-22149-4.
Henrard, Marc (2003). Explicit Bond Option and Swaption Formula in Heath-Jarrow-Morton One Factor Model, International Journal of Theoretical and Applied Finance, 6(1), 57-72. Preprint SSRN.
Henrard, Marc (2009). Efficient swaptions price in Hull-White one factor model, arXiv, 0901.1776v1. Preprint arXiv.
Ostrovski, Vladimir (2013). Efficient and Exact Simulation of the Hull-White Model, Preprint SSRN.
Ostrovski, Vladimir (2016). Efficient and Exact Simulation of the Gaussian Affine Interest Rate Models., International Journal of Financial Engineering, Vol. 3, No. 02.
Puschkarski, Eugen. Implementation of Hull-White´s No-Arbitrage Term Structure Model, Diploma Thesis, Center for Central European Financial Markets
Letian Wang, Hull-White Model, Fixed Income Quant Group, DTCC (detailed numeric example and derivation)

Online utilities

Hull-White Trinomial Tree, Dr. S.H. Man, Turaz.
Short Rates Simulation using Hull White Model, Dr. S.H. Man, Turaz.

Bond market
Bond Debenture Fixed income
Types of bonds by issuer	Agency bond Corporate bond Senior debt Subordinated debt Distressed debt Emerging market debt Government bond Municipal bond
Types of bonds by payout	Accrual bond Auction rate security Callable bond Commercial paper Contingent convertible bond Convertible bond Exchangeable bond Extendible bond Fixed rate bond Floating rate note High-yield debt Inflation-indexed bond Inverse floating rate note Perpetual bond Puttable bond Reverse convertible securities Zero-coupon bond
Bond valuation	Clean price Convexity Coupon Credit spread Current yield Dirty price Duration I-spread Mortgage yield Nominal yield Option-adjusted spread Risk-free bond Weighted-average life Yield curve Yield spread Yield to maturity Z-spread
Securitized products	Asset-backed security Collateralized debt obligation Collateralized mortgage obligation Commercial mortgage-backed security Mortgage-backed security
Bond options	Callable bond Convertible bond Embedded option Exchangeable bond Extendible bond Puttable bond
Institutions	Commercial Mortgage Securities Association (CMSA) International Capital Market Association (ICMA) Securities Industry and Financial Markets Association (SIFMA)

Stochastic processes
Discrete time	Bernoulli process Branching process Chinese restaurant process Galton–Watson process Independent and identically distributed random variables Markov chain Moran process Random walk Loop-erased Self-avoiding Biased Maximal entropy
Continuous time	Bessel process Birth–death process Brownian motion Bridge Excursion Fractional Geometric Meander Cauchy process Contact process Continuous-time random walk Cox process Diffusion process Empirical process Feller process Fleming–Viot process Gamma process Hunt process Interacting particle systems Itô diffusion Itô process Jump diffusion Jump process Lévy process Local time Markov additive process McKean–Vlasov process Ornstein–Uhlenbeck process Poisson process Compound Non-homogeneous Point process Schramm–Loewner evolution Semimartingale Sigma-martingale Stable process Superprocess Telegraph process Variance gamma process Wiener process Wiener sausage
Both	Branching process Galves–Löcherbach model Gaussian process Hidden Markov model (HMM) Markov process Martingale Differences Local Sub- Super- Random dynamical system Regenerative process Renewal process Stochastic chains with memory of variable length White noise
Fields and other	Dirichlet process Gaussian random field Gibbs measure Hopfield model Ising model Potts model Boolean network Markov random field Percolation Pitman–Yor process Point process Cox Poisson Random field Random graph
Time series models	Autoregressive conditional heteroskedasticity (ARCH) model Autoregressive integrated moving average (ARIMA) model Autoregressive (AR) model Autoregressive–moving-average (ARMA) model Generalized autoregressive conditional heteroskedasticity (GARCH) model Moving-average (MA) model
Financial models	Black–Derman–Toy Black–Karasinski Black–Scholes Chen Constant elasticity of variance (CEV) Cox–Ingersoll–Ross (CIR) Garman–Kohlhagen Heath–Jarrow–Morton (HJM) Heston Ho–Lee Hull–White LIBOR market Rendleman–Bartter SABR volatility Vašíček Wilkie
Actuarial models	Bühlmann Cramér–Lundberg Risk process Sparre–Anderson
Queueing models	Bulk Fluid Generalized queueing network M/G/1 M/M/1 M/M/c
Properties	Càdlàg paths Continuous Continuous paths Ergodic Exchangeable Feller-continuous Gauss–Markov Markov Mixing Piecewise deterministic Predictable Progressively measurable Self-similar Stationary Time-reversible
Limit theorems	Central limit theorem Donsker's theorem Doob's martingale convergence theorems Ergodic theorem Fisher–Tippett–Gnedenko theorem Large deviation principle Law of large numbers (weak/strong) Law of the iterated logarithm Maximal ergodic theorem Sanov's theorem
Inequalities	Burkholder–Davis–Gundy Doob's martingale Kunita–Watanabe
Tools	Cameron–Martin formula Convergence of random variables Doléans-Dade exponential Doob decomposition theorem Doob–Meyer decomposition theorem Doob's optional stopping theorem Dynkin's formula Feynman–Kac formula Filtration Girsanov theorem Infinitesimal generator Itô integral Itô's lemma Kolmogorov continuity theorem Kolmogorov extension theorem Lévy–Prokhorov metric Malliavin calculus Martingale representation theorem Optional stopping theorem Prokhorov's theorem Quadratic variation Reflection principle Skorokhod integral Skorokhod's representation theorem Skorokhod space Snell envelope Stochastic differential equation Tanaka Stopping time Stratonovich integral Uniform integrability Usual hypotheses Wiener space Classical Abstract
Disciplines	Actuarial mathematics Econometrics Ergodic theory Extreme value theory (EVT) Large deviations theory Mathematical finance Mathematical statistics Probability theory Queueing theory Renewal theory Ruin theory Statistics Stochastic analysis Time series analysis Machine learning
List of topics Category

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.