Hull–White model

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

:$dr(t)\; =\; \backslash left[\backslash theta(t)\; -\; \backslash alpha(t)\; r(t)\backslash right]\backslash ,dt\; +\; \backslash sigma(t)\backslash ,\; dW(t).$

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:

• $\backslash theta$ has t (time) dependence — the Hull–White model.
• $\backslash theta$ and $\backslash alpha$ are both time-dependent — the extended Vasicek model.

The two-factor Hull–White model {{harvcol|Hull|2006|pp=657–658}} contains an additional disturbance term whose mean reverts to zero, and is of the form:

:$d\backslash ,f(r(t))\; =\; \backslash left\; [\backslash theta(t)\; +\; u\; -\; \backslash alpha(t)\backslash ,f(r(t))\backslash right\; ]dt\; +\; \backslash sigma\_1(t)\backslash ,\; dW\_1(t),$

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

:$du\; =\; -bu\backslash ,dt\; +\; \backslash sigma\_2\backslash ,dW\_2(t)$

For the rest of this article we assume only $\backslash theta$ has t-dependence.

Neglecting the stochastic term for a moment, notice that for $\backslash alpha\; >\; 0$ the change in r is negative if r is currently "large" (greater than $\backslash theta(t)/\backslash alpha)$ 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. 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 $\backslash alpha$, $\backslash theta$, and $\backslash sigma$ are constant, Itô's lemma can be used to prove that

:$r(t)\; =\; e^\{-\backslash alpha\; t\}r(0)\; +\; \backslash frac\{\backslash theta\}\{\backslash alpha\}\; \backslash left(1-\; e^\{-\backslash alpha\; t\}\backslash right)\; +\; \backslash sigma\; e^\{-\backslash alpha\; t\}\backslash int\_0^t\; e^\{\backslash alpha\; u\}\backslash ,dW(u),$

which has distribution

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

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

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

:$r(t)\; =\; e^\{-\backslash alpha\; t\}r(0)\; +\; \backslash int\_\{0\}^\{t\}e^\{\backslash alpha(s-t)\}\backslash theta(s)ds\; +\; \backslash sigma\; e^\{-\backslash alpha\; t\}\backslash int\_0^t\; e^\{\backslash alpha\; u\}\backslash ,dW(u),$

which has distribution

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

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

:$P(S,T)\; =\; A(S,T)\backslash exp(-B(S,T)r(S)),$

where

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

:$A(S,T)\; =\; \backslash frac\{P(0,T)\}\{P(0,S)\}\backslash exp\backslash left(\; \backslash ,\; -B(S,T)\; \backslash frac\{\backslash partial\backslash log(P(0,S))\}\{\backslash partial\; S\}\; -\; \backslash frac\{\backslash sigma^2(\backslash exp(-\backslash alpha\; T)-\backslash exp(-\backslash alpha\; S))^2(\backslash exp(2\backslash alpha\; S)-1)\}\{4\backslash alpha^3\}\backslash right)\; .$

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

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 t of a derivative which has payoff at time S.

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

Here, $\backslash mathbb\{E\}\_S$ is the expectation taken with respect to the forward measure. Moreover, 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

:$F\_V(t,T)\; =\; \backslash mathbb\{E\}\_T[V(T)\backslash mid\backslash mathcal\{F\}(t)].$

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 the Black–Scholes model shows that

:$\{E\}\_S[(K-P(S,T))^\{+\}]\; =\; KN(-d\_2)\; -\; F(t,S,T)N(-d\_1),$

where

:$d\_1\; =\; \backslash frac\{\backslash log(F/K)\; +\; \backslash sigma\_P^2S/2\}\{\backslash sigma\_P\; \backslash sqrt\{S\}\}$

and

:$d\_2\; =\; d\_1\; -\; \backslash sigma\_P\; \backslash sqrt\{S\}.$

Thus today's value (with the P(0,S) multiplied back in and t set to 0) is:

:$P(0,S)KN(-d\_2)\; -\; P(0,T)N(-d\_1).$

Here $\backslash sigma\_P$ is the standard deviation (relative volatility) 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

:$\backslash sqrt\{S\}\backslash sigma\_P$

=\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 the Hull–White model is a monotonic function of today's short rate). Thus knowing how to price caps is also sufficient for pricing swaptions. In the event that the underlying is a compounded backward-looking rate rather than a (forward-looking) LIBOR term rate, Turfus (2020) 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 (2003). 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 (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). An open-source implementation of the exact Monte-Carlo simulation following Fries (2016){{cite journal |last= Fries |first= Christian |date=2016 |title=A Short Note on the Exact Stochastic Simulation Scheme of the Hull-White Model and Its Implementation |url=https://ssrn.com/abstract=2737091 |journal=SSRN |doi=10.2139/ssrn.2737091 |access-date=October 15, 2023}} can be found in finmath lib.{{cite web |url=https://github.com/finmath/finmath-lib/blob/master/src/main/java/net/finmath/montecarlo/interestrate/models/HullWhiteModel.java |title=HullWhiteModel.java |website=finmath lib |publisher=finmath.net |access-date=October 15, 2023}}

Even though single factor models such as Vasicek, CIR and Hull–White model has been devised for pricing, recent research has shown their potential with regard to forecasting. In Orlando et al. (2018,{{cite journal |last1=Orlando |first1=Giuseppe |last2=Mininni |first2=Rosa Maria |last3=Bufalo |first3=Michele |title=A New Approach to CIR Short-Term Rates Modelling |journal=New Methods in Fixed Income Modeling |series=Contributions to Management Science |date=2018 |pages=35–43 |doi=10.1007/978-3-319-95285-7_2 |publisher=Springer International Publishing|isbn=978-3-319-95284-0 }} 2019,{{cite journal |last1=Orlando |first1=Giuseppe |last2=Mininni |first2=Rosa Maria |last3=Bufalo |first3=Michele |title=A new approach to forecast market interest rates through the CIR model |journal=Studies in Economics and Finance |date=1 January 2019 |volume=37 |issue=2 |pages=267–292 |doi=10.1108/SEF-03-2019-0116 |s2cid=204424299 |issn=1086-7376}}{{cite journal |last1=Orlando |first1=Giuseppe |last2=Mininni |first2=Rosa Maria |last3=Bufalo |first3=Michele |title=Interest rates calibration with a CIR model |journal=The Journal of Risk Finance |date=19 August 2019 |volume=20 |issue=4 |pages=370–387 |doi=10.1108/JRF-05-2019-0080 |s2cid=204435499 |language=en |issn=1526-5943}}) was provided a new methodology to forecast future interest rates called CIR#.

The ideas, apart from turning a short-rate model used for pricing into a forecasting tool, lies in an appropriate partitioning of the dataset into subgroups according to a given distribution {{Cite journal|last1=Orlando|first1=Giuseppe|last2=Mininni|first2=Rosa Maria|last3=Bufalo|first3=Michele|date=July 2020|title=Forecasting interest rates through Vasicek and CIR models: A partitioning approach|url=https://onlinelibrary.wiley.com/doi/10.1002/for.2642|journal=Journal of Forecasting|language=en|volume=39|issue=4|pages=569–579|doi=10.1002/for.2642|issn=0277-6693|arxiv=1901.02246|s2cid=126507446 }}).

In there it was shown how the said partitioning enables capturing statistically significant time changes in volatility of interest rates. following the said approach, Orlando et al. (2021) {{Cite journal|last1=Orlando|first1=Giuseppe|last2=Bufalo|first2=Michele|date=2021-05-26|title=Interest rates forecasting: Between Hull and White and the CIR#—How to make a single‐factor model work|journal=Journal of Forecasting|volume=40 |issue=8 |language=en|pages=1566–1580|doi=10.1002/for.2783|issn=0277-6693|doi-access=free}}) compares the Hull–White model with the CIR model in terms of forecasting and prediction of interest rate directionality.

• Vasicek model
• Cox–Ingersoll–Ross model
• Black–Karasinski model

{{Reflist}}

;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

• {{cite book |last= Hull|first= John C.|authorlink= John C. Hull (economist)|title= Options, Futures, and Other Derivatives|url= https://archive.org/details/optionsfuturesot00hull_429|url-access= limited|edition= 6th|year=2006 |publisher= Prentice Hall |location= Upper Saddle River, N.J |isbn= 0-13-149908-4 |oclc= 60321487 |pages= [https://archive.org/details/optionsfuturesot00hull_429/page/n682 657]–658 |chapter= Interest Rate Derivatives: Models of the Short Rate |lccn= 2005047692 }}
• {{cite book | title = Interest Rate Models — Theory and Practice with Smile, Inflation and Credit| author = Damiano Brigo, Fabio Mercurio | publisher = Springer Verlag | year = 2001 | edition = 2nd ed. 2006 | 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. [http://ssrn.com/abstract=434860 Preprint SSRN].
• Henrard, Marc (2009). Efficient swaptions price in Hull–White one factor model, arXiv, 0901.1776v1. [https://arxiv.org/abs/0901.1776 Preprint arXiv].
• Ostrovski, Vladimir (2013). Efficient and Exact Simulation of the Hull–White Model, [https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2304848 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.,[https://ssrn.com/abstract=2837229 Preprint SSRN.]
• Puschkarski, Eugen. [https://web.archive.org/web/*/www.angelfire.com/ny/financeinfo/Diplomnew.ppt Implementation of Hull–White's No-Arbitrage Term Structure Model], Diploma Thesis, Center for Central European Financial Markets
• Turfus, Colin (2020). Caplet Pricing with Backward-Looking Rates., [https://ssrn.com/abstract=3527091 Preprint SSRN.]
• Letian Wang, [https://web.archive.org/web/20110723221130/http://www.letianwang.net/Fixed_Income/09_Hull-White_Model.htm Hull–White Model], Fixed Income Quant Group, DTCC (detailed numeric example and derivation)

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Category:Interest rates

Category:Fixed income analysis

Category:Short-rate models

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