The Hull-White Model

If you implement an interest rate option model, or check the cap, swaption, and barrier option prices it produces, you need to know the assumptions and limits of this model.

Starting from today's market yield curve, the function $\theta(t)$ is chosen for each point in time so that the model reproduces today's zero-coupon bond prices. From there, the products split into two groups: those whose prices can be written as a formula, such as European bond options, caps, and swaptions, and those that cannot, such as barrier options, which are valued numerically.

The equation for the short rate and mean reversion

The Hull-White model is a short-rate model that drives $r_t$, the rate for a very short period, with a stochastic process that has mean reversion. The equation is as follows¹.

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

Here $W_t$ is a Brownian motion. The constant $a$ is the speed of mean reversion, and it is positive. The parameter $\sigma$ is the volatility, the annualized standard deviation of the change in the short rate, and it is treated here as a constant. The function $\theta(t)$ is chosen for each point in time, as covered in the next section.

In 1990, Hull and White presented the model as an extension of the Vasicek model, with a time-dependent function $\theta(t)$ added to the drift. The source also treats a form in which $a$ and $\sigma$ are functions of time. They are written here as constants¹.

The drift is $\theta(t) - a r_t$, so when $r_t$ is above $\theta(t)/a$ it is pulled down, and when it is below, it is pulled up. The short rate is pulled back toward this level at speed $a$ while it moves randomly by the amount $\sigma\,dW_t$. Because the drift is linear in $r_t$ and $\sigma$ is constant, $r_t$ is normally distributed². The probability that the short rate turns negative is therefore not zero³.

With the same $\sigma$, faster mean reversion gives a smaller spread of the short rate. The standard deviation of the short rate after $t$ years is $\sigma\sqrt{(1 - e^{-2at})/(2a)}$⁴. Consider a hypothetical example in which $\sigma = 1\%$ and you compare the standard deviation after 10 years.

In the limit $a = 0$, the standard deviation is $\sigma\sqrt{t}$, which widens to 3.16% after 10 years. With $a = 0.3$, it stays at 1.29%.

Fitting the drift function to today's curve

If $\theta(t)$ is chosen to match today's forward rate derived from market zero-coupon bond prices, the model reproduces all of today's zero-coupon bond prices⁵. When $\sigma$ is constant, $\theta(t)$ can be written as follows⁶.

$$\theta(t) = \frac{\partial f(0,t)}{\partial t} + a f(0,t) + \frac{\sigma^2}{2a}\left(1 - e^{-2at}\right)$$

$f(0,t)$ is the instantaneous forward rate for time $t$, derived from today's yield curve (Zero Rates and Forward Rates). The first term is the slope of today's curve, the second is the level of the curve multiplied by $a$, and the third is the term due to $\sigma$. Where the curve bends sharply, the derivative in the first term gets in the way of numerical computation. The source says that for most uses there is a way of writing the model that avoids using $\theta(t)$ itself⁷.

Consider a hypothetical example in which you fit the model with $a = 0.1$ and $\sigma = 1\%$ to a flat curve with a continuously compounded zero rate of 2%. $f(0,t)$ is constant at 2% and the slope term is 0, so $\theta(t) = 0.002 + 0.0005\,(1 - e^{-0.2t})$.

Even with a flat curve, $\theta(t)$ is not constant. The $\sigma$ term takes effect over time, and $\theta(t)/a$ rises from 2% and approaches $f + \sigma^2/(2a^2) = 2.5\%$ in the far future. When the integral of the rate is normally distributed with mean $m$ and variance $v$, a zero-coupon bond price is $e^{-m + v/2}$, which is higher the larger the variance $v$. To match the price from a flat curve, the mean of the rate therefore has to be higher than 2%.

Deep diveSteps to derive the drift function

Let $G(t,T) = (1 - e^{-a(T-t)})/a$. $\theta(t)$ is found from the condition that the zero-coupon bond price matches the market price $P(0,T)$⁵.

1. Integrate the short-rate equation to express $r_s$ in terms of $r_0$, $\theta$, and the integral of $\sigma\,dW$.
2. The integral of the short rate, $I(0,T) = \int_0^T r_s\,ds$, is normally distributed. Its mean $m$ is $G(0,T)\,r_0 + \int_0^T G(u,T)\,\theta(u)\,du$ and its variance $v$ is $\int_0^T [G(u,T)\,\sigma]^2\,du$⁸.
3. The zero-coupon bond price is $E[e^{-I(0,T)}]$, which by the properties of the normal distribution equals $\exp(-m + v/2)$.
4. Requiring this to match $P(0,T)$ for every $T$, differentiating with respect to $T$, and solving gives the $\theta(t)$ above.

In the flat 2% curve example, for $T$ of 5 and 10 years, computing $\exp(-m + v/2)$ by numerical integration from the $\theta(t)$ above gives $e^{-0.02T}$ (0.904837 and 0.818731).

The zero-coupon bond price formula

A zero-coupon bond price at a future time $t$ can be written as an exponential function of the short rate $r_t$ at that time. The price $P(t,T)$ for maturity $T$ and face value 1 is as follows⁹.

$$P(t,T) = \frac{P(0,T)}{P(0,t)} \exp\left\{ -G(t,T)\,\bigl[r_t - f(0,t)\bigr] - \frac{G(t,T)^2}{2}\,y(t) \right\}$$

$G(t,T)$ is the same as in the section above, and $y(t) = \sigma^2(1 - e^{-2at})/(2a)$⁴. The fraction on the right is the discount from time $t$ to $T$ implied by today's curve. If the short rate $r_t$ is higher than the value $f(0,t)$ on today's curve, the price falls, and if it is lower, the price rises. $G(t,T)$ is a sensitivity that grows as the time to maturity gets longer. $\theta(t)$ does not appear on the right side. The price is determined by today's curve, $a$, and $\sigma$ alone⁹. A model in which the zero-coupon bond price can be written in the form $A\,e^{B r}$ is called an affine model, and this model is one of them¹⁰.

Consider a hypothetical example in which you value, one year from now, a zero-coupon bond with 2 years to maturity, using a flat 2% curve (continuously compounded), $a = 0.1$, and $\sigma = 1\%$. $G(1,2) = 0.951626$, $y(1) = 0.00009063$, and $P(0,2)/P(0,1) = 0.980199$. If the short rate $r_1$ one year from now is 3%, the exponent is $-0.951626 \times (0.03 - 0.02) - 0.452796 \times 0.00009063 = -0.009557$, and the price is $0.980199 \times e^{-0.009557} = 0.970875$.

A holder of the bond gains if the rate falls to 1%, as the price rises above today's forward value of 0.980199, and loses if the rate rises to 3%.

Closed-form prices of bond options and caps

The price of a European bond option with expiry $T_E$ takes the form of the Black formula¹¹. The underlying is a zero-coupon bond with maturity $T_M$ ($T_M \ge T_E$). With strike $K$, $\phi = 1$ for a call and $\phi = -1$ for a put, the payoff at expiry is $\max\{\phi\,(P(T_E,T_M) - K),\,0\}$. The valuation chooses the zero-coupon bond maturing at $T_E$ as the reference asset (numeraire). Under this probability measure (the forward measure for expiry $T_E$, which is different from the risk-neutral measure), $P(T_E,T_M)$ is lognormally distributed, so the formula of the Black model (The Black and Bachelier Models) can be used as it is¹¹.

The price today ($t = 0$) is as follows¹¹.

$$V = P(0,T_E)\,\phi\,\bigl[F\,N(\phi d_1) - K\,N(\phi d_2)\bigr]$$

$F = P(0,T_M)/P(0,T_E)$ is the forward bond price, and $N$ is the cumulative distribution function of the standard normal distribution. $d_1 = (\ln(F/K) + \nu^2/2)/\nu$ and $d_2 = d_1 - \nu$, where $\nu$ is the standard deviation of the logarithm of the bond price (a different quantity from the $\nu$ of the SABR model). It is a value up to expiry $T_E$, not an annualized value, and is given by the following¹¹.

$$\nu = G(T_E,T_M)\sqrt{y(T_E)}$$

Consider a hypothetical example in which you value an option to buy or sell, one year from now, a zero-coupon bond (face value 1) with 2 years to maturity at a strike of 0.98. The curve is a flat 2% (continuously compounded), with $a = 0.1$ and $\sigma = 1\%$. $F = 0.980199$ and $\nu = 0.951626 \times 0.009520 = 0.009060$, so the results are as follows.

The prices are calculated from unrounded values and then rounded. The call buyer receives the difference only if the bond price is above 0.98 one year from now. If it is below, the buyer receives nothing and has simply paid the premium of 0.003570. The put buyer, conversely, receives the difference only if the price is below 0.98. The unrounded difference between the call and the put, 0.000195, equals $P(0,2) - 0.98 \times P(0,1)$.

An interest rate cap is a collection of calls, one for each interest period. One period (a caplet) can be rewritten as a bond put. For a caplet with accrual period $\tau$, paid in arrears, with strike rate $K_c$, the payoff at the start of the period equals $(1 + \tau K_c)\max\{1/(1 + \tau K_c) - P(T_E,T_M),\,0\}$¹². That is, it is $(1 + \tau K_c)$ times a put with strike $1/(1 + \tau K_c)$.

Using the same model as in the example above, value a caplet for a one-year period starting one year from now ($T_E = 1$, $T_M = 2$), with $\tau = 1$ and $K_c = 2\%$ (simple annual rate).

$F_s$ is the simple forward rate from one year to two years from now, derived from today's curve. The caplet price is larger than the value in the last row. The difference is the value of the possibility that rates move.

A swaption can be rewritten as an option on a coupon bond, which has interest and principal, and its price is found as a sum of zero-coupon bond options¹³. In 1989 Jamshidian gave a closed-form solution for European options on coupon bonds, and swaptions can be priced by the same method¹⁴.

Deep diveHow to turn a coupon bond option into a sum of zero-coupon bond options

The payoff of an option on a coupon bond at expiry is $\max\{\sum_i C_i\,P(T_E,T_i) - K,\,0\}$, where $C_i$ is the cash flow on payment date $T_i$. In the Hull-White model, each $P(T_E,T_i)$ is a decreasing function of the state variable $x$ ($r_{T_E} - f(0,T_E)$) at expiry $T_E$. If $C_i \ge 0$, the sum is also a decreasing function¹³.

So there is exactly one state $x^*$ at which the sum equals $K$. Let $K_i = P(x^*;T_E,T_i)$. When $x \le x^*$, $P(T_E,T_i) \ge K_i$ for every $i$. When $x > x^*$, the reverse holds for every $i$. The payoff at expiry therefore equals $\sum_i C_i \max\{P(T_E,T_i) - K_i,\,0\}$, and the price is the sum of zero-coupon bond options $\sum_i C_i\,V^{ZBO}_i$ (with strikes $K_i$)¹³.

The source says that the parameters of the Hull-White model can be fitted to the prices of caps, floors, and European swaptions, or to their volatilities¹⁵.

Using trees and Monte Carlo simulation

European caps and swaptions have formulas. For options without a formula, such as barrier options, numerical computation is used. According to a 2020 paper, no closed-form formula is known for barrier option prices¹⁶.

On a trinomial tree, the short rate moves at each time step $\Delta t$ on lattice points of the form $r_0 + j\Delta r$. Each point has three branches: up, middle, and down¹⁷. $\Delta r$ is set to something like $\sigma\sqrt{3\Delta t}$ so that the probabilities stay between 0 and 1¹⁸. $\theta(t_i)$ is determined step by step, working forward in time. Then the tree reproduces today's zero-coupon bond prices at every time¹⁹.

In the Monte Carlo method, $x(t) = r_t - f(0,t)$ is moved with random numbers. Once the value at the previous time is fixed, the value at the next time follows a normal distribution. The variance is $\sigma^2(1 - e^{-2a(t-s)})/(2a)$, and the mean is also given by a formula²⁰. Because the distribution is known, the path can advance directly from one observation date to the next with a single random number. The zero-coupon bond price on each path is found by putting $x$ into the formula in the section above.

When the assumptions fail, and how the model differs from others

Because the short rate is normally distributed, negative rates can occur within the model. Nothing stops the rate at zero, and the 2020 paper says that this property was long regarded as a drawback but has now become an advantage³.

In the Hull-White model, rate movements are driven by one factor, and every rate moves in the same direction as the short rate¹⁴. A movement in which short-term rates rise while long-term rates fall cannot be produced.

The model can produce only a flat option smile²¹. Raising the speed of mean reversion $a$ lowers the swaption volatility the model produces even with the same $\sigma$, and $a$ also affects the slope of swaption volatilities across expiries and tenors²¹.

The Hull-White model models the short rate. Imposing suitable constraints on the Heath-Jarrow-Morton (HJM) model yields the Hull-White model²². The BGM model, on the other hand, is a model based on forward rates²².

Notes

1. "Interest Rate Modelling and Derivative Pricing" (Humboldt-Universität zu Berlin, Sebastian Schlenkrich, winter semester 2020/21), Part IV, slide p. 254. It gives the extension by Hull and White (1990), the short-rate equation, and the general form with $a$ and $\sigma$ as functions of time. ↩ ↩²

2. "Interest Rate Modelling and Derivative Pricing", Part IV, slide p. 274 (Lemma: distribution of the state variable). The Hull-White model in HJM notation is $r(t) = f(0,t) + x(t)$ on slide p. 270. ↩

3. Andrey Itkin and Dmitry Muravey, "Semi-closed form prices of barrier options in the Hull-White model" (arXiv:2004.09591v2, September 2020), page 1 (Introduction) and page 3 (section 1.1). It states that the model allows negative values, that this was long regarded as a drawback, and that it is now an advantage. ↩ ↩²

4. "Interest Rate Modelling and Derivative Pricing", Part IV, slide p. 269 ($y(t) = \int_0^t \sigma(u)^2 e^{-2a(t-u)}\,du$), p. 275 (variance $\mathrm{Var}[x(t)\mid\mathcal{F}_s]$), and p. 297 (formula for piecewise constant $\sigma$). When $\sigma$ is constant, $y(t) = \sigma^2(1 - e^{-2at})/(2a)$. The standard deviation of the short rate is $\sqrt{y(t)}$. ↩ ↩²

5. "Interest Rate Modelling and Derivative Pricing", Part IV, slide p. 255 (Lemma: Hull-White drift calibration) and pp. 256–259 (derivation). $\theta(t) = \partial f(0,t)/\partial t + a f(0,t) + \int_0^t [e^{-a(t-u)}\sigma(u)]^2\,du$. ↩ ↩²

6. Yuh-Dauh Lyuu, lecture notes (National Taiwan University, June 2, 2023), page 1204 (The Hull-White Model). $\theta(t) = \partial f(0,t)/\partial t + a f(0,t) + \sigma^2(1 - e^{-2at})/(2a)$. When $\sigma$ is constant, the integral in the previous footnote becomes this formula. ↩

7. "Interest Rate Modelling and Derivative Pricing", Part IV, slide p. 260. It states that the derivative term gets in the way of numerical computation, that $\theta(t)$ is not needed in the zero-coupon bond price formula, and that it can be avoided by the HJM way of writing the model. ↩

8. "Interest Rate Modelling and Derivative Pricing", Part IV, slide p. 257 (Proof 2/4). ↩

9. "Interest Rate Modelling and Derivative Pricing", Part IV, slide p. 261 (Theorem: Zero bonds in Hull-White model), p. 265 (the result, and that $\theta(t)$ is not needed), and p. 272 (the form in terms of $x(t)$). The integral in the exponent of the source's formula is $y(t)$ when $\sigma$ is constant. ↩ ↩²

10. Andrey Itkin and Dmitry Muravey, "Semi-closed form prices of barrier options in the Hull-White model" (arXiv, version of September 17, 2020), page 3, equation (6). This paper writes the short-rate equation as $dr = \kappa(t)[\theta(t) - r]\,dt + \sigma(t)\,dW$, so the symbols $\kappa$ and $\theta$ differ from this page. It writes the zero-coupon bond price in the form $A(t,S)\,e^{B(t,S)\,r}$. ↩

11. "Interest Rate Modelling and Derivative Pricing", Part IV, slides pp. 276–278 (Zero coupon bond options, the price by Black's formula and $\nu^2$) and p. 295 (the price at $t = 0$ and $\nu^2 = G(T_E,T_M)^2 y(T_E)$). The source writes Black(F, K, ν, φ) and uses $\nu$ as the square root of the variance of the lognormal distribution of the bond price. ↩ ↩² ↩³ ↩⁴

12. "tf_quant_finance.models.hull_white.cap_floor_price" (Google, TF Quant Finance documentation), the description. It states that a cap is a collection of calls and gives the formula that rewrites the caplet payoff as a zero-coupon bond put. Its reference is Brigo and Mercurio, 2nd edition (2007). ↩

13. "Interest Rate Modelling and Derivative Pricing", Part IV, slides pp. 279–284 (Coupon bond options, Jamshidian's trick), p. 295 (the sum formula), and p. 306 (equivalence of swaptions and coupon bond options). ↩ ↩² ↩³

14. 森本祐司、吉羽要直「BGM金利モデルの実用化に向けて」(Bank of Japan, Institute for Monetary and Economic Studies, Discussion Paper No. 99-J-39) (in Japanese), appendix, page 28. It states that an analytic solution for swaptions can be obtained in one-factor spot-rate models such as Vasicek and Hull-White, where every rate moves in the same direction as the short rate, and its footnote 37 states that Jamshidian (1989) gave a closed-form solution for European options on coupon bonds. ↩ ↩²

15. Andrey Itkin and Dmitry Muravey, "Semi-closed form prices of barrier options in the Hull-White model" (arXiv, version of September 17, 2020), page 1 (Introduction). The model can be fitted to today's term structure and to the prices or implied volatilities of caps, floors, and European swaptions. ↩

16. The same page 1 (Introduction). In the Hull-White model, the prices of zero-coupon bonds and vanilla European options can be written in closed form, while for exotic options such as barrier options no closed form is known and various numerical methods have been used. ↩

17. Yuh-Dauh Lyuu, lecture notes (National Taiwan University, June 2, 2023), pages 1206–1209 (The Hull-White Model: Calibration). Trinomial tree, lattice points $r_0 + j\Delta r$, and three branches from each point. ↩

18. The same lecture notes, pages 1215–1216. The condition $\sigma\sqrt{3\Delta t}/2 \le \Delta r \le 2\sigma\sqrt{\Delta t}$ that keeps the probabilities between 0 and 1, and the example $\Delta r = \sigma\sqrt{3\Delta t}$. ↩

19. The same lecture notes, pages 1210–1211 and 1217–1220. $\theta(t_i)$ is determined by forward induction so that the tree matches the term structure of zero rates. ↩

20. "Interest Rate Modelling and Derivative Pricing", Part IV, slides pp. 274–275 (Lemma: distribution of the state variable) and p. 296. ↩

21. "Interest Rate Modelling and Derivative Pricing", Part IV, slide p. 314 (the only smile shape the model can produce is flat, which is a major limitation of the model; higher mean reversion lowers the model volatility) and p. 315 (mean reversion affects the slope of volatilities across expiries and tenors). ↩ ↩²

22. 森本祐司、吉羽要直「BGM金利モデルの実用化に向けて」(Bank of Japan, Institute for Monetary and Economic Studies, Discussion Paper No. 99-J-39) (in Japanese), page 3 (section 2.1). It states that the Hull-White model is well known as an arbitrage-free model using the spot rate, that imposing suitable constraints on the HJM model yields the Hull-White model, and that the BGM model is a model based on forward rates. ↩ ↩²