Diving deeper on Hull White Model

In a previous post, I made an entry point to the famous Hull-White model that is frequently utilized in rates modeling in the finance industry. In that post, our focus was on its most prominent aspect of called deterministic mean-reversion. The main goal in this post is to provide some details on a (somewhat) general version of the model (e.g focusing on time dependent mean-reversal rate), with the hope to flex some of our mathematical muscles and set the stage for later explorations in derivatives pricing and its calibration in the IR markets.

A (somewhat) general form of Hull-White model

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Changing the notation slightly, we will work with a commonly used form of Hull White model, where the process describing the short rate evolution is given by

\[\begin{equation} \mathrm{d}r_t = \left(\theta_t - a_t\, r_t \right)\,\mathrm{d}t + \sigma_t\, \mathrm{d}W_t^{\mathbb{Q}}, \end{equation}\]

where (denoting the time dependence of the functions with a subscript) the time-dependent mean level can be described by the following combination $\theta_t/a_t$ with $a_t$ denoting the time dependent mean-reversal rate and $\sigma_t$ refers to time dependent volatility. In practice all these parameters should be calibrated (for example using IR derivatives in the market and/or historical data) with a method suitable for the use case of the model.

The stochastic process above can be easily integrated considering the following variable

\[y_t = \exp\left[\int_{0}^{t} a_u\, \mathrm{d}u\right]\, r_t \equiv E(t)\, r_t,\]

such that the process associated with it satisfies

\[\mathrm{d}y_t = E(t)\, \theta_t\, \mathrm{d}t + E(t)\, \sigma_t\, \mathrm{d}W_t^{\mathbb{Q}}.\]

Integration the both hand side of this equation from $s$ to $t$ ($s < t$), the formal solution to the HW model can be written as

\[\begin{equation} \label{sol} r(t) = \frac{E(s)}{E(t)}\, r(s) + \frac{1}{E(t)}\int_{s}^{t} E(u)\theta(u)\mathrm{d}u + \frac{1}{E(t)}\int_{s}^{t} E(u)\sigma(u)\mathrm{d}W_{u}^{\mathbb{Q}}. \end{equation}\]

The ratios of the functions $E(x)$ here actually identifies an exponential kernel that determines the system’s memory to past quantities. To see this notice that in the case of constant mean reversal rate, we have the following simplified expression for the ratio’s that appear in \eqref{sol}:

\[\frac{E(u)}{E(s)} = \exp\left[\int_{t}^{u} a_z \mathrm{d}z\right]\quad \xrightarrow{a = constant} \quad \mathrm{e}^{- a (t-u)}.\]

Considering this expression in the solution \eqref{sol}, we can see that past quantities (such as $\theta_u$, $\sigma_u$ or $r(s)$) make exponentially small contributions to the overall solution due to exponential suppression for $t-u\gg 1$ or $t - s\gg 1$. This is a characteristic property of a mean-reverting process.

Bond Pricing in the Hull-White Model

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Given the form of the Hull-White model, arguably the first quantity of interest is the price of a zero-coupon Bond (which can be considered essentially a derivative on interest rate) in terms of HW model parameters/quantities. For this purpose, we will utilize FTAP(Fundamental Theorem of Asset Pricing) to write the current price of zero coupon that matures at $T$ as

\[P(t,T) = \mathbb{E}_{\mathbb{Q}}\Bigg[\exp\left[-\int_{t}^T\, r(s)\, \mathrm{d}s\right] \Bigg |\, \mathcal{F}_t\,\Bigg], \quad t \leq s \leq T.\]

As should be clear from the expression above, pricing a zero coupon bond boils down to figuring out integral in the exponent:

\[\begin{equation} \label{r_int} \int_{t}^{T} r_s\, \mathrm{d}s = r_t E_t \int_{t}^{T} \frac{\mathrm{d}s}{E_s} + \int_{t}^{T} \frac{\mathrm{d}s}{E_s} \int_{t}^s E_u \theta_u \mathrm{d}u + \int_{t}^{T} \frac{\mathrm{d}s}{E_s} \int_{t}^s E_u \sigma_u \mathrm{d}W^{\mathbb{Q}}_{u}. \end{equation}\]

To simplify this expression, we first define a function

\[\begin{equation} \label{BuT} B(t,T) \equiv E(t) \int_{t}^{T} \frac{\mathrm{d}s}{E(s)}. \end{equation}\]

Next we apply Fubini’s theorem to change the order of the integrals for the second and the third term (stochastic integral). For this purpose, notice that the both integrals cover a rectangle domain

\[D: \{u,s\} \quad \longrightarrow \quad t \leq s \leq T \quad \& \quad t\leq u \leq s \quad \longrightarrow \quad t\leq u \leq s \leq T.\]

Therefore, at fixed $u$, we have $s \in [u,T]$ and at fixed $s$, $u \in [t,T]$ such that we can change the order of the integrals as follows

\[\begin{align} \nonumber \int_{t}^{T} \frac{\mathrm{d}s}{E(s)} \int_{t}^s E(u) \theta(u) \mathrm{d}u &= \int_{t}^T \theta(u)\left( E(u) \int_{u}^T \frac{\mathrm{d}s}{E(s)} \right) \mathrm{d}u = \int_{t}^T \theta(u) B(u,T) \mathrm{d}u,\\ \int_{t}^{T} \frac{\mathrm{d}s}{E(s)} \int_{t}^s E(u) \sigma(u) \mathrm{d}W^{\mathbb{Q}}_{u} &= \int_{t}^T \sigma(u)\left(E(u) \int_{u}^T \frac{\mathrm{d}s}{E(s)}\right)\mathrm{d}W^{\mathbb{Q}}_u = \int_{t}^T \sigma(u) B(u,T) \mathrm{d}W^{\mathbb{Q}}_u, \end{align}\]

where we utilized the definition \eqref{BuT}. The integral of the short rate \eqref{r_int} can be therefore simplified as

\[\int_{t}^{T} r(s)\, \mathrm{d}s = B(t,T) r(t) + \int_{t}^T \theta(u) B(u,T) \mathrm{d}u + \int_{t}^T \sigma(u) B(u,T) \mathrm{d}W^{\mathbb{Q}}_u.\]

This result tells us that the integral of the short rate is Gaussian distributed with the following mean and variance

\[\mu_{t,T} = B(t,T) r(t) + \int_{t}^T \theta(u) B(u,T) \mathrm{d}u, \quad\quad \sigma_{t,T}^2 = \int_{t}^T \sigma^{2}(u) B(u,T)^2 \mathrm{d}u.\]

We can thus simply compute the Zero coupon bond price as:

\[\begin{align} \nonumber P(t,T) &= \mathbb{E}_{\mathbb{Q}}\left[ \mathrm{e}^{-\mu_{t,T} - \sigma_{t,T} Z}\right],\\ \nonumber&= \mathrm{e}^{-\mu_{t,T}} \int_{-\infty}^{\infty} \frac{\mathrm{d}z}{\sqrt{2\pi}} \mathrm{e}^{-\sigma_{t,T}z}\,\mathrm{e}^{-z^2/2},\\ \nonumber &= \mathrm{e}^{-\mu_{t,T} + \sigma^2_{t,T}/2} \int_{-\infty}^{\infty} \frac{\mathrm{d}z}{\sqrt{2\pi}} \mathrm{e}^{-(z + \sigma_{t,T})^2/2},\\ &= \mathrm{e}^{-\mu_{t,T} + \sigma^2_{t,T}/2}, \end{align}\]

where the exponents inside the integral is completed to a square in passing from second the third line and the integral in the third line is simply unity noticing a change of variable $z’ = z + \sigma_{t,T}$. This result give rise to a form commonly referred as the “affine form” for the bond price:

\[\begin{equation} \label{affine} P(t,T) = \exp\bigg[A(t,T) - B(t,T)r(t)\bigg], \end{equation}\]

where

\[\begin{equation} \label{A} A(t,T) \equiv \frac{1}{2}\int_{t}^T \sigma^2(u)B(u,T)^2 \mathrm{d}u - \int_{t}^T \theta(u)B(u,T)\mathrm{d}u. \end{equation}\]

This wraps up the ZCB price in terms of HW model parameters that are embedded in the functions $A(t,T)$, $B(t,T)$ and the state variable $r(t)$. It is instructive to derive the process satisfied by the bond price, which can be written using Ito’s Lemma as

\(\begin{align} \nonumber \mathrm{d}P &= \frac{\partial P}{\partial t} \mathrm{d}t + \frac{\partial P}{\partial r} \mathrm{d}r + \frac{1}{2}\frac{\partial^2 P}{\partial r^2} \mathrm{d}r^2,\\ \label{dP}&= \left[\dot{A} - \dot{B}r - B \theta + B a r + \frac{1}{2} B^2 \sigma^2\right] P \mathrm{d}t - B\, \sigma\, P\mathrm{d}W_{t}^{\mathbb{Q}}, \end{align}\) where we have explicitly omitted the arguments of the functions to reduce clutter. Notice that by virtue of the definition in \eqref{BuT}, we can simplify the drift term by first noticing

\[\dot{B}(t,T) = \dot{E}(t) \int_{t}^T \frac{\mathrm{d}s}{E(s)} - 1 = a(t) B(t, T) - 1,\quad \Rightarrow \quad \dot{B}r = B a r - r \quad \Rightarrow \quad r = aBr - \dot{B}r,\]

so that the second and 4th term in the drift gives $r$. Using the definition of $A(t,T)$ \eqref{A}, it is easy to show that the sum of the odd numbered terms in the drift of \eqref{dP} is vanishing:

\[\dot{A}(t,T) = -\frac{\sigma^2(t)B^2(t,T)}{2} + \theta(t)B(t,T) \quad \Rightarrow \quad \dot{A} - B\theta + \frac{1}{2}\sigma^2B^2 = 0.\]

Therefore under the risk neutral measure, the ZCB price satisfies the following stochastic differential equation

\[\begin{equation} \label{br}\frac{\mathrm{d}P(t,T)}{P(t,T)} = r(t) \mathrm{d}t - \sigma(t) B(t,T) \mathrm{d}W_t^{\mathbb{Q}}, \end{equation}\]

where we can identify the volatility at time $t$ of a ZCB with maturity $T$ as $\sigma(t,T) = \sigma(t) B(t,T)$.

A handy measure for derivative pricing: T-forward measure

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In interest rate derivatives, the payoff and the discount factor are driven by the same rates — making risk-neutral pricing intractable. To see this recall that FTAP tells us that asset prices when discounted by a money market account $M(t) = e^{\int_{0}^t r(s) \mathrm{d}s}$ are martingales, which implies the standard pricing relation

\[\frac{V(t)}{M(t)} = \mathbb{E}_\mathbb{Q}\left[\frac{V(T)}{M(T)}\, \bigg|\, \mathcal{F}_t\right] \quad \Rightarrow \quad V(t) = \mathbb{E}_\mathbb{Q}\left[\mathrm{e}^{-\int_{t}^T r(s)\mathrm{d}s}\, V(T)\, \bigg|\, \mathcal{F}_t\right].\]

The problem here is that for IR derivatives $\mathrm{e}^{-\int_{t}^T r(s)\mathrm{d}s}$ and the payoff $V(T)$ are not independent as interest rates drive both the discount factor and the payoff simultaneously such that we cannot factor the expectation in two digestible pieces. This correlation problem makes the derivation of closed form expressions for IR derivatives very hard under $\mathbb{Q}$. The forward measure is precisely the tool to ease the pain in this context which essentially removes the discount factor from inside the expectation in the above formula.

To accomplish this, the $T$-forward measure $\mathbb{Q}_T$ uses the ZCB $P(t,T)$ as the numeraire instead of the money market account $M(t)$. In the new measure, the pricing simplifies as follows

\[\frac{V(t)}{P(t,T)} = \mathbb{E}_{\mathbb{Q}_T}\left[\frac{V(T)}{P(T,T)}\, \bigg | \, \mathcal{F}_t\right] \quad \Rightarrow \quad V(t) = P(t,T)\, \mathbb{E}_{\mathbb{Q}_T}\left[{V(T)}\, | \, \mathcal{F}_t\right].\]

Notice that the discount factor is represented by the bond price that sits outside the expectation, which is observable at time $t$. This reduces the computation to a single expectation of the payoff under $\mathbb{Q}_T$, which is an enormous simplification.

In light of the discussion, we can consider the following price ratio of ZCB bonds with different maturity: ${P(t,T_F)}/{P(t,T_P)}$ which is of course a martingale under the $T_P$ forward measure $\mathbb{Q}_{T_P}$: $t < T_F < T_P$. Here, we choose a specific subscript notation for the maturity times to reflect a typical IR market structure where $T_F$ denotes fixing/observation time of a floating rate and $T_P$ is the time, where the actual cashflow/payment takes place. We can make more sense of this ratio in this context by deriving its relation to the forward rate $L$ defined for the accrual period between $T_F$ and $T_P$. To see this notice that if we invest one dollar for this period, we would have $1 + \delta L$ (assuming simple compounding) amount at $T_P$ where $\delta \equiv T_P - T_F$. Of course this value should be equal to the reciprocal of discount factor between $T_F$ and $T_P$:

\[1 + \delta\, L(t;T_F,T_P) = \frac{1}{P(T_F,T_P)}.\]

Notice that the expression on the right-hand side is given by the bond ratio we are interested in:

\[\frac{1}{P(T_F,T_P)} = \mathrm{e}^{\int_{T_F}^{T_P} r(s) \mathrm{d}s} = \frac{\mathrm{e}^{-\int_{t}^{T_F} r(s) \mathrm{d}s}}{\mathrm{e}^{-\int_{t}^{T_P} r(s) \mathrm{d}s}} = \frac{P(t,T_F)}{P(t,T_P)}.\]

This prompts us to write the forward rate (as seen from today $t$) in terms of the ratio of the bond prices as

\[L(t;T_F,T_P) = \frac{1}{\delta} \left(\frac{P(t,T_F)}{P(t,T_P)} - 1\right),\]

which in turn establishes the connection between the bond price ratio and forward rate $L$ (fixed/observed at $T_F$ until $T_P$ where it generates a cashflow/payment) that is available during the future time interval $\delta = T_P - T_F$.

Setting this digression on fixing and payment times aside, an important point for our purposes is the fact that the aforementioned bond price ratio is a martingale under the $T_P$ forward measure and hence the underlying stochastic process of the ratio should have no drift term. In order to see this, we let $X = P(t,T_F)$, $Y = P(t,T_P)$ and denote the ratio $R = X/Y$. Using the Ito’s lemma the process describing the ratio can be written as

\[\mathrm{d}R = R \left[ \frac{\mathrm{d}X}{X} -\frac{\mathrm{d}Y}{Y} + \frac{\mathrm{d}Y^2}{Y^2} - \frac{\mathrm{d}X\,\mathrm{d}Y}{X\,Y}\right].\]

Utilizing \eqref{br} and collecting terms to leading order $\mathrm{d}t$, the bond ratio process in the $\mathbb{Q}$ measure gives

\[\mathrm{d}\left(\frac{P(t,T_F)}{P(t,T_P)}\right) = \frac{P(t,T_F)}{P(t,T_P)} \left[\sigma^2(t) \left( B(t,T_P)^2 - B(t,T_F)B(t,T_P) \right)\mathrm{d}t + \sigma(t)\left(B(t,T_P) - B(t,T_F)\right)\mathrm{d}W^{\mathbb{Q}}_t\right].\]

The expression above allows us to infer transformation the Brownian motion must take when we change the measure from $\mathbb{Q}$ to $T_P$ forward measure $\mathbb{Q}^{T_P}$ as the above process should drift-less in the latter measure to be able to qualify as a martingale. This tells us the following transformation rule under the change of measure:

\[\mathrm{d}W^{\mathbb{Q}}_t = \mathrm{d}W^{\mathbb{Q}^{T_P}}_t - \sigma(t)B(t,T_P)\mathrm{d}t,\]

so that the bond price ratio satisfies:

\[\begin{equation} \mathrm{d}\left(\frac{P(t,T_F)}{P(t,T_P)}\right) = \frac{P(t,T_F)}{P(t,T_P)}\, \sigma(t)\left(B(t,T_P) - B(t,T_F)\right)\mathrm{d}W^{\mathbb{Q}^{T_P}}_t. \end{equation}\]

We can identify the instantaneous volatility of this drift-less GBM as

\[\nu(t) \equiv \sigma(t)\left(B(t,T_P) - B(t,T_F)\right).\]

Integrating the (log) ratio process by applying Ito’s lemma again we have

\[\mathrm{d}\ln R = - \frac{\nu(t)^2}{2} \mathrm{d}t + \nu(t) \mathrm{d}W^{T_P} \quad \Rightarrow \quad \ln\left(\frac{R(T_F)}{R(t)}\right) \sim \mathcal{N}\left(-\frac{1}{2}V_p, V_p\right)\]

where we defined the integrated variance of the log process as

\[\begin{equation} \label{varbr}V_p(t,T_F,T_P) = \int_t^{T_F} \nu^2(u) \mathrm{d}u = \int_t^{T_F} \sigma^2(u)\left(B(u,T_P) - B(u,T_F)\right)^2\, \mathrm{d}u. \end{equation}\]

This expression can be factorized in terms of the integrated variance of the rate process \eqref{sol} itself by noticing

\[B(u,T_P) - B(u,T_F) = \frac{E(u)}{E(T_F)}B(T_F,T_P).\]

Finally, defining the variance of the rate process (see eq. \eqref{sol}) as

\[V_r(t,T_F) \equiv \frac{1}{E(T_F)^2}\int_{t}^{T_F} E(u)^2 \sigma(u)^2 \mathrm{d}u,\]

we can factorize bond ratio variance \eqref{varbr} as

\[V_p(t,T_F,T_P) = V_r(t,T_F) B(T_F,T_P).\]

Conclusions

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Focusing on a general form of Hull White model of interest rates, we have derived some fundamental (mathematical) building blocks that will help us derive closed form formulas for basic IR derivatives such as caps and swaptions. For the sake of brevity, I will leave these explorations to another post which will arm us further for our ultimate goal on understanding various calibration methods of Hull-White model.

References

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1. “Calibration methods of Hull-White Model”, Risk Management Department, Mizuho Securities. Sebastien Gurrieri1, Masaki Nakabayashi and Tony Wong