This post derives the expression of zero coupon bond price of Hull-White 2-factor model.
Earlier posts on Hull-White 2-factor model
Let \(P(t,T)\) denotes the price of zero-coupon bond with maturity \(T\) at time \(t\). Assuming \(\mathscr{F_t}\) as the information generated by \(x(t)\) and \(y(t)\) up to time \(t\), \(P(t,T)\) have the following form.
\[\begin{align}
P(t,T) &= E \left[\exp \left(-\int_{t}^{T} r(u)du \right)|\mathscr{F_t} \right] \\
&= E \left[\exp \left(-\int_{t}^{T} x(u)+y(u)+\varphi(u) du \right)|\mathscr{F_t}\right]
\end{align}\]
To solve for \(P(t,T)\), we need to know the implementable expresssion for \(\int_{t}^{T} x(u)+y(u) du\) except for \(\varphi(u)\) because \(\varphi(u)\) is not stochastic but deterministic process. Integrating \(x(u)+y(u)\) from time \(t\) to \(T\), we can get the following result.
\[\begin{align}
\int_{t}^{T} &{x(u) + y(u)} du = x(t) B_{1} (t,T) + y(t)B_{2} (t,T) \\
&+ \int_{t}^{T} \sigma(u) B_{1} (u,T) dW_{1} (u) \\
&+ \rho \int_{t}^{T} \eta(u) B_{2} (u,T) dW_{1}(u) \\
&+ \sqrt{1-\rho^2} \int_{t}^{T} \eta(u) B_2 (u,T) dW_2 (u)
\end{align}\]
Here, \(B_1(t,T) = \int_{t}^{T} e^{-\int_{t}^{u} a(v) dv} du \) and \(B_2(t,T) = \int_{t}^{T} e^{-\int_{t}^{u} b(v) dv} du \). The derivation above is of the same logic for the case of Hull-White 1-factor model.
\(V(t,T)\) is also given using Itô isometry like Hull-White 1-factor model.
\[\begin{align}
V(t,T)&=\int_{t}^{T} \sigma(u)^2 B_ 1(u,T)^2 du \\
&+ \int_{t}^{T} \eta(u)^2 B_2(u,T)^2 du \\
&+2 \rho \int_{t}^{T} \sigma(u)\eta(u) B_1(u,T)B_2(u,T) du
\end{align}\]
Similar to HW 1-factor model, we can find that \(\int_{t}^{T} x(u) + y(u) du\) follows the normal distribution of a mean \(x(t)B_1(t,T)+y(t)B_2(t,T)\) and a variance \(V(t,T)\).
Using the fact that \(E[\exp(Y)]=\exp \left( \mu + \frac{1}{2}\sigma^2 \right) \) with a normally distributed random variable \(Y\) which has a mean \(\mu\) and a variance \(\sigma^2\), the price of zero-coupon bond becomes
\[\begin{align}
P(t,T) &= \exp \left( -\int_{t}^{T} \varphi(u)du \right) E \left[\exp \left(-\int_{t}^{T} x(u) + y(u) du \right)|\mathscr{F_t} \right] \\
&= \exp \left( -\int_{t}^{T} \varphi(u)du \right) \exp \left( -x(t)B_1(t,T) - y(t)B_2(t,T) + \frac{1}{2}V(t,T) \right) \\
&= \exp \left( -\int_{t}^{T} \varphi(u)du -x(t)B_1(t,T) - y(t)B_2(t,T) + \frac{1}{2}V(t,T) \right)
\end{align}\]
It is argued that Hull-White model is consistent to the no-arbitrage assumption (perfect fit) if market discount factor \(P(0,T)\) satisfies the following condition.
\[\begin{align}
&P(0,T) = \exp \left( -\int_{0}^{T} \varphi(u)du + \frac{1}{2}V(0,T) \right)\\
&\rightarrow \exp \left( -\int_{0}^{T} \varphi(u)du \right) = P(0,T) \exp \left( - \frac{1}{2}V(0,T) \right)
\end{align}\]
Using the above relationship,
\[\begin{align}
&\exp \left( -\int_{t}^{T} \varphi(u)du \right) = \frac{\exp \left( -\int_{0}^{T} \varphi(u)du \right)}{\exp \left( -\int_{0}^{t} \varphi(u)du \right)} \\
&= \frac{P(0,T)}{P(0,t)} \exp \left( -\frac{1}{2}\{V(0,T)-V(0,t)\} \right)
\end{align}\]
As the above expression for function \( \varphi(.)\) holds, the price of zero-coupon bond has the following form.
\[\begin{align}
P(t,T) &= \frac{P(0,T)}{P(0,t)} \exp \left( -x(t)B_1(t,T) - y(t)B_2(t,T) + \frac{1}{2}\Omega(t,T) \right) \\
\Omega(t,T) &= V(t,T)-V(0,T)+V(0,t)
\end{align}\]
Substituting \(V(t,T)\) into the equation, the price of zero coupon bond \(P(t,T)\) is reformulated as
\[\begin{align}
P(t,T) &= \frac{P(0,T)}{P(0,t)} \exp \left( -x(t)B_1(t,T) - y(t)B_2(t,T) + \frac{1}{2}\Omega(t,T) \right)\\
\Omega(t,T) &= \int_{0}^{t} \sigma(u)^2 \{B_1(u,t)^2-B_2(u,T)^2\} du \\
&+\int_{0}^{t} \eta(u)^2 \{B_2(u,t)^2-B_2(u,T)^2\} du \\
&+2\rho\int_{0}^{t} \sigma(u)\eta(u) \{B_1(u,t)B_2(u,t)-B_1(u,T)B_2(u,T)\} du
\end{align}\]
From this post, we know that most of derivation regarding HW 2-factor model is similar to the HW 1-factor model. \(\blacksquare\)
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