Zero-Coupon Bond Pricing in the Extended Hull-White Model

Summary

The document presents a closed-form expression for a zero-coupon bond price under an extended Hull-White short-rate model with time-varying drift, mean-reversion, and volatility parameters. The bond price is exponential-affine in the current short rate: one term captures the deterministic components of the model, while another multiplies the current rate by an integrated loading. The formulas define both terms through integrals of the model parameters.

The derivation is motivated by solving the short-rate process with an integrating factor, which shows that the rate is normally distributed and provides expressions for its mean and variance. The bond price then follows from the risk-neutral conditional expectation of the exponential of the negative integrated short rate. This gives a pricing formula rather than a numerical solution procedure for the bond-pricing equation. The document provides no worked numerical example or calibration discussion, and its result depends on the stated Gaussian risk-neutral model assumptions and parameter specification.

Key ideas

• The extended Hull-White model allows drift, mean-reversion, and volatility to vary over time.
• The short rate can be represented using an integrating factor, yielding its conditional mean and variance.
• The zero-coupon bond price has an exponential-affine form in the current short rate.
• Bond valuation is obtained as a risk-neutral expectation of the discount factor over the remaining term.
• The formula applies under the specified Gaussian short-rate model and does not include a numerical example.

Tags

# Zero coupon bond pricing under Extended Hull & White


# Zero coupon bond pricing under Extended Hull & White












How do you price zero coupon bond in extended Hull & White model by solving the Bond Pricing Equation??

## Answer by user16891 (score 3)

https://quant.stackexchange.com/a/18959

In the Hull-White model, $r_t$ follows the Ito process as described by the following stochastic differential equation $$d{{r}_{t}}=(\alpha (t)-\beta (t)\,{{r}_{t}})dt+\sigma (t)d{{W}_{t}^Q}$$ let $$K(t)=\int_{0}^{t}{\beta (u)\,du}$$ then the zero coupon bond price is given by equation $$p(t\,,T)=exp\,[-A(t\ ,T)-r(t)B(t,T)\,]$$ where \begin{align} & A(t\,,T)=\int_{t}^{T}{\left[ \alpha (u)\,{{e}^{K(u)}}\left( \int_{u}^{T}{{{e}^{-K(v)}}dv} \right)-\frac{1}{2}{{e}^{2K(u)}}{{\sigma }^{2}}(u){{\left( \int_{u}^{T}{{{e}^{-K(v)}}dv} \right)}^{2}}\, \right]}\,du \\ & B(t\,,T)=\int_{t}^{T}{{{e}^{-K(v)}}}dv \\ \end{align}

### Hint

- By application of Ito's lemma, we have $$r(t)={{e}^{-K(t)}}\left[ r(0)+\int_{0}^{t}{{{e}^{K(u)}}\alpha (u)\,du+\int_{0}^{t}{{{e}^{K(u)}}\sigma (u)\,d{{W}_{u}}}} \right]$$

- $r_t$ is a normal process and

\begin{align} & \mathbb{E^Q}[{{r}_{t}}]={{e}^{-K(t)}}\left[ r(0)+\int_{0}^{t}{{{e}^{K(u)}}\alpha (u)\,du} \right] \\ & Var^Q({{r}_{t}})={{e}^{-2K(t)}}\int_{0}^{t}{{{e}^{2K(u)}}{{\sigma }^{2}}(u)\,du} \\ \end{align} 3. Compute $\int_{0}^{T}{r(t)\,dt}$ and use $$P(t,T)=\mathbb{E^Q}\left[ exp\left( -\int_{t}^{{{T}^{{}}}}{r(u)du} \right)\left| \,{\mathcal {F}_{t}} \right. \right]$$

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.