How do you price zero coupon bond in extended Hull & White model by solving the Bond Pricing Equation??
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$\begingroup$ What is the specific form of the extended Hull & White model? $\endgroup$– GordonMay 15, 2015 at 13:19
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$\begingroup$ See answer at quant.stackexchange.com/questions/21513/… $\endgroup$– GordonNov 18, 2015 at 2:25
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1 Answer
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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 $$P(t,T)=\mathbb{E^Q}\left[ exp\left( -\int_{t}^{{{T}^{{}}}}{r(u)du} \right)\left| \,{\mathcal {F}_{t}} \right. \right]$$