# Hull White Equation Derivation

Hello I need your help. I found the formula for deriving $$A(t,T)$$ and $$B(t,T)$$ in Hull White paper is like this

$$BB_{tT} - B_{t}B_{T} - B_{T} = 0$$ and $$ABA_{tT} - BA_{t}A_{T} - AA_{t}B_{T} + \frac{1}{2}\sigma^2(t)A^2B^2B_{T} = 0$$

and they are resulting in this $$B(t,T) = \frac{B(0,T) - B(0,t)}{\frac{\partial B(0,t)}{\partial t}}$$ and $$\hat{A}(t,T) = \hat{A}(0,T) - \hat{A}(0,t) - B(t,T)\frac{\partial \hat{A}(0,t)}{\partial t} - \frac{1}{2}\Bigg[B(t,T)\frac{\partial B(0,t)}{\partial t}\Bigg]^{2}\int_{0}^{t} \Bigg[\frac{\sigma(\tau)}{\frac{\partial{B(0,\tau)}}{\partial \tau}}\Bigg]^{2}d\tau$$

where $$\hat{A}(t,T) = \ln{|A(t,T)|}$$. Anyone know how to derive the solution? Please this is for my thesis

• See this and this for an alternative derivation. – Gordon Jun 17 at 13:08