# Solutions to the Bond Pricing Equation

Consider a spot rate of the form:

$$dr = (\eta - \gamma r) dt + \sqrt{\alpha r + \beta} dW$$

where all parameters are constants.

Lets look for a solution of the form $$Z(r; t) = e^{A(t;T) - r B(r; T)}$$ of the Bond Pricing Equation. I can show that the pair of first order differential equation are:

$$d{A(t)}/{dt} = \eta B(t) - \frac{1}{2} \beta B(t)^2$$

$$d{B(t)}/{dt} = \frac{1}{2} \alpha B(t)^2 + \gamma B(t) - 1$$

I know that $$A(T;T) = B(T;T) = 0$$

What I want to solve is the equation below:

$$d{B(t)}/{dt} = \frac{1}{2} \alpha B(t)^2 + \gamma B(t) - 1$$

I have the solution in this thesis (pag 20 - eq 3.9): https://ro.uow.edu.au/cgi/viewcontent.cgi?referer=https://www.google.com.br/&httpsredir=1&article=3882&context=theses

I can also find the solution in this book (pag 431): Wilmott P., DERIVATIVES: The Theory and Practice of Financial Engineering, John Wiley & Sons Ltd, 1998.

Both references shows the answer, but none shows how to solve the differential equation for $$B(\tau)$$.

Thank you.