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.


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