## Glossary

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# Affine term structure

For all short rate models of the form $$dr(t)=\alpha(t)r(t)+\beta(t)dt+(\gamma(t)r(t)+\delta(t))^{1/2}dW(t)$$ the Zero coupon bonds values have an affine term structure that is $$p(t,T)=\exp(A(t,T)-B(t,T)r(t)),$$ where A, B are deterministic functions which do not depend on r.
The functions A and B satisfy the following system of ordinary differential equations (ODE:s) \begin{eqnarray} B'_t(t,T)&=&-\alpha(t)B(t,T)+\frac{1}{2}\gamma(t)B^2(t,T)-1, B(T,T)=0\\ A'_t(t,T)&=&\beta(t)B(t,T)-\frac{1}{2}\delta(t)^2B(t,T), A(T,T)=0. \end{eqnarray}