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<math>p(t,T)=F^T(t,r(t))</math>
and <math>F</math> has the form
<math>F^T(t,r)=e^{A(t,T)-B(t,T)r}</math>
where <math>A</math> and <math>B</math> are deterministic functions, then the short rate model is said to have an '''affine term structure'''.
== Existence ==
Using Ito's formula we can determine the constraints on <math>\mu</math> and <math>\sigma</math> which will result in an affine term structure. Assuming the bond has an affine term structure and <math>F</math> satisfies the [[term structure equation]], we get
<math>A_t(t,T)-(1+B_t(t,T))r-\mu(t,r)B(t,T)+\frac{1}{2}\sigma^2(t,r)B^2(t,T)=0</math>
The boundary value
<math>F^T(T,r)=1</math>
implies
<math>
\begin{align}
A(T,T)&=0\\
B(T,T)&=0
\end{align}
</math>
Next, assume that <math>\mu</math> and <math>\sigma^2</math> are affine in <math>r</math>:
<math>
\begin{align}
\mu(t,r)&=\alpha(t)r+\beta(t)\\
\sigma(t,r)&=\sqrt{\gamma(t)r+\delta(t)}
\end{align}
</math>
The differential equation then becomes
<math>
A_t(t,T)-\beta(t)B(t,T)+\frac{1}{2}\delta(t)B^2(t,T)-\left[1+B_t(t,T)+\alpha(t)B(t,T)-\frac{1}{2}\gamma(t)B^2(t,T)\right]r=0
</math>
Because this formula must hold for all <math>r</math>, <math>t</math>, <math>T</math>, the coefficient of <math>r</math> must equal zero.
<math>
1+B_t(t,T)+\alpha(t)B(t,T)-\frac{1}{2}\gamma(t)B^2(t,T)=0
</math>
Then the other term must vanish as well.
<math>
A_t(t,T)-\beta(t)B(t,T)+\frac{1}{2}\delta(t)B^2(t,T)=0
</math>
Then, assuming <math>\mu</math> and <math>\sigma^2</math> are affine in <math>r</math>, the model has an affine term structure where <math>A</math> and <math>B</math> satisfy the system of equations:
<math>\begin{align}
1+B_t(t,T)+\alpha(t)B(t,T)-\frac{1}{2}\gamma(t)B^2(t,T)=0\\
B(T,T)&=0\\
A_t(t,T)-\beta(t)B(t,T)+\frac{1}{2}\delta(t)B^2(t,T)=0\\
A(T,T)&=0
\end{align}</math>
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