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An '''affine term structure model''' is a specific type of financial model which relates zero coupon bond prices (i.e. the discount curve) to a spot rate model. It is particularly useful for ''inverting the yield curve'' - the process of determining spot rate model inputs from observable bond market data. This
== Background ==
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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>
== Models with ATS ==
=== Vasicek ===
The [[Vasicek model]] <math>dr=(b-ar)dt+\sigma dW</math> has an affine term structure where
<math>\
p(t,T)&=e^{A(t,T)-B(t,T)r(T)}\\
B(t,T)&=\frac{1}{a}\left(1-e^{-a(T-t)}\right)\\
A(t,T)&=\frac{(B(t,T)-T+t)(ab-\frac{1}{2}\sigma^2)}{a^2}-\frac{\sigma^2B^2(t,T)}{4a}
</math>
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