An affine term structure model is a financial model that 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.
Using Ito's formula we can determine the constraints on and which will result in an affine term structure. Assuming the bond has an affine term structure and satisfies the term structure equation, we get
The boundary value
The differential equation then becomes
Then the other term must vanish as well.
Models with ATS
The Vasicek model has an affine term structure where
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