Bahcall–Wolf cusp

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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.


Start with a stochastic short rate model with dynamics

and a risk-free zero-coupon bond maturing at time with price at time . If

and has the form

where and are deterministic functions, then the short rate model is said to have an affine term structure.


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


Next, assume that and are affine in :

The differential equation then becomes

Because this formula must hold for all , , , the coefficient of must equal zero.

Then the other term must vanish as well.

Then, assuming and are affine in , the model has an affine term structure where and satisfy the system of equations:

Models with ATS


The Vasicek model has an affine term structure where


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