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This article needs attention from an expert on the subject. The specific problem is: Confirmation, details on the Affine Term Structure Model.. When placing this tag, consider associating this request with a WikiProject.(December 2012)
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
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.
Existence
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
implies
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
Vasicek
The Vasicek model has an affine term structure where
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